Fracture trace length and number distributions from fracture mapping

Abstract

Statistical distributions of fracture trace length and density determined from two‐dimensional fracture mapping generally provide biased estimates of the underlying distributions due in part to edge effects of the finite window of observation (censoring) and the protocol adopted for recording short traces (lower truncation limit). Although methods for estimating the parameters of an assumed underlying distribution exist, validation of the inferred models by forward prediction of the observed length distribution is rarely undertaken since mathematical formulations of the required distributions are not reported in the literature. This paper presents formulae that can be used simply to obtain these distributions, with results for rectangular windows given as a specific example. Formulae relating the observed and underlying fracture density are also given. The formulae are presented for underlying trace lengths that can be characterized by exponential, lognormal, or finite range power law distributions. Results for the semi‐infinite range power law can also be obtained simply. The formulae are based upon the assumption that fracture locations can be adequately described by a uniform Poisson process, although the methodology is described more generally. The distributions of uncensored, singly censored and doubly censored trace lengths are also derived. The effects of the window of observation and the lower truncation limit for the frequently assumed semi‐infinite range power law are outlined.