Simulation of Fluid Flow Through Rough Fractures
Authors: Muralidharan Vivek , Chakravarthy Deepak , Putra Erwinsyah , Schechter David S. 
Published: December 2007

Abstract
Flow through a fracture is usually assumed to take place between two smooth parallel plates. However, it is widely accepted that the fracture has tortuous paths and roughness and hence the flow behaviour in these paths compared to that in parallel plates is different. Although previous studies have shown that the fracture aperture follows lognormal distribution, studies have not been conducted to determine the distribution of fracture aperture with changes in stress conditions. In this paper, we present fracture aperture measurements under different stress conditions using an X-Ray CT scanner. We developed a calibration curve to obtain a correlation between CT numbers and fracture aperture since there is no direct calculation of aperture from CT scanner data. Aperture distribution patterns from about six thousand aperture measurements were obtained for each stress condition evaluated. The results of this study show that the apertures follow lognormal distribution even at elevated stress conditions. We then performed waterflood experiments to validate the use of distributed apertures in simulators. A sensitivity analysis was also performed to analyze the effect of injection rates and fracture roughness on oil recovery.

Introduction
Modeling of fluid flow through rough fractures has gained importance over the years. This can be attributed to the extremely low ultimate recoveries obtained from naturally fractured reservoirs, in spite of their huge reserves. Attempts are being made to develop efficient models to better formulate depletion plans. The first comprehensive work on flow through open fractures was done by Lomize1, in which he used parallel glass plates and demonstrated the validity of cubic law for laminar flow. He modeled fluid flow with different fracture shapes and investigated the effects of changing the fracture walls from smooth to rough.
Witherspoon et al.2 conducted laboratory experiments to validate parallel plate theory and they showed that the parallel plate approximation tends to break down at higher normal stress (&gt;10 MPa) across the fracture. Alfred3 also confirmed that the parallel plate assumption is not valid to adequately model the fluid flow experiments when overburden pressure is significant. The flow through a single fracture does not progress uniformly as assumed by parallel plate theory; rather, it flows through a limited number of channels4,5. Hence, the fluid flow in these tortuous channels tends to follow a preferred path. Pyrak et al.6 (1985) performed laboratory experiments wherein they injected molten wood's metal into single fractures at different applied stress conditions. The direct evidence of tortuous paths was observed upon opening the cooled metal in the fracture. The fluid flow in these paths will be through the larger apertures which offer least resistance to flow.
When the parallel plate approach was proved invalid, Tsang and Witherspoon5 accounted for the variation of apertures in a rough fracture. Later, Tsang4 modeled the variation of fracture apertures by electrical resistors with different resistance values placed on a two-dimensional grid. The results indicated that smaller apertures play a key role in restricting fluid flow. When the fracture contact area increases, tortuosity and connectivity of fractures become important. The flow through a single fracture took place in a limited number of channels, which was evident from the field experiment carried out in a single fracture7. Gentier8 measured fracture surface roughness profiles in a granite fracture. Upon plotting the apertures, the aperture density distribution was approximated by a gamma function. The density distribution is given byEquation 1
where bo represents the distribution peaks, and the mean aperture is 2bo. The same distribution was assumed when considering the channeling of flow through fractured media9. Tsang and Tsang9 assumed the channel width to be a constant of the same order as the correlation length ?, where correlation length is the spatial length within which the apertures have similar values. The reduction in channel apertures affected the tracer breakthrough curves when normal stress across a fracture was increased.