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.
