Re-injection of drill cuttings is considered in several areas as a mean of disposing solid wastes. Injection in sands or in shales has been considered. In sands, fracture mechanics as used in hydraulic fracturing should apply. A new problem arises in shales where the stress contrast no more contains the fracture and a vertical propagation is expected. In this paper, we derive analytical solutions of the vertical propagation problem in order to better analyze the stability/unstability of fracture propagation. The shape of the fracture is assumed to be elliptical in the propagation plane and the ellipticity is obtained as a solution of the mechanical problem. The case of a circular fracture is firstly considered in order to analyze and illustrate the implication of the evolution of the minor horizontal stress as a function of depth. Considerations on the influence of the spatial discretization in numerical modelling are derived. The elliptical fracture is then dealt with. Finally, the case of a variable Young modulus is considered and the analytical solution is compared with numerical results obtained in the literature with the Terrafrac model. The analytical solution and the numerical one are similar for the case considered by BP.

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