A comparison of a fractional derivative model with an empirical model for non-linear shock waves in swelling shales

The control of drilling parameters, such as fluid pressure, mud weight, salt concentration, etc., is essential to avoid instabilities when drilling through shale sections. To investigate shale deformation, which is fundamental for deep oil drilling and hydraulic fracturing for gas extraction (“fracking”), a non-linear model of mechanical and chemo-poroelastic interactions among fluids, solutes and the solid matrix is discussed here. The two equations of this model describe the isothermal evolution of fluid pressure and solute density in a fluid-saturated porous rock. If the non-linear term in these equations is larger than the diffusive term the solutions are quick, non-linear Burgers solitary waves, which are potentially destructive for deep operations. In our study the role of diffusion in presence of these solitary waves has been also analyzed. Then, following Civan (1998), both diffusive and shock waves are applied to fine particle filtration and their effects on the adjacent rocks. Finally, the resulting time-delayed evolution is discussed. Because time delays in simple porous media dynamics have recently been analysed using a fractional derivative approach, we insert fractional time derivatives, i.e., a type of time-average of the fluid-rock interactions, to make a tentative comparison of these two deeply different methods in our model. The delaying effects of fine particle filtration à la Civan (1998) are then compared with the time delay of the fractional derivative model; thus, the fractional derivative order for this filtration is realistically estimated. Such a comparison can be seen as a heuristic determination of “natural” time averages for fine particle related phenomena in this model.