Performance of vertical fractured wells with multiple finite-conductivity fractures

Multiple fractures originating from a vertical wellbore are usually observed in hydraulic fracturing treatments. Most of the previous models for vertical fractured wells are based on single-planar fractures that may be symmetric or asymmetric about the wellbore, but few studies have been devoted to multiple-planar fractures. This paper presents a new semi-analytical solution in Laplace space for the pressure responses of a vertical fractured well, producing at a constant flow rate through multiple finite-conductivity fractures. The solution is presented in Laplace space so that the effects of the wellbore storage and skin factor can be easily incorporated by Duhamel’s principle, and then the solution in real space can be obtained using the numerical inversion algorithm proposed by Stehfest. Pressure response curves are plotted and the effects of the relevant parameters on these are analyzed. It is found that with the increase of the fracture number, smaller pressure depletion will appear under the same conditions, and the interaction between fractures takes place much earlier and becomes much stronger. We find that with the increase of the fracture number, the duration of the wellbore storage period becomes shorter, and the bilinear flow appears earlier. Decreasing the angles between fractures and increasing the fracture asymmetry coefficient will lead to a stronger interaction between them, and will then affect the bilinear flow and linear flow behaviors. The present model can be used to interpret the pressure data of the vertical fractured wells with multiple fractures and to provide more accurate dynamic parameters.