Montecarlo DLA-type simulations of wetting effects in fluid displacement in porous media

In this work, we report diffusion-limited aggregation (DLA)-type Montecarlo computations of a stochastic model of displacement of a viscous fluid by another that preferentially wets a porous medium, for the case when both fluids are immiscible in the absence of buoyancy forces. The model has the aim to simulate cooperative invasion processes found in experiments of immiscible wetting displacement. The model considers the nonlocal effects of the Laplacian pressure field and the capillary forces via hydrodynamic equations in the Darcy regime with a boundary condition for the pressure at the interface. The boundary condition contains two different types of disorder: the capillary term, which constitutes an additive random disorder, and a term containing an effective random surface tension, which couples to a curvature (it constitutes a multiplicative random term that carries nonlocal information of the whole pressure). We generate different displacement patterns for different setting of the parameters of the model. We analyze these patterns by studying the scaling properties of the interface that separate the two fluids and calculating the fractal dimension of the interface. The results show the existence of three distinct regimes of scaling. One regime at the smallest-length scales is due to the multiplicative random disorder together with the nonlocal coupling; it reveals itself in a roughness exponent α ≈ 0.80. Additionally, we find a DLA-type scaling regime with a roughness exponent α ≈ 0.60 at the largest scales and intermediate scaling regime with α ≈ 0.70 corresponding to invasion percolation with trapping. Each regime has definite scaling ranges that depend on the capillary number and the relative wetting tendency of the fluids. The behavior of the fractal dimensions of the interfaces of the aggregates constitutes a further confirmation of the existence of three scaling regimes and the multi-self-affinity of the perimeter of the interface boundaries.