How surface tension in two-phase flow leads to anisotropic effective stress, and affects soil erodability and slope stability

Effective stress in partly saturated porous media is a crucial question to understand the mechanical stability and the erodability of soils. In general, two-phase flow in unconsolidated granular media is a common process. It takes place during rain infiltration in soils, in sandcastles, and numerous situations in the critical zone.
The mechanical stability of slopes and materials is expressed by considering stability envelope of the stress tensor supported by the solid material.  In one phase flow, this leads to criterias on Terzaghi stress, or effective stress when the contacts between solid elements are not reduced to points.
In two-phase flow, the stress carried by solids is usually expressed using an effective average fluid pressure in the effective stress formulation, following Bishop. We show that this approach does not take the explicit stress carried by the two-dimensional interface between the two fluids into account: the explicit effect of surface tension is missing. This term is called Bachelor stress in the framework of foam mechanics, but is usually not incorporated in two-phase flow in porous media formulation
We evaluate the importance of this effect from a micromechanical perspective, and show how to incorporate it in a generalized large scale effective stress formulation. We show how this formulation can take into account an anisotropic tensor reflecting the stress carried by the fluids and the fluid/fluid interfaces, depending on the anisotropy of the fabrics of these interfaces.
We bridge the gap between microscopic interactions and macroscopic behavior, offering a robust model for evaluating and predicting forces in multi-phase systems. Numerical simulations comparing the standard model with the new framework demonstrate that incorporating surface tension significantly refines slope stability predictions, especially during intense rainfall events.