Water Propagation in the Porous Media, Self-Organized Criticality and Ising Model

TL;DR

This paper models water propagation in porous media using an Ising model and self-organized criticality, revealing critical behavior at specific probabilities and comparing geometric properties with reservoir simulations.

Contribution

It introduces a novel application of the Ising model and self-organized criticality to water movement in porous media, connecting geometric analysis with reservoir simulation results.

Findings

At p ≈ 0.59, the model aligns with the Ising universality class.

The self-organized critical model matches reservoir simulation behaviors.

Percolation probability peaks around p=0.68, contrary to common beliefs.

Abstract

In this paper we propose the Ising model to study the propagation of water in 2 dimensional (2D) petroleum reservoir in which each bond between its pores has the probability $p$ of being activated. We analyze the water movement pattern in porous media described by Darcy equations by focusing on its geometrical objects. Using Schramm-Loewner evolution (SLE) technique we numerically show that at $p=p_c\simeq 0.59$, this model lies within the Ising universality class with the diffusivity parameter $\kappa=3$ and the fractal dimension $D_f=\frac{11}{8}$. We introduce a self-organized critical model in which the water movement is modeled by a chain of topplings taking place when the amount of water exceeds the critical value and numerically show that it coincides with the numerical reservoir simulation. For this model, the behaviors of distribution functions of the geometrical quantities and the Green function are investigated in terms of $p$. We show that percolation probability has a maximum around $p=0.68$, in contrast to common belief.