Linear hydraulic fracture with tortuosity: Conservation laws and fluid extraction M. R. R. Kgatle and D. P. Mason School of computational and applied mathematics.

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Presentation transcript:

Linear hydraulic fracture with tortuosity: Conservation laws and fluid extraction M. R. R. Kgatle and D. P. Mason School of computational and applied mathematics University of the Witwatersrand Energy Postgraduate Conference 2013 Introduction Modelling process Conservation laws Fluid extraction Conclusion

Introduction  Hydraulic fracturing Natural fractures and man-made fractures Introduction Modelling process Conservation laws Fluid extraction Conclusion

Modeling process  Problem description Propagation of a pre-existing hydraulic fracture with length x=L(t) and width z=H at the fracture entry Introduction Modelling process Conservation laws Fluid extraction Conclusion ooo

 Modelling formulation

Introduction Modelling process Conservation laws Fluid extraction Conclusion ooo Constant quantityα 1/(n+2) 0.5 (n+1)/(n+2) 1

Conservation laws Introduction Modelling process Conservation laws Fluid extraction Conclusion ooo

Introduction Modelling process Conservation laws Fluid extraction Conclusion ooo  Interpretation of working conditions obtained from conservation laws Obtained from conservation laws Working conditions curves

Fluid extraction Introduction Modelling process Conservation laws Fluid extraction Conclusion ooo

Fluid extraction Introduction Modelling process Conservation laws Fluid extraction Conclusion ooo

Introduction Modelling process Conservation laws Fluid extraction Conclusion ooo

Introduction Modelling process Conservation laws Fluid extraction Conclusion ooo

Conclusion  Conservation laws suggest parameters that lead to analytical solutions.  A new solution is obtained.  Fluid always flows out of the closed fracture at the fracture entry.  No pockets of fluid left in the fracture.  Analysis of flux shows that there’s fluid flow in the forward and backward directions during fluid extraction.  Maximum rate for fluid extraction is the limiting working condition. Introduction Modelling process Conservation laws Fluid extraction Conclusion

References 1.A. D. Fitt, A. D. Kelly and C. P. Please. Crack propagation models for rock fracture in a geothermal energy reservoir. SIAM J Appl Math, 55: , A. P. Oron and B. Berkowitz. Flow in rock fractures: The local cubic law assumption re-examined. Water Resources Research, 34: , A. G. Fareo and D. P. Mason. Group invariant solutions for a pre-existing fluid-driven fracture in permeable rock. Nonlinear Analysis: Real World Applications, 12: , Acknowledgements Gratitude to NRF, SANHARP, for funding and to the School of Computational and Applied Mathematics, Wits University, for academic support.