On the Density Variability of Poissonian Discrete Fracture Networks, with application to power-law fracture size distributions

Etienne Lavoine,Caroline Darcel,Romain Le Goc,Philippe Davy

doi:10.5194/adgeo-49-77-2019

Abstract

Abstract. This paper presents analytical solutions to estimate at any scale the fracture density variability associated to stochastic Discrete Fracture Networks. These analytical solutions are based upon the assumption that each fracture in the network is an independent event. Analytical solutions are developed for any kind of fracture density indicators. Those analytical solutions are verified by numerical computing of the fracture density variability in three-dimensional stochastic Discrete Fracture Network (DFN) models following various orientation and size distributions, including the heavy-tailed power-law fracture size distribution. We show that this variability is dependent on the fracture size distribution and the measurement scale, but not on the orientation distribution. We also show that for networks following power-law size distribution, the scaling of the three-dimensional fracture density variability clearly depends on the power-law exponent.

Highlights

Characterizing fracture networks in geosciences is a key challenge for many industrial projects such as deep waste disposal, hydrogeology or petroleum resources, because it may change the mechanical (Davy et al, 2018; Grechka and Kachanov, 2006) and hydrological (Bogdanov et al, 2007; De Dreuzy et al, 2001a, b) behaviour of the rock mass
This paper presents analytical solutions to estimate at any scale the fracture density variability associated to stochastic Discrete Fracture Networks
Those analytical solutions are verified by numerical computing of the fracture density variability in three-dimensional stochastic Discrete Fracture Network (DFN) models following various orientation and size distributions, including the heavytailed power-law fracture size distribution

Summary

Introduction

Characterizing fracture networks in geosciences is a key challenge for many industrial projects such as deep waste disposal, hydrogeology or petroleum resources, because it may change the mechanical (Davy et al, 2018; Grechka and Kachanov, 2006) and hydrological (Bogdanov et al, 2007; De Dreuzy et al, 2001a, b) behaviour of the rock mass. Darcel et al (2013) proposed an analytical solution to quantify at any scale, the standard deviation associated to fracture frequency p10, considering fracture-borehole crossing as a one-dimensional Poisson point process (random positions with fixed density). They show that the standard deviation associated to the number of intersections per unit length p10 is inversely proportional to the square root of the measurement scale. We show that this variability depends on the measurement scale and the fracture size distribution, but not on the orientation distribution These solutions are validated by numerical simulations, computing the associated fracture densities mean and variance at various scales for three-dimensional Poissonian Discrete Fracture Networks

Theoretical development

Fracture number density p30

Percolation parameter p

Conclusion