Estimation of Tortuosity of Fluid Flow Through a Single Fracture
Authors: Murata S. , Saito T. 
Published: March 2010

Abstract
Tortuosity of fluid flow through a fracture is mainly caused by the geometry of the fracture surface and contact condition. In addition, it causes the deviation from the cubic law of fracture permeability. In this study, to estimate the tortuosity, several physical experiments and numerical simulations to visualize the streamlines were carried out for the water flow through interlocking fractures and non-interlocking fractures. These fractures were artificially prepared by using a fractal model to control the fractal parameters and interlocking parameter. Furthermore, the tortuosity was discussed in relation to the fractal parameters, interlocking parameter, and contact area ratio. Consequently, the behaviour of fluid flow through the fractures was clarified, and the effects of those parameters on tortuosity were clarified. The obtained results can give a basic knowledge of the fluid flow through a fracture and will be helpful to estimate the fracture permeability used for reservoir simulation and well test analysis.

Introduction
In order to estimate the condition of fluid flow in a fractured reservoir, it is necessary to understand the behaviour of fluid flow through a single fracture and to estimate fracture permeability correctly as well as to estimate the heterogeneous distribution of the fractures in the reservoir. This is because the fractures have considerably higher permeability than the intact parts of the reservoir and they govern the fluid flow in the reservoir.
Generally, fracture permeability has been estimated by the cubic law that is valid for the laminar flow between two perfectly smooth parallel plates. However, the topography of actual fracture surfaces is very complex and some parts on them are contacting. This makes the flow paths tortuous and causes the deviation from the cubic law of fracture permeability. Moreover, aperture of the fracture estimated from the cubic law, called hydraulic aperture, is smaller than the actual aperture. This would lead to some errors in reservoir simulation and well test analysis for a highly fractured reservoir. Therefore, many research studies(1–5) have been carried out to determine the behaviour of fluid flow through a fracture and to estimate the fracture permeability correctly.
Among this research, Walsh and Brace(2) presented the following equation to represent the permeability of rocks:
Equation (available in full paper)
They discussed the permeability in relation to the formation factor without estimating the tortuosity directly. In their study, they assumed that electrical current and fluid follow the same paths and so tortuosity for the two processes is the same. But, Brown(5) showed that the paths of the two processes are different by using computer simulations. Therefore, a parameter other than the formation factor or the direct estimation of tortuosity is necessary to estimate the fracture permeability.
In this study, we try to estimate the tortuosity directly from several physical experiments and numerical simulations in order to visualize the streamlines of water flow through an interlocking fracture and a non-interlocking fracture. These fractures are prepared artificially by using a fractal model.