Shape factors for pseudo-steady state flow of a fractured-well in reservoirs of square shape

Pseudo-steady state (PSS) flow is a dominant time-dependent flow regime during constant rate production from a closed reservoir. Recently Chen (2016) has obtained an exact analytical solution for the PSS flow of a fully-penetrated fractured vertical well with finite conductivity in an elliptical drainage area. The availability of this analytical solution shortens the computational time required for such a solution by several orders of magnitude. This paper correlates the PSS flow of a fully-penetrated fractured vertical well in square drainage areas to Chen’s solution for an elliptical drainage area using shape factors. Specifically such a shape factor is established by equating the dimensionless productivity index of the PSS flow in a square domain to that in an elliptical domain of identical area. The shape factor was dependent on the proppant number and fracture penetration ratio. Productivity index data for fractured wells with finite conductivity in square drainage area and no skin from Romero et al. (2003) was compared to Chen’s solution assuming equivalent drainage areas and identical proppant numbers, with the penetration ratio as a parameter. A non-linear multi-variable regression analysis results in a unified shape factor function with a correlation coefficient of 0.80 and a minimized sum of squared error of 36.1. The achieved shape factor allows the analytical solution for PSS flow of fractured well in an elliptical drainage area to be applied to square drainage areas. This generalization of the PSS flow solution is of practical significance in fracture design optimization and production rate decline analysis. Future recommendations including testing the accuracy of the shape factor in predictions of proppant numbers not used in analysis using COMSOL™, and increasing the dataset pool to increase the model accuracy.