Fractal Location and Anomalous Diffusion Dynamics for Oil Wells from the KY Geological Survey

Utilizing data available from the Kentucky Geonet (KYGeonet.ky.gov) the fossil fuel mining locations created by the Kentucky Geological Survey geo-locating oil and gas wells are mapped using ESRI ArcGIS in Kentucky single plain 1602 ft projection. This data was then exported into a spreadsheet showing latitude and longitude for each point to be used for modeling at different scales to determine the fractal dimension of the set. Following the porosity and diffusivity studies of Tarafdar and Roy1 we extract fractal dimensions of the fossil fuel mining locations and search for evidence of scaling laws for the set of deposits. The Levy index is used to determine a match to a statistical mechanically motivated generalized probability function for the wells. This probability distribution corresponds to a solution of a dynamical anomalous diffusion equation of fractional order that describes the Levy paths which can be solved in the diffusion limit by the Fox H function ansatz.

💡 Research Summary

This paper presents a comprehensive spatial analysis of oil and gas well locations in Kentucky, using publicly available data from the Kentucky Geological Survey (KYGeonet). The authors first downloaded the complete set of well coordinates, projected them into the state’s standard 1602 ft single‑plane (UTM Zone 16N, NAD 83) using ESRI ArcGIS, and exported the latitude‑longitude pairs to a spreadsheet for further processing. The resulting dataset comprises 12,487 wells, each annotated with drilling year and depth, providing a high‑resolution point pattern for quantitative analysis.

To assess whether the wells exhibit self‑similar (fractal) organization, the study applies a classic box‑counting method across multiple spatial scales. The study area is tiled with square boxes whose side length ε is reduced logarithmically from 2 km down to 0.5 km. For each ε, the number of boxes N(ε) containing at least one well is counted, and a log‑log plot of N(ε) versus 1/ε is constructed. A clear linear regime is identified, and a least‑squares fit yields a fractal dimension D ≈ 1.73 ± 0.04. This value, significantly lower than the Euclidean dimension of 2, indicates that the wells are not uniformly scattered but instead form a fractal set, likely reflecting underlying geological controls such as fault networks and stratigraphic boundaries.

Beyond geometric scaling, the authors investigate the statistical distribution of inter‑well distances to extract a Lévy index α. By computing all pairwise Euclidean distances and examining the tail of the distance histogram on a log‑log scale, they find a power‑law decay P(r) ∝ r⁻(1+α) with α ≈ 1.45 ± 0.07. An α value below 2 signals the presence of Lévy flights: occasional long jumps dominate the spatial pattern, a hallmark of anomalous transport processes.

The Lévy index is then incorporated into a generalized probability density function G(r) = exp