Pressure transient analysis of multiple fractured horizontal wells in naturally fractured unconventional reservoirs based on fractal theory and fractional calculus

- Fractal theory is used to describe the heterogeneous, complex natural fractures.
- The anomalous behavior of diffusion process is considered by employing the temporal fractional derivatives.
- A new FFDM for MFHWs in NFURs is established based on fractal theory and fractional calculus.
- The type curves of MFHW with three different outer boundaries are plotted.
- Different flowing regimes are identified, and the characteristics of pressure response are then studied in depth.

Currently, most models for multiple fractured horizontal wells (MFHWs) in naturally fractured unconventional reservoirs (NFURs) are based on classical Euclidean models which implicitly assume a uniform distribution of natural fractures and that all fractures are homogeneous. While fractal theory provides a powerful method to describe the disorder, heterogeneity, uncertainty and complexity of the NFURs. In this paper, a fractally fractional diffusion model (FFDM) for MFHWs in NFURs is established based on fractal theory and fractional calculus. Particularly, fractal theory is used to describe the heterogeneous, complex fracture network, with consideration of anomalous behavior of diffusion process in NFURs by employing fractional calculus. The Laplace transformation, line source function, dispersion method, and superposition principle are used to solve this new model. The pressure responses in the real time domain are obtained with Stehfest numerical inversion algorithms. The type curves of MFHW with three different outer boundaries are plotted. Sensitivity analysis of some related parameters are discussed as well. This new model provides the relatively more accurate and appropriate evaluation results for pressure transient analysis for MFHWs in NFURs, which could be applied to accurately interpret the real pressure data of an MFHW in field.