The Application of Fractal Geostatistics to Oil And Gas Property Evaluation And Reserve Estimates
Authors: Poon D.C. , McCormack M. , Thimm H.F. 
Published: March 2010

Abstract
A fractal is a geometrical object which seems to have similarshapes when observed at different ranges of scales. For example natural fractures in a reservoir, permeability data, porosity data and the viscous fingers resulting from unstable miscible flood have been found to demonstrate such fractal characteristics. Turcotte introduced the concept of renormalization group and derived the fractal dimension for the fragmentation of a cubic cell into subsequent elements with decreasing volume. This concept has been successfully applied to analyse the tonnage and grade relationship using the production dam from the mining of minerals. In this study, the concept of renormalization group has been extended to predict the discovery rate of oil and gas pools, which is based on the success history of exploration drilling in a certain region. As a secondary objective, a model for predicting the ultimate reserve in a petroleum bearing region has also been developed.

Introduction
It has long been recognized that many processes in nature demonstrate fractal characteristics. The concept of fractal geometry provides a means of quantifying these processes. Basically, a power law relationship between number and size defines a fractal. The classic example of a fractal is the measured length of a complex coastline. If the length L(t) of the coastline is related to the length of the measuring stick, r, by: Equation (1) (Available in full paper)
then the coastline is a fractal and D is the fractal dimension.
The application of fractal geometry to the petroleum problems is not new. Hewett (1) applied the techniques of fractal geostatistics and conditional simulation to investigate the effect of reservoir heterogeneity on waterflood. Tang et. al. (2) applied the same techniques to forecast the performances of waterflood and carbon dioxide flood processes in a low permeability sandstone reservoir characterized by alternating sand/shale sequences. Multiphase flow in a porous media also demonstrates fractal characteristics. Displacement of a fluid in porous media by another fluid of a much lower viscosity usually results in a displacement front which is notoriously unstable with the development of a so called viscous fingers. Maloy et al (3) analysed the viscous lingers in two-dimensional fluid flow in a porous media and found that the viscous lingers for various viscosity ratios generally have a fractal dimension of about 1.6.
The objective of this study was to extend the techniques of fractal geostatistics to oil and gas property evaluation and reserve estimates. We will begin with a review of the conventional methods for forecasting the ultimate reserves and discovery rates of oil and gas exploration. A reliable estimate of the amount of oil and gas in a geographic area would be of great value in formulating governmental and corporate policies.
The term "reserve" used in this paper means the probable amount of petroleum deposits in a geologic region based on exploration drilling data only. We further define a pool as the body of oil and/or gas occurring in a separate reservoir and under a single pressure system.