Some Applications of Fractal Mathematics In the Evaluation of Environmental Problems
Authors: Thimm H.E. , Poon D.C. , McCormack M. 
Published: March 2010

Abstract
Recent advances in the development of the techniques of fractal mathematics permit applications to environmental problems commonly encountered in the petroleum industry and other industries. Examples are given whereby the use of Turcotte's theory of renormalization groups leads, to simple equations for 'use in the interpretation of data sets and in practical decision making. Specifically; the applications are illustrated, by examples. Mercury contamination data at a gas plant are interpreted and the optimal volume of soil excavation is determined at a contaminated site.

Introduction
The relatively new theories of fractal mathematics were introduced as recently as 1967 by Manctelbrot(1). He noticed that the apparent or measured length of a complex coastline increased with the decreasing length, of the measuring rod, in accordance with a power law Equation (Available in full paper)
where P is the length of the coastline, and r is the length of the measuring rod. The term D in the exponent is referred to as the fractal dimension.
Since Mandelbrot's initial work, many natural phenomena have been shown to have fractal properties. Applications have been published in petroleum engineering by: Hewett in investigating the effect of reservoir heterogeneities on a water flood(2), by Tang et al., as a means of forecasting the performance of carbon dioxide and water floods(3), and by the present authors in reserve estimation of hydrocarbon(4). Maloy et al., have analysed the viscous fingers in two-dimensional fluid flow in porous media and have demonstrated fractal properties(5). In heterogeneous catalysis, it was realized that the molecular accessibility of a rough catalyst surface is dependent of molecular size, and an equation of the type Equation (Available in full paper)
is obeyed, where N is the number of molecules needed for monolayer coverage, S is the molecular size, and Da is the fractal dimension of the surface available for coverage. A short review of this subject has recently been published by Avnir(6).
In addition to the practical engineering applications, Turcotte has applied the theory to geology and geophysics(7), and it is these applications which are considered here for the solution to environmental problems.
Turcotte shows that the cumulative average weight grade of an ore,
Equation(1)(Available in full paper)
Equation(2) (Available in full paper)
Equation(3) (Available in full paper)
The derivations may be found in Reference (6). Turcotte illustrated these equations by considering the total tonnage and grades of mercury ore mined in the United States. He obtained a fractal dimension of D =2.01. On extrapolation, by setting Mo equal to the tonnage of soil in the United States and arbitrarily choosing a data pair from the line, he obtained a value of Co = 80 ppb. This value is the same as the generally accepted mean concentration of mercury in U.S. soils, as required.

Applications
Case 1: Mercury Contamination at a Gas Plant
At a sour gas plant in Southern Alberta, which is currently being decommissioned, a limited number of soil samples were analysed for the presence of mercury.