Abstract
Non-Newtonian fluids having Bingham or power-law rheology are common in many applications within drilling and reservoir engineering. Examples of such fluids are drilling muds, foams, heavy oil, hydraulic-fracturing and other stimulation fluids, and cement slurries. Despite the importance of non-Newtonian rheology, it is rarely used in reservoir simulators and fracture flow simulations. We study two types of non-Newtonian rheology: the truncated power-law (Ostwald-de Waele) fluid and the Bingham fluid. For either of the two types of non-Newtonian rheology, we construct relationships between the superficial fluid velocity and the pressure gradient in fractures and porous media. The Bingham fluid is regularized by means of Papanastasiou-type regularization for porous media and by means of a simple hyperbolic function for fracture flow. Approximation by Taylor expansion is used to evaluate the fluid velocity for small pressure gradients to reduce rounding errors. We report simulations of flow in rough-walled fractures for different rheologies and study the effect of fluid parameters on the flow channelization in rough-walled fractures. This effect is known from previous studies. We demonstrate how rheologies on different domains can be included in a fully-unstructured reservoir simulation that incorporates discrete fracture modeling (DFM). The above formulation was implemented in the open-source MATLAB Reservoir Simulation Toolbox (MRST), which uses fully implicit discretization on general polyhedral grids, including industry standard grids with DFM. This robust implementation is an important step towards hydro-mechanically coupled simulation of hydraulic fracturing with realistic non-Newtonian fluid rheology since most hydraulic fracturing models implemented so far make use of oversimplified rheological models (e.g., Newtonian or pure power-law).
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References
Alassi, H.T.: Modeling Reservoir Geomechanics Using Discrete Element Method: Application to Reservoir Monitoring, Ph.D. thesis, Norwegian University of Science and Technology (NTNU (2008)
Auradou, H., Boschan, A., Chertcoff, R., Angelo, M.V., Hulin, J.P., Ippolito, I.: Miscible transfer of solute in different model fractures: From random to multiscale wall roughness. Compt. Rendus Geosci. 342(7), 644–652 (2010)
Auradou, H., Boschan, A., Chertcoff, R., Gabbanelli, S., Hulin, J., Ippolito, I.: Enhancement of velocity contrasts by shear-thinning solutions flowing in a rough fracture. J. Non-Newtonian Fluid Mech. 153(1), 53–61 (2008)
Balhoff, M., Sanchez-Rivera, D., Kwok, A., Mehmani, Y., Prodanovic, M.: Numerical algorithms for network modeling of yield stress and other non-newtonian fluids in porous media. Transp. Porous Media 93(3), 363–379 (2012)
Bird, R., Armstrong, R., Hassager, O.: Dynamics of Polymer Liquids. Wiley, New York (1987)
Brown, S.R.: Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. Solid Earth 92 (B2), 1337–1347 (1987)
Carreau, P.J.: Rheological equations from molecular network theories. Trans. Soc. Rheol. (1957-1977) 16(1), 99–127 (1972)
Darby, R.: Chemical Engineering Fluid Mechanics, 2nd Edition. CRC Press, New York (2001)
Duda, A., Koza, Z., Matyka, M.: Hydraulic tortuosity in arbitrary porous media flow. Phys. Rev. E 84 (3), 036,319 (2011)
Frigaard, I., Nouar, C.: On the usage of viscosity regularisation methods for visco-plastic fluid flow computation. J. Non-Newtonian Fluid Mech. 127(1), 1–26 (2005)
Hanssen, A.R.: Numerical Modelling of Bingham Fluid Flow and Particle Transport in a Rough-Walled Fracture,Master’s thesis, Norwegian University of Science and Technology (NTNU) (2013)
Irgens, F.: Continuum mechanics Springer Berlin Heidelberg (2008)
Karimi-Fard, M., Durlofsky, L., Aziz, K., et. al: An efficient discrete-fracture model applicable for general-purpose reservoir simulators. SPE J. 9(02), 227–236 (2004)
Koponen, A., Kataja, M., Timonen, J.: Tortuous flow in porous media. Phys. Rev. E. 54(1), 406 (1996)
Lavrov, A.: Non-newtonian fluid flow in rough-walled fractures: A brief review. Rock Characterisation. Modelling and Engineering Design Methods 363 (2013)
Lavrov, A.: Numerical modeling of steady-state flow of a non-newtonian power-law fluid in a rough-walled fracture. Comput. Geotech. 50, 101–109 (2013)
Lavrov, A.: Redirection and channelization of power-law fluid flow in a rough-walled fracture. Chem. Eng. Sci. 99, 81–88 (2013)
Lavrov, A.: Radial flow of non-newtonian power-law fluid in a rough-walled fracture: effect of fluid rheology. Transp. Porous Media 105(3), 559–570 (2014)
Lavrov, A.: Flow of truncated power-law fluid between parallel walls for hydraulic fracturing applications. J. Non-Newtonian Fluid Mech. 223, 141–146 (2015)
Lavrov, A.: Lost circulation: Mechanisms and solutions. Elsevier (2016)
Lavrov, A., Larsen, I., Bauer, A.: Numerical modelling of extended leak-off test with a pre-existing fracture. Rock Mech. Rock. Eng. 49(4), 1359–1368 (2016)
Lavrov, A., Larsen, I., Holt, R., Bauer, A., Pradhan, S., et al.: Hybrid FEM/DEM Simulation of Hydraulic Fracturing in Naturally-Fractured Reservoirs 48Th US Rock Mechanics/Geomechanics Symposium. American Rock Mechanics Association (2014)
Lie, K.A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open-source matlab implementation of consistent discretisations on complex grids. Comput. Geosci. 16(2), 297–322 (2012)
Lipscomb, G., Denn, M.: Flow of bingham fluids in complex geometries. J. Non-Newtonian Fluid Mech. 14, 337–346 (1984)
O’Donovan, E., Tanner, R.: Numerical study of the bingham squeeze film problem. J. Non-Newtonian Fluid Mech. 15(1), 75–83 (1984)
Papanastasiou, T.C.: Flows of materials with yield. J. Rheol. 31(5), 385–404 (1987)
Perkowska, M., Wrobel, M., Mishuris, G.: Universal hydrofracturing algorithm for shear-thinning fluids: Particle velocity based simulation. Comput. Geotech. 71, 310–337 (2016)
Rossen, W., Gauglitz, P.: Percolation theory of creation and mobilization of foams in porous media. AIChE J 36(8), 1176–1188 (1990)
Sandve, T., Berre, I., Nordbotten, J.M.: An efficient multi-point flux approximation method for discrete fracture–matrix simulations. J. Comput. Phys. 231(9), 3784–3800 (2012)
Shah, C., Yortsos, Y.: Aspects of flow of power-law fluids in porous media. AIChE J. 41(5), 1099–1112 (1995)
Sochi, T.: Modelling the flow of yield-stress fluids in porous media. Transp. Porous Media 85(2), 489–503 (2010)
Talon, L., Auradou, H., Hansen, A.: Effective rheology of bingham fluids in a rough channel. Front. Phys. 2, 24 (2014)
Valk, P., Economides, M.J.: Hydraulic Fracture Mechanics. Wiley, New York (1995)
White, F.M.: Fluid Mechanics. McGraw-Hill, New York (2003)
Wu, Y., Pruess, K., Witherspoon, P., et al.: Flow and displacement of bingham non-newtonian fluids in porous media. SPE Reserv. Eng. 7(03), 369–376 (1992)
Zimmerman, R., Kumar, S., Bodvarsson, G.: Lubrication theory analysis of the permeability of rough-walled fractures. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 28(4), 325–331 (1991)
Acknowledgements
The authors are grateful to two anonymous reviewers, whose comments have helped improve the manuscript. This publication has been produced with support from the KPN project “Controlled Fracturing for Increased Recovery.” The authors acknowledge the following partners for their contributions: Lundin and the Research Council of Norway (244506/E30).
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Bao, K., Lavrov, A. & Nilsen, H.M. Numerical modeling of non-Newtonian fluid flow in fractures and porous media. Comput Geosci 21, 1313–1324 (2017). https://doi.org/10.1007/s10596-017-9639-y
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DOI: https://doi.org/10.1007/s10596-017-9639-y