The Impact of Fractal Characteristics on the Performance of Structurally Complex Reservoirs
Authors: Al-Rbeawi Salam 
Published: November 2020

Abstract
The main objective of this paper is understanding the impact of the fractal characteristics of fractured reservoirs on their pressure behavior, flow rate decline, and productivity index. The paper aims to propose a new methodology for developing several analytical models that could be used for describing the wellbore pressure behaviors at a constant sandface flow rate and the flow rate decline trends at a constant wellbore pressure. Considering the fractal characteristics in the proposed analytical models may help eliminating significantly the uncertainties in predicting reservoir performance in highly disordered and heterogeneous formations. The target of this study is the reservoirs characterized by a complex structure and undergoing an anomalous diffusing flow mechanism such as naturally fractured reservoirs with a nonuniform distribution of embedded fractures and depleted by multiple hydraulic fractures.
The proposed methodology considers including the fractal characteristics such as the mass fractal dimension, conductivity index of anomalous diffusion flow mechanism, fractal-network parameters, fractional-derivative order, and matrix/fracture-interaction index as well as dual-porosity media characteristics such as the storativity and interporosity flow coefficient in the analytical models of the pressure, rate, and productivity index. These analytical models are derived for the porous media with the fractal networks of fractures that are embedded in the matrix and connected to hydraulic fractures. The porous media in this study are represented by two volumes. The first is the stimulated reservoir volume with a rectangular-shape area of a length equal to the hydraulic fracture length, while the second is the unstimulated reservoir volume that extends away from the tips of hydraulic fractures towards the reservoir boundaries. The solutions of the two approaches of the inner wellbore conditions are demonstrated in this study. The first is the wellbore pressure drop assuming constant sandface flow rate while the second is the decline rate and cumulative production assuming constant wellbore pressure.
The outcomes of this study are: 1) Understanding the impact of the fractal characteristics on pressure behavior, flow rate declining pattern, and productivity index scheme during early and late production time. 2) Developing analytical models that consider the fractal characteristics of structurally complex reservoirs depleted by multiple hydraulic fractures. 3) The study has found that some of the fractal characteristics such as mass fractal dimension has a significant impact on reservoir performance while others such as matrix/fracture interaction index may not have a significant impact. 4) Fractal reservoirs exhibit better performance than the standard geometry reservoirs of single porosity media or naturally fractured reservoir with dual-porosity media.
The novel point presented in this study is developing a new methodology for including the impact of the fractal characteristics in the pressure, rate, and productivity index models. In these models, the stimulated reservoir volume is assumed as a vertical wellbore with an equivalent radius equal to the fracture half-length. This approach is verified by comparing the results of pressure, rate, and productivity index obtained for a very short fracture half-length with the result of the vertical wellbore wherein an excellent matching is observed.