Propagation of a linear hydraulic fracture with tortuosity

• Tortuosity is modelled by a symmetric fracture with modified flow and crack laws.
• Sufficiently strong tortuosity can remove the singularity at the fracture tip.
• For a very tortuous fracture the length depends weakly on the operating conditions.
• The width averaged velocity increases approximately linearly along the fracture.
• The approximate solution for a hydraulic fracture may be useful in practice.

The propagation of a pre-existing hydraulic fracture with tortuosity in the fluid flow is investigated. The tortuosity is caused by the roughness of the crack walls and by areas of contact between asperities (deformations) on opposite crack walls. The normal stress at the crack walls is distributed between the fluid pressure and the contact areas of touching asperities. The tortuous fracture is replaced by a symmetric open fracture without asperities but with a modified Reynolds flow law and modified stress in the fracture. For a partially open tortuous fracture the linear crack law is used in which the half-width is related to the effective pressure by a piecewise linear law. The Perkins–Kern–Nordgren approximation is made in which the normal stress at the crack walls is proportional to the half-width of the symmetric model fracture. A Lie point symmetry analysis is used to formulate a group invariant solution for the length, volume and half-width of the pre-existing fracture. Exact analytical solutions are derived for fractures with constant volume and constant speed of propagation and a numerical solution is developed for general operating conditions at the fracture entry. It is found that tortuosity can remove the singularity in the spatial gradient of the half-width at the fracture tip of the model fracture and that the length of a partially open hydraulic fracture becomes less dependent on the operating conditions at the fracture entry as the tortuosity increases. From the numerical solution it is found that the fluid velocity averaged over the width of the model fracture increases approximately linearly along the fracture and this observation is used to derive an approximate analytical solution for the length and half-width which agrees well with the numerical solution.