Nonlinear Fractal Flow Model of Fractured Vertical Wells With Stimulated Reservoir Volume Considering the Quadratic Gradient Term

Abstract

Although various fractal flow models have recently been developed to investigate pressure responses of fractured vertical wells, almost all of the existing fractal models ignore the quadratic gradient term (QGT), which makes them violate mass conservation. In this paper, fractal theory is introduced to develop a nonlinear flow model of fractured vertical wells with stimulated reservoir volume (SRV). The QGT is reserved so that the present model fully obeys material balance. Function transforms are used to linearize the nonlinear flow model, and then, Laplace transform and Laplace numerical inversion algorithm are employed to derive the pressure solution. Type curves are provided to analyze the flow characteristic and identify the flow regimes. The effects of some parameters on the pressure responses are discussed in detail. It is found that the existence of the QGT leads to the decrease of the pressure drop, especially at a large nonlinear coefficient and a large time scale. Fractal parameters and SRV radius not only affect type curves but also affect the relative error caused by neglecting the QGT.