A Comparative Study of Mathematical Models for Fractured Reservoirs: Anomalous Diffusion and Continuum Approach.

Abstract

This study aims to determine an appropriate representative flow-model of a fractured reservoir after comparing two existing approaches: the anomalous diffusion and the continuum approach. A fractured reservoir is assumed in this paper that drains the fluid in transient condition, to a hydraulically fractured horizontal well. To investigate the comparison, dimensional consistency is maintained for both the anomalous diffusion and the continuum approach. Chen and Raghavan's (2015) model is considered as the anomalous diffusion model with modified boundary conditions. Continuum approach model considers the linear flow in a triple continuum structure that consists of matrix slab, micro-fracture, and hydraulic fracture. An analysis of the pressure response curves and the field data evaluates the proper approach for the analysis of the flow behavior. The solution of the wellbore pressure is derived in Laplace domain and is inverted by the Stehfest algorithm. Slope of the pressure response curve depends on the order of differentiation at the anomalous diffusion model. Conversely, the permeability of the hydraulic fracture controls the transient behavior at the continuum approach. The first set of analyses states that the continuum-based model considers the physical structure of the reservoir and increases the accuracy in the prediction of the reservoir behavior; however, more reservoir parameters are required for new continuum those are difficult to be determined. Alternatively, anomalous diffusion approach requires less parameter compare to the continuum approaches, but a high uncertainty exists in the precise determination of the order of the differentiation or the fractal exponent. The anomalous diffusion shows a good agreement with the synthesized field data at the early time whereas the continuum approach matches better at late time response.