This work explains how capillary forces affect the strength and deformation of rock that is partially-saturated with two immiscible fluids. Using the fracture mechanics approach applied to a partially-saturated crack, the empirical Bishop’s effective stress , capillary pressure and the effective stress coefficients are explained and calculated theoretically. The calculations provide the closed-form analytical solutions for effective stress coefficients and capillary pressure curves at three different regimes: failure, deformation during imbibitions and deformation during drainage (drying). In particular, it is explained how capillary pressure curves and effective stress coefficients are dependent on saturation degree, contact angle, surface tension between fluids, elastic parameters and fracture toughness of the rock and far-field stresses.
Fluids can affect the strength and deformation of porous rocks by different physical-chemical processes. Previous investigations have shown that the strength and compressibility of the rock can be significantly changed by dissolution and precipitation of minerals , and stimulated by chemically active fluids like oxidants or deoxidants. The strength and deformation of the rock can also be significantly altered by the osmotic diffusion , which is driven by the difference in salinity between water and clay-containing rocks, where clays act as osmotic membranes and thus give rise to osmotically induced hydrostatic pressure, causing clay swelling effects . The capillary forces have been experimentally shown to cause a major change in the strength in various rocks (such as shale, chalk, limestone, sandstone, gypsum, mudrock, etc.) [e.g., 5- 13]. Capillary effects occur whenever the pore space is partially-saturated with at least two immiscible fluids, like gas and water, or oil and water, for instance. The surface tension between fluids and wettability of the rock gives rise to the capillary forces, which are the focal point of investigation of this paper. From the theoretical point of view, it is clear that capillary phenomena and osmotic diffusion may have a coupled effect on the strength and deformation of the rock, because the osmotic diffusion is changing the degree of water saturation of the rock, which could be pre-saturated with oil or gas. Therefore the osmotic diffusion does change capillary pressure as well. However for simplicity reasons, coupling effects are not considered in this paper.This work presents a closed-form analytical solution for the effective stress coefficients and capillary pressure curves (at three different regimes: failure, deformation during imbibitions and deformation during drying) of rock that is partially-saturated with two immiscible fluids. The calculations provide a theoretical explanation of Bishop’s effective stress using a fracture-mechanics approach. The main difference between the crack-based approach used in this paper and grain-based approaches as used in unsaturated soil mechanics is that the crack-based approach is clearly more sensible for more compacted or denser rocks, where the pore space is not predominantly the spaces between grains. Using a crack-based approach makes it possible to predict strength, capillary pressure curves and effective stress coefficients for the whole range of water saturation, in contrast to grain-based model, where the maximum saturation limit is about 25%.