Use of curved scanlines and boreholes to predict fracture frequencies

Abstract

We advance the method of Hudson and Priest (Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 20 (1983) 73–89) to develop a method for a curved scanline to be used to predict the numbers of fractures that would be observed in any direction. When sampling along a scanline, the probability of intersecting a fracture is influenced by the relative orientations of the fracture and of the scanline at that location. This sampling bias can be minimised by the use of the Terzaghi correction, w=(cosχ)⁻¹, where χ is the angle between the scanline and the normal to the fracture. These corrected frequencies are used to simulate fracture frequencies for all other orientations by doubly-correcting the data. Modelled fracture frequency is contoured on a graph of simulated scanline plunge against simulated scanline azimuth. This method is based upon the assumption that the data collected along the scanline is representative of the fracture population when the Terzaghi correction has been applied.

A graph of cumulative frequency of fractures against distance along a scanline provides a simple method for determining whether the scanline crosses differently fractured areas. Frequencies are corrected for dip, strike, and both dip and strike, with data from homogeneously fractured areas plotting as straight lines. These frequencies can be normalised for ease of comparison.

Introduction

Sampling along a scanline is an important approach for collecting fracture information. In many situations it is either the easiest or the only way to collect data. For example, image log or core data from oil wells may be the only source of information available about fractures in reservoirs. Knowledge of fracture patterns and distributions is of great importance in hydrocarbon recovery, and so scanline data from borehole image logs need to be exploited to the fullest extent. For example, to enhance oil recovery within a fractured reservoir, drilling a well with an orientation to intersect the maximum number of fractures is typically desirable.

Scanlines are a quick and systematic collection technique for fracture data (LaPointe and Hudson, 1985). Yet the analysis will be complicated by the use of curved scanlines, or curved boreholes, and by the line crossing differently fractured areas.

Several approaches have been adopted to deal with sampling biases, particularly related to straight scanlines. Straight scanlines have been used to characterise fracture orientations and to make predictions about fracture frequencies, which is the reciprocal of fracture spacing along the line. Fracture sets are progressively under-sampled as the angle between the scanline and the fracture set decreases. Terzaghi (1965) introduced a factor w to correct this under-sampling of fracture frequencies from straight scanlines, namely:

$$\text{w=(}\text{cos}\text{χ)}{}^{\mathrm{-1}}\text{,}$$

where χ is the angle between the scanline and the normal to a fracture. Eq. (1) is commonly applied to fracture data that are divided into sets, with the weighting factor calculated for the mean orientation of each set. Linear fracture frequency, which is the number of fractures observed or predicted to occur in a unit length, is the simplest and most commonly used measure of fracture frequency (Priest, 1993).

LaPointe and Hudson (1985, fig. 26a) used the Terzaghi correction to develop a fundamental rosette of corrected fracture frequencies, showing predicted frequencies as if all of the fractures on a plane on a plane. They verified the result using two sets of orthogonal scanlines to compare predicted and measured results. LaPointe and Hudson (1985, fig. 26b) also developed a rose diagram of fracture frequencies that would be encountered along straight scanlines in different directions. Their method is essentially two-dimensional, using strikes of fracture traces measured along a straight scanline.

Hudson and Priest, 1983, Priest, 1993 used straight scanlines to predict fracture frequencies in all directions in three dimensions. Priest (1993, p. 112) presented a complex method to predict fracture maxima and minima, but stated that a more direct, but less elegant, method for determining fracture frequencies is to simply compute frequencies for the complete range of possible sampling directions. The latter approach is used here (Section 2). Priest (1993) did not discuss the use of curved scanlines. Lacazette (1991) expanded the method to model scanlines of all orientations, and showed how predicted fracture frequencies can be plotted on a stereogram.

Several approaches have been adopted to deal with sampling biases, particularly related to straight scanlines. Mauldon and Mauldon (1997) developed correction factors for sampling fractures along a borehole of non-zero radius, enabling predictions to be made about fracture frequencies from boreholes and tunnels. These factors require, however, prior knowledge of fracture size. Consequently, these correction factors are particularly useful when the borehole radius is large in comparison with the length of fracture intersects, which is the situation in which fracture size may be better approximated. Their approach reduces to the Terzaghi correction factor for a borehole of zero radius, i.e. a scanline. Martel (1999) analysed fracture orientation data from boreholes using the mean orientation of fractures, spherical variance and the moment of inertia to analyse fracture pole orientations distributed on a hemisphere. The method considers the effect of borehole sampling bias on measured orientations of fractures with a pre-assumed orientation distribution, such as uniform, and then modifies the distribution using the mismatch between observations and predictions for a particular case. The significance of the mismatch can be evaluated visually on a stereogram, or quantitatively with chi-square or Kolmogorov–Smirnov tests (e.g. Davis, 1986, Martel, 1999).

Grossenbacher et al. (1997) present a method for determining fracture frequencies from data collected along circular scanlines, which they suggest can be expanded and adapted to the more general case of irregularly curved scanlines. Mauldon et al., 1999, Mauldon et al., 2001 use circular scanlines to avoid sampling biases. They also develop methods for quantifying the intensity, density and mean trace-lengths of fracture traces within the circle. This method is useful when fracture traces can be measured on surfaces.

All of these approaches offer improvements in resolving bias issues, but none exploit the strategy proposed in this paper of considering fracture frequency along curvilinear lines.

This paper describes a method that uses data collected along a curved scanline to estimate the numbers of fractures that would be intersected along straight scanlines of any orientation through the same statistically homogeneous rock mass (Section 2). In this paper, the term fracture is used to describe any brittle planar discontinuity in rock, including faults, joints, veins and dykes. The method identifies the scanline orientations that intersect the maximum and minimum numbers of fractures. The method is a more generalised version of the methods of LaPointe and Hudson, 1985, Priest, 1993 for predicting spatial distributions of fracture frequencies. Both of these methods require straight scanlines and the first only considers fracture strikes. By contrast, we give a mathematical formulation for the three-dimensional case where the dip and dip direction of fractures are recorded along straight or curved scanlines. This method is particularly useful for the analysis of borehole data, since boreholes are commonly curved.

We also present a graphical representation of the cumulative number of fractures versus distance along the scanline as a simple way to identify fracture domains with different frequencies and patterns (Section 3). This representation tests the assumption that the fracture distribution is spatially uniform.