Scaling Laws for the Distribution of Natural Resources

Tom BlenkinsopSchool of Earth and Ocean Sciences

Cardiff University

NP3 – Scales, Scaling and Nonlinear VariabilityNP3.1/CL6.12/SSS0.6 Scales, scaling and extremes in the geosciences

• Scaling Relationships for Natural Resources• Data sets: Gold, Gas• Results• Percolation Theory

The Mass Dimension, DmM(r) = C1rDm

cf. The radial density function:r(r) = CM(r)/pr2 = C1r Dm-2

The Mass-Radius Scaling Exponent, Dmr M(r) = C2rDmr

Understanding the Exponents

• Space filling, random or uniform patterns: Dm = 2• A point: Dm = 0• Dm and Dmr measure how mass varies as a function of

distance from a point, or the degree of clustering. • Dm values less than 2 indicate a decrease in density

with distance, a characteristic of fractal patterns.• Dmr can have values > 2

Data Sets

Gold deposits, Zimbabwe craton

Dm = 1.05Dmr = 1.02

Roll-off:Sampling from a fractaldistribution

Clustering near edges of study area

Conventional Gas Wells, Pennsylvannia

Dm = 1.63Dmr = 1.72

Unconventional Gas Wells, Pennsylvannia

Dm = 1.26Dmr = 1.32

Summary of Results

Gold and Gas:Structural controls

and fluid flow

Golden Pig Mine, Western Australia

Pennsylvannia Gas wellsIn the Marcellus shales

The crust as a Percolation Network

Percolation clusters

P = 0.26P<PcD = 1.56

P = 0.37P~ PcD = 1.9

PercolationThreshold, Pc

Conclusions• The percolation threshold has not been reached for gas or gold

fluid flow networks in the crust• Dm, Dmr (gold) < Dm, Dmr (gas)• The less clustered pattern of the gas distribution reflects the

more pervasive source/trap geology of the gas deposits compared to the stronger structural control of the gold, which localises deposits

• Mass Radius-Scaling (Dmr) exponents are similar to Mass Dimensions (Dm) for Archean Gold deposits and gas wells in Pennsylvannia

• Percolation theory is an attractive hypothesis to explain distributions of these natural resources