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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
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• Scaling Relationships for Natural Resources• Data sets: Gold, Gas• Results• Percolation Theory
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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
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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
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Data Sets
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Gold deposits, Zimbabwe craton
Dm = 1.05Dmr = 1.02
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Roll-off:Sampling from a fractaldistribution
Clustering near edges of study area
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Conventional Gas Wells, Pennsylvannia
Dm = 1.63Dmr = 1.72
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Unconventional Gas Wells, Pennsylvannia
Dm = 1.26Dmr = 1.32
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Summary of Results
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Gold and Gas:Structural controls
and fluid flow
Golden Pig Mine, Western Australia
Pennsylvannia Gas wellsIn the Marcellus shales
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The crust as a Percolation Network
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Percolation clusters
P = 0.26P<PcD = 1.56
P = 0.37P~ PcD = 1.9
PercolationThreshold, Pc
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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