Geometrical clusters of Darcy’s reservoir model and Ising universality class

Abstract

In this paper the geometrical features of the fluid propagation in two-dimensional petroleum reservoir described by Darcy equations are studied. The porous media are considered to be tuned by the occupancy parameter p being the probability that a pore is occupied. We analyze the statistical geometrical observables of the Darcy model. To this end we let the water to be injected into random sites of the porous media and solve numerically the Darcy equations to describe the flow motion pattern, using the control volume finite difference (CVFD) method. The fractal dimension of the frontier of the avalanches (defined as the set of the sites through which the fluid passed) and the distribution functions of gyration radius, loop length and cluster mass are numerically obtained revealing that at p=pc (the critical occupancy parameter above which there is definitely a spanning cluster in the system) this model lies within a universality class compatible with the Ising model. We observe that for p>pc, although the model shows critical behaviors, this duality is broken. The mentioned exponents are reported in this paper.