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.


Set of lines on which all division surfaces between three parts are in contact for a typical random medium.In addition to the backbone spanning the medium from left to right (shown in red), the set also contains a cloud of disconnected loops (green). The transparent planes are guides to the eye.
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f1: Set of lines on which all division surfaces between three parts are in contact for a typical random medium.In addition to the backbone spanning the medium from left to right (shown in red), the set also contains a cloud of disconnected loops (green). The transparent planes are guides to the eye.

Mentions: Both problems consist in dividing space into parts: either the shares of the companies in the oil field or the different tissues in the image processing. In both cases, regions are separated by division surfaces. Here we consider three regions and find that their division surfaces join in a fractal thread that crosses the medium, being surrounded by a cloud of disconnected loops (see Fig. 1). In the case of oil exploration these points, where all three division surfaces merge, are the places where water should be injected to assure that no oil is pushed out on the wrong side. In medical image processing, the simultaneous boundary between three parts might indicate, for example, the region where a tumor is nested between two other tissues.


How to share underground reservoirs.

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

Set of lines on which all division surfaces between three parts are in contact for a typical random medium.In addition to the backbone spanning the medium from left to right (shown in red), the set also contains a cloud of disconnected loops (green). The transparent planes are guides to the eye.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f1: Set of lines on which all division surfaces between three parts are in contact for a typical random medium.In addition to the backbone spanning the medium from left to right (shown in red), the set also contains a cloud of disconnected loops (green). The transparent planes are guides to the eye.
Mentions: Both problems consist in dividing space into parts: either the shares of the companies in the oil field or the different tissues in the image processing. In both cases, regions are separated by division surfaces. Here we consider three regions and find that their division surfaces join in a fractal thread that crosses the medium, being surrounded by a cloud of disconnected loops (see Fig. 1). In the case of oil exploration these points, where all three division surfaces merge, are the places where water should be injected to assure that no oil is pushed out on the wrong side. In medical image processing, the simultaneous boundary between three parts might indicate, for example, the region where a tumor is nested between two other tissues.

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.