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

TL;DR

This study maps Kentucky oil and gas wells, analyzes their fractal dimensions, and models their distribution using anomalous diffusion equations to understand underlying scaling laws and Levy processes.

Contribution

It introduces a fractal and anomalous diffusion framework to model oil well distributions, linking geological data with fractional calculus and Levy statistics.

Findings

Fractal dimensions of oil well locations were calculated.

Evidence of scaling laws in the distribution of wells.

Modeling suggests Levy flights describe well placement patterns.

Abstract

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.