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The effects of fluid transport on the creation of a dense cluster of activated fractures in a porous medium

Published online by Cambridge University Press:  21 May 2018

Mohammed G. Alhashim  and
Donald L. Koch Open the ORCID record for Donald L. Koch [Opens in a new window]
Mohammed G. Alhashim
Affiliation:
Robert Frederick Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
Donald L. Koch*
Affiliation:
Robert Frederick Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
*
†Email address for correspondence: [email protected]
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Abstract

The formation of a cluster of activated fractures when fluid is injected in a low permeability rock is analysed. A fractured rock is modelled as a dual porosity medium that consists of a growing cluster of activated fractures and the rock’s intrinsic porosity. An integro-differential equation for fluid pressure in the developing cluster of fractures is introduced to account for the pressure-driven flow through the cluster, the loss of fluid into the porous matrix and the evolution of the cluster’s permeability and porosity as the fractures are activated. Conditions under which the dependence of the permeability and porosity on the fluid pressure can be derived from percolation theory are discussed. It is shown that the integro-differential equation admits a similarity solution for the fluid pressure and that the cluster radius grows as a power law of time in two regimes: (i) a short-time regime, when many fractures are activated but pressure-driven flow in the network still dominates over fluid loss; and (ii) a long-time regime, when fluid loss dominates. The power law exponents in the two regimes are functions of the Euclidean dimension of the cluster, percolation universal exponents and the injection protocol. The model predicts that the effects of the fluid properties on the morphology of the formed network are different in the two similarity regimes. For example, increasing the injection rate with time, in the flow dominant regime, produces a smaller cluster of activated fractures than that formed by injecting the fluid at a constant rate. In the fluid loss dominated regime, however, ramping up the injection rate produces a larger cluster.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Gravity currents Low-Reynolds-number flows: Low-Reynolds-number flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 847 , 25 July 2018 , pp. 286 - 328
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Copyright
© 2018 Cambridge University Press 

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References

Acton, J. M., Huppert, H. E. & Worster, M. G. 2001 Two-dimensional viscous gravity currents flowing over a deep porous medium. J. Fluid Mech. 440, 359–380.Google Scholar
Adachi, J., Sieberits, E., Peirce, A. & Desroches, J. 2007 Computer simulation of hydraulic fractures. J. Rock Mech. Mining Sci. 44, 739–757.CrossRefGoogle Scholar
Asanuma, H., Nozaki, H., Niitsuma, H. & Wyborn, D. 2005 Interpretation of micro-seismic events with larger magnitude collected at Cooper Basin, Australia. GRC Trans. 29, 87–91.Google Scholar
Balberg, I., Anderson, C. H., Alexander, S. & Wagner, N. 1984 Excluded volume and its relation to the onset of percolation. Phys. Rev. B 30, 3933–3943.Google Scholar
Cleary, M. P. 1988 The engineering of hydraulic fractures-State of the art and technology of the future. J. Petrol. Tech. 40, 13–21.Google Scholar
Detournay, E. 2016 Mechanics of hydraulic fracture. Annu. Rev. Fluid Mech. 48, 311–339.Google Scholar
Elmo, D. & Stead, D. 2010 An integrated numerical modelling-discrete fracture network approach applied to the characterization of rock mass strength of naturally fractured pillars. Rock Mech. Rock Engng 43, 3–19.Google Scholar
Ellsworth, W. L. 2013 Injection-induced earthquakes. Science 341, 1225942.Google Scholar
Fu, P., Johnson, S. M. & Carrigan, C. R. 2012 An explicitly coupled hydro-geomechanical model for simulating hydraulic fracturing in arbitrary discrete fracture networks. Intl J. Numer. Anal. Meth. Geomech. 37, 2278–2300.Google Scholar
Geertsma, J. & de Klerk, F. 1969 A rapid method of predicting width and extend of hydraulically induced fractures. J. Petrol. Tech. 21, 1571–1581.Google Scholar
Gelmi, C. A. & Jorquera, H. 2014 IDSOLVER: a general purpose solver for the th-order integro-differential equations. Comput. Phys. Commun. 185, 392–397.Google Scholar
Gischig, V. S. & Wiemer, S. 2013 A stochastic model for induced seismicity based on non-linear pressure diffusion and irreversible permeability enhancement. Geophys. J. Intl 194, 1229–1249.Google Scholar
Gor, G. Y., Stone, H. A. & Prevost, J. H. 2013 Fracture propagation driven by fluid outflow from a low-permeability aquifer. Trans. Porous Med. 100, 69–82.Google Scholar
Grasselli, G. & Egger, P. 2002 Constitutive law for the shear strength of rock joints based on three-dimensional surface parameters. Intl J. Rock Mech. Mining Sci. 40, 25–40.Google Scholar
Hewitt, D. R., Neufeld, J. A. & Balmforth, N. J. 2015 Shallow, gravity-driven flow in a poro-elastic layer. J. Fluid Mech. 778, 335–360.Google Scholar
Howard, G. C. & Fast, C. R. 1957 Optimum fluid characteristic for fracture extension. J. Petrol. Tech. 24, 261–270.Google Scholar
Huppert, H. E. 2006 Gravity currents: a personal perspective. J. Fluid Mech. 554, 299–322.CrossRefGoogle Scholar
Huppert, H. E. & Neufeld, J. A. 2014 The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46, 255–272.Google Scholar
Huppert, H. E. & Woods, A. W. 1995 Gravity-driven flows in porous layers. J. Fluid Mech. 292, 55–69.Google Scholar
Hummel, N. & Shapiro, S. 2013 Nonlinear diffusion-based interpretation of induced microseismicity: a Barnett shale hydraulic fracturing case study. Geophysics 78, B211–B226.Google Scholar
Hunt, A., Ewing, R. & Ghanbarian, B. 2014 Properties relevant for transport and transport applications. In Percolation Theory for Flow in Porous Media, pp. 37–57. Springer.Google Scholar
Kozlov, B. & Lagues, M. 2010 Universality of 3D percolation exponents and first-order corrections to scaling for conductivity exponents. Physica A 389, 5339–5346.CrossRefGoogle Scholar
Lai, C. Y., Zheng, Z., Wexler, J. S. & Stone, H. A. 2016 Fluid driven cracks in an elastic matrix in the toughness dominated limit. Trans. R. Soc. Lond. A 374, 20150425.Google Scholar
Lai, C. Y., Zhong, Z., Dressaire, E., Wexler, J. S. & Stone, H. A. 2015 Experimental study on penny-shaped fluid-driven cracks in an elastic matrix. Proc. R. Soc. Lond. A 471 (2182), 20150255.Google Scholar
Lei, X., Ma, S., Chenn, W., Pang, C., Zeng, J. & Jiang, B. 2013 A detailed view of the injection-induced seismicity in a natural gas reservoir in Zigong, southwestern Sichuan Basin, China. J. Geophys. Res. 118, 4296–4311.CrossRefGoogle Scholar
Lolon, E., Shaoul, J. R. & Mayerhofer, J. M.2007 Application of 3-D reservoir simulator for hydraulically fractured rocks. SPE 110093, presented at the Asia Pacific Oil and Gas Conference and Exhibition, Jakarta, Indonesia, 30 October–1 November 2007.Google Scholar
Majer, E. L., Baria, R., Stark, M., Oates, S., Bommer, J., Smith, B. & Asanuma, H. 2007 Induced seismicity associated with enhanced geothermal systems. Geothermics 36, 185–222.CrossRefGoogle Scholar
Montgomery, C. 2013 Fracturing fluids. In Effective and Sustainable Hydraulic Fracturing (ed. Jeffery, R.), pp. 4–24. InTech.Google Scholar
Murphy, H. D., Tester, J. W., Grigsby, C. O. & Potter, R. M. 1981 Energy extraction from fractured geothermal reservoirs in low-permeability crystalline rock. J. Geophys. Res. 86, 7145–7158.Google Scholar
Nagel, N. B., Sanchez-Nagel, M. A., Zhang, F., Garcia, X. & Lee, B. 2013 Coupled numerical evaluations of the geomechanical interactions between a hydraulic fracture stimulation and a natural fracture system in shale formations. Rock Mech. Rock Engng 46, 581–609.Google Scholar
Neufeld, J. A. & Huppert, H. E. 2009 Modelling carbon dioxide sequestration in layered strata. J. Fluid Mech. 625, 353–370.Google Scholar
Nordgren, R. P. 1972 Propagation of a vertical hydraulic fracture. SPE 12, 306–314.Google Scholar
Olsson, R. & Barton, N. 2001 An improved model for hydromechanical coupling during shearing of rock joints. Intl J. Rock Mech. Mining Sci. 38, 317–329.Google Scholar
Pegler, S. S., Huppert, H. E. & Neufeld, J. A. 2014 Fluid injectino into a confined porous layer. J. Fluid Mech. 745, 592–620.CrossRefGoogle Scholar
Perkins, T. K. & Kern, L. R. 1961 Widths of hydraulic fracture. SPE 89, 937–949.Google Scholar
Philips, W. S. 2000 Precise microearthquake locations and fluid flow in the geothermal reservoir at Soultz-sous-Forets, France. Bull. Seismol. Soc. Am. 90, 212–228.CrossRefGoogle Scholar
Pritchard, D. 2007 Gravity currents over fractured substrates in a porous medium. J. Fluid Mech. 584, 415–431.Google Scholar
Pritchard, D., Woods, A. W. & Hogg, A. J. 2001 On the slow draining of a gravity current moving through a layered permeable medium. J. Fluid Mech. 444, 23–47.Google Scholar
Rutldege, J. T. & Phillips, W. S. 2003 Hydraulic stimulation of natural fractures as revealed by induced microearthquakes, Carthage Cotton Valley gas field, east Texas. Geophysics 68, 441–452.Google Scholar
Sahimi, M. 2011 Characterization of fractures, fracture networks, and fractured porous media. In Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches, pp. 143–175. Wiley-VCH.Google Scholar
Sahimi, M., Robertson, M. C. & Sammis, C. G. 1993 Fractal distribution of earthquake hypocenters and its relation to fault pattern and percolation. Phys. Rev. Lett. 70, 2186–2189.Google Scholar
Sasaki, S. 1998 Characteristics of microseismic events induced during hydraulic fracturing experiments at the Hijiori hot dry rock geothermal energy site, Ymagata, Japan. Tectonophysics 289, 171–188.Google Scholar
Savitski, A. A. & Detournay, E. 2002 Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions. Intl J. Solids Struct. 39, 6311–6337.Google Scholar
Shapiro, S. & Dinske, C. 2009 Scaling of seismicity induced by nonlinear fluid-rock interaction. J. Geophys. Res. 114, B09307.Google Scholar
Sminchak, J., Gupta, N., Byrer, C. & Bergman, P. 2002 Issues related to seismic activity induced by the injection of CO2 in deep saline aquifers. J. Energy Environ. Res. 2, 32–46.Google Scholar
Stauffer, D. 1979 Scaling theory of percolation clusters. Phys. Rep. 54, 1–74.Google Scholar
Stauffer, D. & Aharony, A. 1994 Cluster numbers. In Introduction to Percolation Theory, pp. 19–53. CRC Press.Google Scholar
Tayeb, A. T., Sahimi, M., Aminzadeh, F. & Sammis, C. G. 2013 Use of microseismicity for determining the structure of the fracture network of large-scale porous media. Phys. Rev. E 87, 032152.Google Scholar
Tezuka, K. & Niitsuma, H. 2000 Stress estimated using microseismic clusters and its relationship to the fracture system of the Hijiori hot dry rock reservoir. Engng Geol. 56, 47–62.Google Scholar
Wang, C. Y. & Mao, N. H. 1979 Shearing of saturated clays in rock joints at high confining pressures. Geophys. Res. Lett. 6, 825–828.Google Scholar
Wang, J., Zhou, Z., Zhang, W., Garoni, T. M. & Deng, Y. 2013 Bond and site percolation in three dimensions. Phys. Rev. E 87, 052107.Google Scholar
Warpinski, N. R., Kramm, R. C., Heinze, J. R. & Waltman, C. K.2005 Comparison of single- and dual-array microseismic mapping technique in the barnett shale. SPE 95568, presented at the SPE Annual Technology Conference and Exhibition, Dallas, 9–12 October.Google Scholar
Warpinski, N. R. & Teufel, L. W. 1987 Influence of geologic discontinuities on hydraulic fracture propagation. SPE 39, 209–220.Google Scholar
Warren, J. E. & Root, P. J. 1963 The behavior of naturally fractured reservoirs. SPE 3, 245–255.Google Scholar
Willis-Richards, J., Watanabe, K. & Takahashi, H. 1996 Progress toward a stochastic rock mechanics model of engineered geothermal systems. J. Geophys. Res. 101, 481–496.Google Scholar
Wilkinson, D. & Willemsen, J. F. 1983 Invasion percolation: a new form of percolation theory. J. Phys. A: Math. Gen. 16, 3365–3376.Google Scholar
Zimmerman, R. W. & Bodvarsson, G. S. 1996 Hydraulic conductivity of rock fractures. Trans. Porous Med. 23, 1–30.Google Scholar
Zimmerman, R. W., Chen, G., Hadgu, T. & Bodvarsson, G. S. 1993 A numerical dual-porosity model with semianalytical treatment of fracture/matrix flow. Water Resour. Res. 29, 2127–2137.Google Scholar
Zhang, X., Jeffrey, R. G. & Thiercelin, M. 2007 Deflection and propagation of fluid-driven fractures at frictional bedding interfaces: a numerical investigation. J. Struct. Geol. 29, 396–410.Google Scholar
Zheng, Z., Christov, I. C. & Stone, H. A. 2014 Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents. J. Fluid Mech. 747, 218–246.Google Scholar
Zheng, Z., Guo, B., Christov, I. C., Celia, M. A. & Stone, H. A. 2015 Flow regimes for fluid injection into a confined porous medium. J. Fluid Mech. 767, 881–909.Google Scholar