A review of growing interfaces in quenched disordered media

TL;DR

This review compares two principal models of fluid imbibition in porous media within the directed percolation depinning universality class, highlighting differences in their growth mechanisms and the limitations of the QKPZ equation in describing these models.

Contribution

It clarifies the differences between DPD models and the QKPZ equation, providing analytical insights into their distinct growth mechanisms and limitations.

Findings

DPD models increase roughness near criticality due to lateral contributions.

QKPZ equation tends to flatten roughness, not capturing DPD behavior.

Analytical equations confirm the limitations of QKPZ in describing DPD models.

Abstract

We make a review of the two principal models that allows to explain the imbibition of fluid in porous media. These models, that belong to the directed percolation depinning (DPD) universality class, where introduced simultaneously by the Tang and Leschhorn [Phys. Rev A 45, R8309 (1992)] and Buldyrev et al. [Phys. Rev. A 45, R8313 (1992)] and reviewed by Braunstein et al. [J. Phys. A 32, 1801 (1999); Phys. Rev. E 59, 4243 (1999)]. Even these models have been classified in the same universality class than the Kardar-Parisi-Zhang equation [Phys. Rev. Lett. 56, 889, (1986)] with quenched noise (QKPZ), the contributions to the growing mechanisms are quite different. The lateral contribution in the DPD models, leads to an increasing of the roughness near the criticality while in the QKPZ equation this contribution always flattens the roughness. These results suggest that the QKPZ equation does not describe properly the DPD models even when the exponents derived from this equation are similar to the one obtained from the simulations of these models. This fact is confirmed trough the deduced analytical equation for the Tang and Leschhorn model. This equation has the same symmetries than the QKPZ one but its coefficients depend on the balance between the driving force and the quenched noise.