On the Analytical Estimation of Fracture Porosity from Fracture Permeability - Influence of Fracture Roughness, Tortuosity and Connectivity
Authors: Duperay P.A. , Kamp A.M. 
Published: April 2015

Abstract
One of the specific properties of a fractured reservoir that is hard to obtain is the pore volume of the fractures per unit bulk volume, also called the effective fracture porosity. Under some assumptions, such as the absence of vugs or karstic features, it may be possible to estimate fracture effective porosity from fracture effective permeability. The latter can be obtained from a well test. The theory is simple; it is based on Poiseuille's equation and an assumption of the geometry of the fracture network. It was outlined in a 1966 SPE paper by Parsons and it has been included in often-used books on fractured reservoirs such as the one by Nelson (1985). We will refer to it as the cubic law. The subject of our paper is the influence of fracture tortuosity, roughness, and connectivity on this porosity estimation through the cubic law. Papers have been published on these three topics, but never in the context of fracture porosity estimation.
Concerning roughness, we use the classical Joint Roughness Coefficient to generate representative 1D fracture surfaces. Using previously published works in literature it is possible to estimate the influence of surface roughness on the pressure drop without going through a full 3-D Stokes flow calculation in the domain between the two parallel fracture surfaces. Tortuosity is the effect of non-straight flow along the fracture planes. Its effect can be estimated through a formalism developed by Yamatomo et al. (2011). In order to investigate the impact of connectivity, Discrete Fracture Models (DFN's) were generated. Varying apertures, a relation between fracture permeability and porosity was obtained. Connectivity was varied either by varying the angle between different fracture sets, or by varying fracture length. Finally, extension of Parson's law was made beyond the classical parallel plane or cube geometry, to the general parallelepiped matrix block case, limited to three not-necessarily-perpendicular fracture families.
The calculation yields in case of neglect of roughness an underestimation of fracture porosity in the order of 20%, depending on kinematic fracture aperture and roughness. For tortuosities between 2 and 5, in case of overlooking of this effect, the porosity may be underestimated by about 50%, again depending on the kinematic aperture. The neglect of connectivity may lead to porosity underestimation depending on the degree of connectivity. The extension of the cubic law breaks down in the case of different apertures for each family, in which case additional information is needed for porosity estimation from permeability.
As a conclusion, doing quick-look estimates of fracture porosity from fracture permeability may be a good solution in case no time is available to build DFN's. The neglect of the influence of tortuosity, roughness, and connectivity may lead to a considerable underestimation of fracture effective porosity. If additional information were available (e.g. joint roughness), analytical corrections might be applied to correct this bias. The correct estimation of fracture porosity is important to correctly predict for example water break-through in oil- and gas reservoirs.