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.


Related in: MedlinePlus

Illustration of the model.(a) In the initial state (p = 0) the vertical faces of the cubic lattice are divided into three sets. (b) Example of the final state of the invasion (p = 1), dividing the medium into three parts: R, G, and B. RGB edges and nodes are shown as thick black lines and spheres, respectively. In (c) we separate the three pieces to be able to look inside. Solid lines represent the edges of the dual lattice of the pore network. The color of each cube corresponds to the one of the fluid in the pore at the center of the cube. The channels connecting the pores are perpendicular to the faces of the cubes and for clarity they are not shown. The RGB edges and nodes are part of the dual lattice.
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f2: Illustration of the model.(a) In the initial state (p = 0) the vertical faces of the cubic lattice are divided into three sets. (b) Example of the final state of the invasion (p = 1), dividing the medium into three parts: R, G, and B. RGB edges and nodes are shown as thick black lines and spheres, respectively. In (c) we separate the three pieces to be able to look inside. Solid lines represent the edges of the dual lattice of the pore network. The color of each cube corresponds to the one of the fluid in the pore at the center of the cube. The channels connecting the pores are perpendicular to the faces of the cubes and for clarity they are not shown. The RGB edges and nodes are part of the dual lattice.

Mentions: To find the partitioning of the medium into three parts, we divide the (four) vertical boundaries of a cubic system in three parts of about the same area. Each part corresponds to a different invading fluid distinguished by dyeing them with different colors: red (R), green (G), and blue (B) [see Fig. 2(a)]. In the illustration of Fig. 2 we see a medium of 5 × 5 × 5 pores. The pores are in the center of each cube and the edges are the bonds of the dual lattice of the pore network. The cubes have the color of the fluid contained in the corresponding pore. We invade the system simultaneously from all vertical walls.


How to share underground reservoirs.

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

Illustration of the model.(a) In the initial state (p = 0) the vertical faces of the cubic lattice are divided into three sets. (b) Example of the final state of the invasion (p = 1), dividing the medium into three parts: R, G, and B. RGB edges and nodes are shown as thick black lines and spheres, respectively. In (c) we separate the three pieces to be able to look inside. Solid lines represent the edges of the dual lattice of the pore network. The color of each cube corresponds to the one of the fluid in the pore at the center of the cube. The channels connecting the pores are perpendicular to the faces of the cubes and for clarity they are not shown. The RGB edges and nodes are part of the dual lattice.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f2: Illustration of the model.(a) In the initial state (p = 0) the vertical faces of the cubic lattice are divided into three sets. (b) Example of the final state of the invasion (p = 1), dividing the medium into three parts: R, G, and B. RGB edges and nodes are shown as thick black lines and spheres, respectively. In (c) we separate the three pieces to be able to look inside. Solid lines represent the edges of the dual lattice of the pore network. The color of each cube corresponds to the one of the fluid in the pore at the center of the cube. The channels connecting the pores are perpendicular to the faces of the cubes and for clarity they are not shown. The RGB edges and nodes are part of the dual lattice.
Mentions: To find the partitioning of the medium into three parts, we divide the (four) vertical boundaries of a cubic system in three parts of about the same area. Each part corresponds to a different invading fluid distinguished by dyeing them with different colors: red (R), green (G), and blue (B) [see Fig. 2(a)]. In the illustration of Fig. 2 we see a medium of 5 × 5 × 5 pores. The pores are in the center of each cube and the edges are the bonds of the dual lattice of the pore network. The cubes have the color of the fluid contained in the corresponding pore. We invade the system simultaneously from all vertical walls.

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.


Related in: MedlinePlus