Limits...
How to share underground reservoirs.

Schrenk KJ, Araújo NA, Herrmann HJ - Sci Rep (2012)

Bottom Line: We identify and characterize these lines, showing that they form a fractal set consisting of a single thread spanning the medium and a surrounding cloud of loops.While the spanning thread has fractal dimension 1.55 ± 0.03, the set of all lines has dimension 1.69 ± 0.02.The size distribution of the loops follows a power law and the evolution of the set of lines exhibits a tricritical point described by a crossover with a negative dimension at criticality.

View Article: PubMed Central - PubMed

Affiliation: Computational Physics for Engineering Materials, IfB , ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland. jschrenk@ethz.ch

ABSTRACT
Many resources, such as oil, gas, or water, are extracted from porous soils and their exploration is often shared among different companies or nations. We show that the effective shares can be obtained by invading the porous medium simultaneously with various fluids. Partitioning a volume in two parts requires one division surface while the simultaneous boundary between three parts consists of lines. We identify and characterize these lines, showing that they form a fractal set consisting of a single thread spanning the medium and a surrounding cloud of loops. While the spanning thread has fractal dimension 1.55 ± 0.03, the set of all lines has dimension 1.69 ± 0.02. The size distribution of the loops follows a power law and the evolution of the set of lines exhibits a tricritical point described by a crossover with a negative dimension at criticality.

No MeSH data available.


Mass scaling analysis.The total mass Mtot (, total number of RGB nodes) and the mass of the spanning cluster Msc (, number of nodes in largest RGB cluster) are shown as function of the lattice length L. One observes that for large lattices the masses scale as power laws of the lattice size, i.e.,  and . The fractal dimensions are dtot = 1.69 ± 0.02 and dsc = 1.55 ± 0.03. In the inset we see the area of the division surfaces Msur () as function of the lattice size L. The fractal dimension of the division surfaces is dsur = 2.49 ± 0.02. Straight lines are guides to the eye.
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f3: Mass scaling analysis.The total mass Mtot (, total number of RGB nodes) and the mass of the spanning cluster Msc (, number of nodes in largest RGB cluster) are shown as function of the lattice length L. One observes that for large lattices the masses scale as power laws of the lattice size, i.e., and . The fractal dimensions are dtot = 1.69 ± 0.02 and dsc = 1.55 ± 0.03. In the inset we see the area of the division surfaces Msur () as function of the lattice size L. The fractal dimension of the division surfaces is dsur = 2.49 ± 0.02. Straight lines are guides to the eye.

Mentions: The surfaces dividing two colors are fractal objects of fractal dimension dsur = 2.49 ± 0.02, as seen in the inset of Fig. 3, numerically consistent with what has been reported for watersheds in three dimensions56. While these boundaries are singly connected, the RGB set consists of one spanning cluster connecting the two sides of the system surrounded by a cloud of smaller disconnected loops (see Fig. 1). As shown in Fig. 3, the entire RGB set is fractal of dimension dtot = 1.69±0.02, while the spanning cluster has a smaller fractal dimension dsc = 1.55 ± 0.03. To analyze the topology of the spanning cluster, we used the burning method proposed in Ref. [7]. We found that the spanning cluster has loops, however its backbone, elastic backbone, shortest path, and its set of singly connected RGB edges all have fractal dimensions consistent with dsc.


How to share underground reservoirs.

Schrenk KJ, Araújo NA, Herrmann HJ - Sci Rep (2012)

Mass scaling analysis.The total mass Mtot (, total number of RGB nodes) and the mass of the spanning cluster Msc (, number of nodes in largest RGB cluster) are shown as function of the lattice length L. One observes that for large lattices the masses scale as power laws of the lattice size, i.e.,  and . The fractal dimensions are dtot = 1.69 ± 0.02 and dsc = 1.55 ± 0.03. In the inset we see the area of the division surfaces Msur () as function of the lattice size L. The fractal dimension of the division surfaces is dsur = 2.49 ± 0.02. Straight lines are guides to the eye.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3475994&req=5

f3: Mass scaling analysis.The total mass Mtot (, total number of RGB nodes) and the mass of the spanning cluster Msc (, number of nodes in largest RGB cluster) are shown as function of the lattice length L. One observes that for large lattices the masses scale as power laws of the lattice size, i.e., and . The fractal dimensions are dtot = 1.69 ± 0.02 and dsc = 1.55 ± 0.03. In the inset we see the area of the division surfaces Msur () as function of the lattice size L. The fractal dimension of the division surfaces is dsur = 2.49 ± 0.02. Straight lines are guides to the eye.
Mentions: The surfaces dividing two colors are fractal objects of fractal dimension dsur = 2.49 ± 0.02, as seen in the inset of Fig. 3, numerically consistent with what has been reported for watersheds in three dimensions56. While these boundaries are singly connected, the RGB set consists of one spanning cluster connecting the two sides of the system surrounded by a cloud of smaller disconnected loops (see Fig. 1). As shown in Fig. 3, the entire RGB set is fractal of dimension dtot = 1.69±0.02, while the spanning cluster has a smaller fractal dimension dsc = 1.55 ± 0.03. To analyze the topology of the spanning cluster, we used the burning method proposed in Ref. [7]. We found that the spanning cluster has loops, however its backbone, elastic backbone, shortest path, and its set of singly connected RGB edges all have fractal dimensions consistent with dsc.

Bottom Line: We identify and characterize these lines, showing that they form a fractal set consisting of a single thread spanning the medium and a surrounding cloud of loops.While the spanning thread has fractal dimension 1.55 ± 0.03, the set of all lines has dimension 1.69 ± 0.02.The size distribution of the loops follows a power law and the evolution of the set of lines exhibits a tricritical point described by a crossover with a negative dimension at criticality.

View Article: PubMed Central - PubMed

Affiliation: Computational Physics for Engineering Materials, IfB , ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland. jschrenk@ethz.ch

ABSTRACT
Many resources, such as oil, gas, or water, are extracted from porous soils and their exploration is often shared among different companies or nations. We show that the effective shares can be obtained by invading the porous medium simultaneously with various fluids. Partitioning a volume in two parts requires one division surface while the simultaneous boundary between three parts consists of lines. We identify and characterize these lines, showing that they form a fractal set consisting of a single thread spanning the medium and a surrounding cloud of loops. While the spanning thread has fractal dimension 1.55 ± 0.03, the set of all lines has dimension 1.69 ± 0.02. The size distribution of the loops follows a power law and the evolution of the set of lines exhibits a tricritical point described by a crossover with a negative dimension at criticality.

No MeSH data available.