Measuring the effective stress parameter using the multiphase lattice Boltzmann method and investigating the source of its hysteresis

The effective stress parameter, $\chi$, is essential for calculating the effective stress in unsaturated soils. Experimental measurements have captured different relationships between $\chi$ and the degree of saturation, $S_r$; however, they have not been able to justify the particular shapes of the $\chi$-$S_r$ curves. Theoretical solutions express $\chi$ as a function of $S_r$ and the air-water interfacial area, $a_{wn}$; however, $a_{wn}$ is difficult to predict, limiting further investigation of $\chi$ variation. We seek an alternative approach for studying $\chi$ by simulating the pore-scale distribution of the two fluid phases in unsaturated soils using the multiphase lattice Boltzmann method (LBM). We develop an algorithm for measuring $\chi$ based on the suction and surface tension forces applied to each grain. Using this algorithm, we simulate the $\chi$-$S_r$ curve over a full hydraulic cycle for a synthetic 3D granular soil column with immobile grains. We find that $\chi=1$ at $S_r=1$ and $\chi=0$ at $S_r=0$, while $\chi>S_r$ for all other saturations. The maximum divergence of $\chi$ from $S_r$ happens at the transition from/to the pendular regime. We also observe that the $\chi$-$S_r$ curve is hysteretic; $\chi$ is larger during wetting (imbibition) compared to drying (drainage) due to larger contribution of surface tension forces.