Abstract

Conventional porous-medium-flow models are not suitable for naturally fractured, nano-porous reservoirs where the conditions of the continuum assumption do not hold. Due to the abrupt pore-scale variations and severe contrast between matrix and fracture characteristics, the velocity fields over nano-porous unconventional reservoirs display sharp discontinuities and irregular fluctuations. These heterogeneous velocity fields cannot be represented by Darcy's Law, which relates fluid velocity to an instantaneous, local pressure-gradient. An alternative approach is to use anomalous diffusion over a highly heterogeneous velocity field. This paper considers transient, single-phase production from a fractured horizontal well in an unconventional reservoir and uses anomalous diffusion to represent flow in the naturally fractured region between hydraulic fractures. The anomalous-diffusion model combines a time-fractional flux law and classical mass balance to describe sub-diffusive flow in natural fractures, in matrix, or both. The solution of the model is obtained within the framework of fractional calculus. Comparison with established and published solutions shows excellent agreement. Sensitivity analyses indicate that the proposed model can be used to describe a wider range of flow heterogeneity without need for the intrinsic details of the matrix and natural fracture properties.

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