Estimating fracture density from a linear or areal survey

Fracture density is an important parameter of a fracture network. There are several definitions of fracture density and we are concerned here with the average fracture surface per unit volume. When fractures are not distributed equally among all directions, this parameter can be considered as a function of fracture pole. It can be evaluated from areal and/or linear surveys. This evaluation must account for the geometric bias due to the fact that fractures oblique to the scanline or surface are less easily observed than fractures orthogonal to it. A well-known technique to achieve this is the Terzaghi correction. We propose here an improved correction when survey outcrops and/or boreholes with different orientations are available. INTRODUCTION Fractures have a large impact on the recovery of oil, the safety of underground storage of CO2 or nuclear waste, the assessment of high-enthalpy geothermal installations, the safety of mining exploitations, the efficiency of in-situ leaching operations or of extraction by block-caving. Fractures is a generic term which includes a variety of objects occurring at all scales: faults (with a clear lateral displacement of one surface with respect to the other), joints, veins, etc. The methodology which follows is not specific to a type of fractures but in applications shall be applied separately to the various fracture types considered. Usually we study separately the fracture network and the single fracture. In the latter case we consider the exact shape of a fracture: topography of its two walls, roughness, geometry and continuity of the void space, filling material, traces of movement, reaction to mechanical stress or water injection. In the former case a fracture is considered as a planar object and stress is laid on the statistical and geometric distribution of the objects: fracture density, distribution of fracture orientation and fracture size, relationships between the various fracture sets, etc. (see, e.g., Chilès, 2005). J.P. CHILÈS, H. WACKERNAGEL, H. BEUCHER, C. LANTUÉJOUL, AND P. ÉLION GEOSTATS 2008, Santiago, Chile In practical applications we have to choose a sensible fracture network model and fit its parameters. Most models have a large number of parameters, because several fracture sets are present, each one with its own characteristics, not to speak of their interrelationships. Some of them allow a direct evaluation of their parameters, such as the regionalized disc-cluster model designed for granitic rocks (Chilès, 1988), and a hierarchical model designed for layered rocks (Chilès et al., 2000). Important parameters of nearly all models are fracture density and fracture size. The difficulty of their inference is that a fracture network is a 3D entity whereas observations are limited to 1D (boreholes) and 2D (outcrops, drift walls), and moreover very often to rather short stations, so that the data sets are subject to geometric bias, truncation, censoring, etc. Many authors have shown how to infer the distribution of fracture traces from the sampling of outcrops, bench faces or drift walls, in particular in the framework of a maximum likelihood or Bayesian approach; see, among others, Baecher (1980), Laslett (1982), and Lantuéjoul et al. (2005). Once the shape of the fractures is known, the distribution of fracture size can be deduced from that of trace length, for example as shown by Warburton (1980) for disc-shaped fractures. We will focus here on the other parameter, fracture density, which is also of prime importance. There are several definitions of fracture density (see Dershowitz and Herda, 1992), the main ones being the mean number of fracture centres per unit volume (λ) and the mean fractured surface per unit volume (μ). (Since we will not consider the fractures as three-dimensional objects – except by introducing attributes such as thickness or opening – we will not consider the definition of fracture density as the average fractured volume per unit volume.) When fractures are considered as finite objects, these parameters are related by μ = λ A (1) where A denotes the average fracture surface. If fractures are not distributed equally in all orientations, the above parameters shall be considered as functions of fracture pole orientation ω (the normal to the fracture plane). There are many situations where the direct estimation of fracture density λ and fracture size is associated with a large uncertainty, whereas density μ can be estimated with more robustness. The problem originates in the stereological component of the problem, and also in the difficulty of identifying a fracture, as shown in Figure 1: what is observed as a series of three fractures on the outcrop can be considered as three distinct fractures, or as a single en-échelon fracture, or as a standard fracture whose observation is affected by erosion, vegetation, etc. According to the interpretation, we record either a single large fracture, or three short fractures. This has a large impact on fracture density λ and fracture size A but does not alter fracture density μ. It is thus safe to first estimate μ, which can be done in rather good conditions, even if only borehole data are available. The fracture size distribution will be inferred – or simply assumed if outcrops are not available – and fracture density λ, which is required by most models (e.g., Boolean models), will then be deduced from formula (1). ESTIMATING FRACTURE DENSITY VIII International Geostatistics Congress 3D interpretation outcrop view mapping 3 distinct fractures single en-échelon fracture single rectangular fracture Figure 1. Three possible interpretations of the same outcrop view.