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INTRODUCTION ABSTRACT: This paper presents a simple derivation of the unbiased fracture trace intensity estimator n/4r, where n is the number of intersections between fracture traces and a circular scanline of radius r. The use of circular scanlines eliminates the orientation bias associated with straight scanlines while still achieving the time efficiency of scanlines. Application of the estimator to synthetic and real fracture trace maps is discussed. The utility and efficiency of the circular scanline intensity estimator make it a valuable tool for fracture studies. Fracture traces, for present purposes, are the intersections of planar fractures with an exposed planar surface such as a bedding plane, rock face or mine drive wall. The traces appear as straight line segments with ends that may be either exposed or hidden beyond the boundaries of the exposure (i.e. censored). Fracture trace intensity, with /timension L-', is defined as the mean length of fracture traces per unit area of the planar sampling region. For parallel fractures of infinite length, trace intensity may be considered equal to frequency (Priest & Hudson 1981). Intensity can be considered as roughly the product of mean size and density. Estimators for mean fracture trace length and trace density, using circular windows are discussed in a companion paper (Mauldon et al. 1999). Fracture trace intensity is useful for characterizing fractured rocks to determine their hydraulic and mechanical behavior (Brown 1970, Oda 1982, 1985, 1993, Barton & Larson 1985, Narr 1991, Wu & Pollard 1995, Kulatilake et al. 1996). Trace intensity is determined from a two- dimensional (2-d) sample and as such, serves as a proxy for the volumetric fracture intensity. Relationships between fracture trace intensity (2-d) and fracture intensity (3-d) are discussed by Dershowitz & Herda (1992) and Mauldon (1994). This paper discusses an unbiased estimator of fracture trace intensity based on a count of fracture trace intersections with a circular scanline. In current geologic and rock engineering practice, straight scanlines and fracture trace maps (Fig. 1) are commonly used to estimate fracture intensity (Priest & Hudson 1981, LaPointe & Hudson 1985, Schaeffer 1991, Dershowitz & Herda 1992, Wu & Pollard 1995, Becker & Gross 1996). Straight scanlines provide a fast method for recording fracture attributes, but yield a sample biased by fracture orientation with respect to the scanline (Terzaghi 1965, Wathugala et al. 1990, Priest 1993, Mauldon 1994, Mauldon & Mauldon 1997). Fracture trace maps reduce this sampling bias by sampling an area rather than a single direction, but are highly labor-intensive and therefore costly.
INTRODUCTION ABSTRACT: Mean fracture trace length and trace density are important parameters in the characterization of fractured rock with applications in hydrogeology, oil recovery and rock engineering. Estimating these parameters from field data is beset with problems due to censoring and length bias. The paper presents simple distribution-free estimators for mean trace length and density that automatically correct for sampling bias. The estimators use samples collected with circular windows placed at random on exposed pavements or rock surfaces. Use of the estimators is demonstrated by application to synthetic and real data sets. This paper describes easy-to-use estimators for fracture trace density and mean fracture trace length using circular sampling windows (Fig. 1). Trace density is here defined as the mean number of trace centers per unit area of the sampling plane. Mean fracture trace length is the mean length of the entire population of traces. Unless both ends of a trace are visible within a sampling window, the location of the trace center is unknown. Therefore, mean trace length and trace density must both be inferred, which is the purpose of these new estimators. The estimators are distribution independent and automatically correct for errors from censoring and length bias, making the estimators an improvement on existing methodologies. The results of these techniques yield useful information about the trace density and mean trace length of a fracture population, which may be used Figure 1. Ci?ular window, scanline and irregular rock pavement window samples of fracture traces. Solid traces where visible on the pavement. to tackle many problems in geology and engineering. For example, trace density and mean trace length are used for estimating elastic rock properties, fracture porosity, path length and connectivity for fluid flow, and mechanical behavior of fractured rock (Segall & Pollard 1983, Long et al. 1985, Amadei & Savage 1993, Elsworth & Mase 1993, Hu & Huang 1993, Dershowitz & LaPointe 1994, Kulafilake et al. 1996, Zhang et al. 1996, Odling 1997). The estimators are simplified from special cases of more general models developed for sampling windows of arbitrary convex shape (Mauldon 1998). When circular windows are used as the sampling domain, the estimators are independent of both the trace orientation distribution and of the trace length distribution (each of which may be either discrete or continuous). Thus, they are entirely distribution independent. The performance of these estimators is demonstrated by Monte Carlo simulation of synthetic fracture sets, and also by application to fracture trace maps from rock pavements (Fig. 2b, c) in Wales, UK, and South Carolina, USA.
- North America > United States > Tennessee (0.29)
- North America > United States > South Carolina (0.25)
- Europe > United Kingdom > Wales (0.25)
- Law > Civil Rights & Constitutional Law (0.58)
- Energy > Oil & Gas > Upstream (0.54)