Fractal Transient Pressure Behavior of Naturally Fractured Reservoirs
Authors: Flamenco-Lopez F. , Camacho-Velazquez R. 
Published: December 2007

Abstract
The objective of this paper is to investigate the transient pressure behavior of naturally fractured reservoirs with fractal characteristics. This work is based on the findings of previous studies where it is shown that the networks of fractures in some reservoirs are fractals. Thus, using this assumption, an approximate analytical solution for dual-porosity systems is derived in the Laplace space.
The approximate solution presented in this work uses a pseudo-steady state matrix-to-fractal fracture transfer function. This solution is compared with a finite element solution and good agreement is found. Short- and long-time approximations are used to obtain procedures in time to determine some fractal parameters. These approximations are compared to the appropriate expressions when an unsteady-state matrix-to-fractal fracture transfer function is used.
Synthetic and field examples are presented to illustrate the methodology proposed in this work.
This paper also presents an analytical solution for the pseudo-steady-state flow period and demonstrates the importance of analyzing both transient and pseudo-steady-state flow pressure data for a single-well situation in order to fully characterize a naturally fractured reservoir with a fractal geometry.

Introduction
Reservoir heterogeneity influences fluid flow trajectories. These heterogeneities are present on a wide range of scales. Fractal theory provides a method to describe the network of fractures in a rock and to connect heterogeneities at smaller scales to those at larger scales and vice versa. The simplest fractal models assume a power law scaling procedure where the exponent is related to the fractal dimension, which is the main tool of fractal geometry. The dimension provides a description of how much space the set of fractures fills, it contains much information about the geometrical properties of the set of fractures. Thus, this fractal parameter describes variations over a range of scales and if it can be estimated, reservoir properties such as porosity and permeability can be properly defined.
Chang and Yortsos1 applied a fractal model to pressure transient analysis. This model describes a naturally fractured system which has different scales, poor fracture connectivity and disorderly spatial distribution in a proper fashion. Acuña et. al.2 applied this model to analyze pressure transient tests and they (as Ref. 1 did) found out that the change in wellbore pressure is a power-law function of time, where the fracton or spectral dimension can be obtained. This parameter is a function of both fractal dimension and the conductivity index, which is related to the topology of fractures network. Both Refs. 1 and 2 point out that to determine the values of fractal parameters additional information to the transient test is needed.
Olarewaju3 also examined the pressure transient response of naturally fractured reservoirs by using a fractal model, but instead of assuming a pseudo-steady state transfer function between matrix and fracture systems as Refs. 1 and 2 did, he used a transient interporosity flow assumption.
Beier4 extended the fractal model of Chang and Yortsos1 to consider a hydraulic fractured well. He also observed a power-law behavior during the linear and "radial" flow periods.