The propagation of a linear hydraulic fracture with tortuosity and fluid leak-off at the fluid-rock interface

Abstract

In this work, the propagation of a pre-existing hydraulic fracture in permeable rock is investigated. Apartiallyopentortuousfracturewithleak-offisreplacedbyatwo-dimensional symmetric model fracture with a modified Reynolds’ flow law to account for the effect of asperities on the fluid flow. The model is closed by considering a linear crack law, which considers the presence of touching asperities, and a Perkins-Kern and Nordgren (PKN) approximation, which relates the half-width of the model fracture to the normal stress at the fracture walls. The result is a nonlinear diffusion equation that accounts for leak-off atthefluid-rockinterfaceasaresultoftherock’spermeability. The leak-off velocity is not specified a priori and its functional form is determined by calculating Lie point symmetries of the governing non-linear diffusion equation for a model fracture which leads to a group invariant solution of the half-width of the fracture and the leak-off velocity respectively. We consider different forms of the Lie point symmetry of the governing equations by taking some of the constants in the generator to be zero. This leads to three cases of the group invariant solution, namely the general case, thetravelingwavesolutionandtheexponentialsolution. TwoexactanalyticalsolutionsareobtainedasaresultofassociationofaLiepointsymmetry with a conserved vector. However, the constant volume working condition is not furtherinvestigatedastherequirementsforittoholdarenotsufficientforthemodelconsidered. Other operating conditions at the fracture entry are also obtained by analysing the different properties of the partially open fracture. Numerical solutions of the halfwidthandtheleak-offdeptharecomputed. Thepropagationofalinearhydraulicfracture withtortuosityforthreecasesoftheLiepointsymmetryarefullyanalysedbynumerically solvingforthehalf-widthandthelengthofthefracture. Thederivationandanalysisofthewidthaveragedfluidvelocityleadstothederivation of an approximate analytical solution for the half-width of the model fracture which will then be compared to the numerical solution obtained. The approximate solution could prove useful when analytical solutions are unattainable or when numerical solutions are difficult to compute. The effects of leak-off and no leak-off in tortuous hydraulic fracture are compared to gain insight on the effect porosity has on the characteristics of the model fracture with tortuosity