A Method to Estimate Length Distributions from Outcrop Data

Dimensions of shales and other geologic bodies that affect fluid flow through reservoirs and aquifers are often estimated from analogous deposits exposed in outcrops. Shale lengths observed in outcrops are biased because the finite length of outcrops truncates longer shales and long shales tend to be overrepresented in the sample. Shale length distributions can be modeled using an Erlangian probability density function based on termination frequency. Termination frequency can be calculated from outcrop observations even if the shales are incompletely exposed. Termination frequency is unbiased regardless of underlying shale length distribution and outcrop size and shape. Complex length distributions can be represented by a weighted sum of Erlangian models, each with a distinct termination frequency. The proportions and termination frequencies of the component models can be estimated using Bayes' theorem. Subsamples of the outcrop area can be analyzed to quantify spatial trends in termination frequency and thus shale length. These methods can be applied to estimate length distributions of any geologic object exposed in outcrops or other spatially limited samples. In many cases estimated termination frequency is normally distributed, and its variance can be estimated using closed-form expressions. Shale length distributions for the Frewens Sandstone were modeled using one- and two-component Erlangian probability density functions. The Frewens Sandstone is a tide-influenced deltaic sandstone body within the Cretaceous-age Frontier Formation, exposed in central Wyoming, USA. Length observations and Erlangian models agree reasonably well. Because the models correct for the effects of shale truncation in limited outcrop exposures, predicted means and proportions of long shales are substantially greater than observed values. There is a vertical trend and pronounced vertical cyclicity in termination frequency, indicating that shales become shorter upward and intervals of longer and shorter shales alternate. Along strike, termination frequency decreases away from the sandstone-body axis, reflecting greater preserved shale lengths toward sandstone-body margins. No correlation was observed between distance along dip and termination frequency. Termination frequencies and histograms of shale length for adjacent, perpendicular walls are similar, indicating that there is no anisotropy in shale length.