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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.


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Scale-free behavior.(a) Number of loop sizes s divided by  as function of s for different lattice sizes L (256 (), 362 (), and 512 ()). For intermediate sizes the data follows a power law, p(s) ~ s−a with a = 2.04 ± 0.04. (b) Size dependence of the total mass Mtot at and above the threshold pressure pt. At pt () the total mass decreases according to a power law, Mtot ~ L−t, where t = 0.69 ± 0.08. In contrast, above the threshold () the slope is given by the fractal dimension dtot. (c) Rescaled total mass MtotLt as a function of the scaling variable (p − pt)Lθ for different lattice sizes L. Close to the threshold , where ζT = 2.0 ± 0.3. The solid lines are guides to the eye.
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f5: Scale-free behavior.(a) Number of loop sizes s divided by as function of s for different lattice sizes L (256 (), 362 (), and 512 ()). For intermediate sizes the data follows a power law, p(s) ~ s−a with a = 2.04 ± 0.04. (b) Size dependence of the total mass Mtot at and above the threshold pressure pt. At pt () the total mass decreases according to a power law, Mtot ~ L−t, where t = 0.69 ± 0.08. In contrast, above the threshold () the slope is given by the fractal dimension dtot. (c) Rescaled total mass MtotLt as a function of the scaling variable (p − pt)Lθ for different lattice sizes L. Close to the threshold , where ζT = 2.0 ± 0.3. The solid lines are guides to the eye.

Mentions: The difference between dtot and dsc is due to the cloud of disconnected loops. These loops result from the entanglement of three compact regions, as illustrated in Fig. 4, which shows a transversal cross section of a medium, where the three regions are simultaneously in contact at different locations. In this particular case, the lower location (dashed circle) is where the spanning cluster intersects the cross section. The upper location (dotted circle) shows the cut through a disconnected loop: The G and B regions are in contact in an area completely surrounded by the R region, thus the contact line between the three forms a closed loop (discretization effects are discussed in the Supplementary Information). The size distribution of the loops is shown in Fig. 5(a), where the size s is defined as the number of RGB nodes forming the loop. A power-law distribution is observed, p(s) ~ s–a, with a = 2.04 ± 0.04, revealing the absence of a characteristic size. The distribution of distances of disconnected loops from the spanning cluster decays exponentially, i.e., the loop cloud is mainly localized in the neighborhood of the spanning cluster (see Supplementary Information for data).


How to share underground reservoirs.

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

Scale-free behavior.(a) Number of loop sizes s divided by  as function of s for different lattice sizes L (256 (), 362 (), and 512 ()). For intermediate sizes the data follows a power law, p(s) ~ s−a with a = 2.04 ± 0.04. (b) Size dependence of the total mass Mtot at and above the threshold pressure pt. At pt () the total mass decreases according to a power law, Mtot ~ L−t, where t = 0.69 ± 0.08. In contrast, above the threshold () the slope is given by the fractal dimension dtot. (c) Rescaled total mass MtotLt as a function of the scaling variable (p − pt)Lθ for different lattice sizes L. Close to the threshold , where ζT = 2.0 ± 0.3. The solid lines are guides to the eye.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f5: Scale-free behavior.(a) Number of loop sizes s divided by as function of s for different lattice sizes L (256 (), 362 (), and 512 ()). For intermediate sizes the data follows a power law, p(s) ~ s−a with a = 2.04 ± 0.04. (b) Size dependence of the total mass Mtot at and above the threshold pressure pt. At pt () the total mass decreases according to a power law, Mtot ~ L−t, where t = 0.69 ± 0.08. In contrast, above the threshold () the slope is given by the fractal dimension dtot. (c) Rescaled total mass MtotLt as a function of the scaling variable (p − pt)Lθ for different lattice sizes L. Close to the threshold , where ζT = 2.0 ± 0.3. The solid lines are guides to the eye.
Mentions: The difference between dtot and dsc is due to the cloud of disconnected loops. These loops result from the entanglement of three compact regions, as illustrated in Fig. 4, which shows a transversal cross section of a medium, where the three regions are simultaneously in contact at different locations. In this particular case, the lower location (dashed circle) is where the spanning cluster intersects the cross section. The upper location (dotted circle) shows the cut through a disconnected loop: The G and B regions are in contact in an area completely surrounded by the R region, thus the contact line between the three forms a closed loop (discretization effects are discussed in the Supplementary Information). The size distribution of the loops is shown in Fig. 5(a), where the size s is defined as the number of RGB nodes forming the loop. A power-law distribution is observed, p(s) ~ s–a, with a = 2.04 ± 0.04, revealing the absence of a characteristic size. The distribution of distances of disconnected loops from the spanning cluster decays exponentially, i.e., the loop cloud is mainly localized in the neighborhood of the spanning cluster (see Supplementary Information for data).

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