Osmotic Ion Concentration Control of Steady-State Subcritical Fracture Growth in Shale

Abstract

ABSTRACT: The mechanism of formation of natural cracks in sedimentary rocks in the geologic past is an important problem in hydraulic fracturing. Why are the natural cracks roughly parallel and equidistant, and why is the spacing in the order of 10 cm rather than 1 cm or 100 cm? Fracture mechanics alone cannot answer these questions. Here it is proposed that fracture mechanics must be coupled with the diffusion of solute ions (Na+ and Cl− are considered here), driven by an osmotic pressure gradient. Parallel equidistant cracks are considered to be subcritical and governed by the Charles-Evans law. The evolution in solute concentration also affects the solvent pressure in the pores and cracks, altering the resistance to frictional sliding. Only steady-state propagation and periodic cracks are studied. An analytical solution of the crack spacing as a function of the properties of the rock as well as the solvent and solute, and the imposed far-field deformation is obtained. Finally, the stability of the growth of parallel cracks is proven by examining the second variation of free energy. Stability of the periodic growth state is also considered. 1 INTRODUCTION The deep layers of sedimentary rocks such as shale and sandstone are usually intersected by systems of nearly parallel natural cracks either filled by mineral deposits or closed by creep over a million year life span. Their spacing is roughly uniform and is on the order of 0.1 m (rather than 1 m or 0.01m). These cracks likely play an important role in hydraulic fracturing for gas or oil recovery (aka fracking, fraccing or frac) (Rahimi-Aghdam et al., 2019, e.g.). Therefore, understanding the mechanism of their formation in the distant geologic past is of interest. What controls the spacing of the nearly parallel cracks in shale? According to the fracture mechanics alone, the crack spacing is arbitrary. If propagating parallel equidistant cracks are in a critical state, stability analysis shows that many cracks would have to stop growing, causing a great increase of their average spacing, which was obviously not the case (Bažant et al., 2014).