Water propagation in two-dimensional petroleum reservoirs

In the present paper we investigate the problem of water propagation in 2 dimensional (2D) petroleum reservoir in which each site has the probability p of being occupied. We first analyze this propagation pattern described by Darcy equations by focusing on its geometrical features. We find that the domain-walls of this model at p=pc≃0.59 are Schramm-Loewner evolution (SLE) curves with κ=3.05∓0.1 consistent with the Ising universality class. We also numerically show that the fractal dimension of these domain-walls at p=pc is Df≃1.38 consistent with SLEκ=3. Along with this analysis, we introduce a self-organized critical (SOC) model in which the water movement is modeled by a chain of topplings taking place when the water saturation exceeds the critical value. We present strong indications that it coincides with the reservoir simulation described by Darcy equation. We further analyze the SOC model and show numerically that for this model the spanning cluster probability has a maximum around p=0.65.