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A non-singular integral equation formulation of permeable semi-infinite hydraulic fractures driven by shear-thinning fluids

University of British Columbia

This thesis considers the problem of semi-infinite hydraulic fractures driven by shear-thinning power-law fluids through a permeable elastic medium. In the recent work by Dontsov and Peirce [Journal of Fluid Mechanics, 784:R1 (2015)], the authors reformulated the governing equations in a way that avoids singular integrals for the case of a Newtonian fluid. Moreover, the authors constructed an approximating ordinary differential equation (ODE) whose solutions accurately describe the fracture opening at little to no computational cost. The present thesis aims to extend their work to the more general case where the fracture is driven by a shear-thinning power-law fluid. In the first two chapters of this thesis we outline the relevant physical modelling and discuss the asymptotic propagation regimes typically encountered in hydraulic fracturing problems. This is followed by Chapters 4 and 5 where we reformulate the governing equations as a non-singular integral equation, and then proceed to construct an approximating ODE. In the final chapter we construct a numerical scheme for solving the non-singular integral equation. Solutions obtained in this way are then used to gauge the accuracy of solution obtained by solving the approximating ODE. The most important results of this thesis center on the accuracy of using the approximating ODE. In the final chapter we find that when the fluid's power-law index is in the range of 0.4 ≤ n ≤ 1, an appropriate method of solving the approximating ODE yields solutions whose relative errors are less than 1%. However, this relative error increases with decreasing values of n so that in the range 0 ≤ n < 0.4 it reaches a maximum value of approximately 6%. Thus, at least for values of 0.4 ≤ n ≤ 1 the approximating ODE presents an accurate and computationally fast alternative to solving the semi-infinite problem. The same can't be said for values of 0 ≤ n < 0.4, but the methods presented in this thesis may be used as a starting point for future work in this direction.

Text

2016-08-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58976

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/