# A mathematical model for fractured reservoirs using anomalous diffusion and multi-continuum approach

Karim, Md Mostafijul (2019) A mathematical model for fractured reservoirs using anomalous diffusion and multi-continuum approach. Masters thesis, Memorial University of Newfoundland.

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## Abstract

An accurate analysis of the characteristic behavior of a fractured reservoir is challenging due to the complex reservoir formation; furthermore, the irregular flow patterns in the discrete domains add to the computational complicacy. The study aims to develop a mathematical model for the fractured reservoir through the utilization of the anomalous diffusion approach and the multi-continuum approach. Firstly, the study reviews both the concepts in details to make a better understanding of the limitations, formulations and the application criteria. A comparative study is done to determine the relative impacts of two approaches in the reservoir at different flow periods. Consequently, a linear model is developed for the reservoir flow towards a hydraulically fractured horizontal well at the transient condition. The derivation considers a modified tri-linear model with different arrangements of the matrix and fractures. The solution is derived in the Laplace domain, and numerically it is inverted to the real-time domain by the Stehfest algorithm. The study shows that the continuum-based approaches differ for the different fracture network, inter-flow condition, continuum-number, and the interface transfer function whereas the anomalous diffusion approach captures the heterogeneity of the reservoir by the fractional time or space derivative. The evidence from the comparative study suggests that a combination of the continuum approach and the anomalous diffusion is recommended as an alternative approach for the modelling of fluid flow in a fractured reservoir. In the developed model, the influence of the super-diffusion in the hydraulic fracture is remarkable as it alters the pressure response during the whole life of the reservoir. However, the sub-diffusion impact increases with the time and is significant at the late stage. The study also shows that Macro-fracture permeability regulates the pressure drop in the reservoir as it is the primary conduit in the inner reservoir. The combination of the approaches and the logical distribution of the flow conditions are shown as the better alternative to the conventional method.

Item Type: Thesis (Masters) http://research.library.mun.ca/id/eprint/13755 13755 Includes bibliographical references (pages 144-145). Anomalous Diffusion, Continuum Approach, Fractured Reservoir, Mathematical Model, Tri-linear Model Engineering and Applied Science, Faculty of May 2019 Submission Oil fields--Mathematical models.