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Boundary-induced shear and tracer transport in heterogeneous porous rock

Published online by Cambridge University Press:  17 December 2020

Neeraja Bhamidipati Open the ORCID record for Neeraja Bhamidipati [Opens in a new window]  and
Andrew W. Woods Open the ORCID record for Andrew W. Woods [Opens in a new window]
Neeraja Bhamidipati*
Affiliation:
BP Institute, University of Cambridge, Madingley Road, CambridgeCB3 0EZ, UK
Andrew W. Woods
Affiliation:
BP Institute, University of Cambridge, Madingley Road, CambridgeCB3 0EZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

We study the longitudinal dispersion of a passive tracer by a two-dimensional pressure-driven flow through a layer of heterogeneous porous rock which is bounded above and below by impermeable seal rock. We assume there are localised regions of different permeability at random vertical positions within an otherwise uniform layer. This leads to a Fickian-type dispersion (Eames & Bush, Proc. R. Soc. Lond. A, vol. 455, issue 1990, 1999, pp. 3665–3686) and a shear near the boundaries (Rabinovich et al., Phys. Rev. E, vol. 86, issue 4, 2012, 046601). We illustrate this effect in a long two-dimensional layer, consisting of $(a)$ long and thin lenses and $(b)$ elliptical lenses, using expressions for depth-dependent mean speed and dispersivity to derive a depth-averaged transport equation for the flow. We derive asymptotic solutions for the layer-averaged dispersal of tracer, showing that at long times the spreading of the tracer is controlled by the shear. We discuss the importance of our results for the interpretation of tracer dispersal tests in layered permeable rocks.

JFM classification

Low-Reynolds-number flows: Porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , A45
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Bentham, M., Williams, G., Vosper, H., Chadwick, A., Williams, J. & Kirk, K. 2017 Using pressure recovery at a depleted gas field to understand saline aquifer connectivity. Energy Procedia 114, 2906–2920.CrossRefGoogle Scholar
Berkowitz, B., Scher, H. & Silliman, S. E. 2000 Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resour. Res. 36 (1), 149–158.CrossRefGoogle Scholar
Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D. & Brown, B. P. 2019 Dedalus: a flexible framework for numerical simulations with spectral methods. arXiv:1905.10388.CrossRefGoogle Scholar
Cala, M. A. & Greenkorn, R. A. 1986 Velocity effects on dispersion in porous media with a single heterogeneity. Water Resour. Res. 22 (6), 919–926.CrossRefGoogle Scholar
Dagan, G. 1982 Stochastic modeling of groundwater flow by unconditional and conditional probabilities: 1. Conditional simulation and the direct problem. Water Resour. Res. 18 (4), 813–833.CrossRefGoogle Scholar
Dagan, G. & Fiori, A. 2003 Time-dependent transport in heterogeneous formations of bimodal structures: 1. The model. Water Resour. Res. 39 (5), 1396.CrossRefGoogle Scholar
Dagan, G. & Lessoff, S. C. 2001 Solute transport in heterogeneous formations of bimodal conductivity distribution: 1. Theory. Water Resour. Res. 37 (3), 465–472.CrossRefGoogle Scholar
Darvini, G. 2016 The boundaries influence in trending hydraulic conductivity fields. Transp. Porous Med. 113 (3), 471–490.CrossRefGoogle Scholar
Dietrich, C. R. & Newsam, G. N. 1993 A fast and exact method for multidimensional gaussian stochastic simulations. Water Resour. Res. 29 (8), 2861–2869.CrossRefGoogle Scholar
Eames, I. & Bush, J. W. M. 1999 Longitudinal dispersion by bodies fixed in a potential flow. Proc. R. Soc. Lond. A 455 (1990), 3665–3686.CrossRefGoogle Scholar
Fiori, A., Janković, I. & Dagan, G. 2003 Flow and transport in highly heterogeneous formations: 2. Semianalytical results for isotropic media. Water Resour. Res. 39 (9), 1719.CrossRefGoogle Scholar
Freedman, D. & Diaconis, P. 1981 On the histogram as a density estimator: L2 theory. Probab. Theory Relat. Fields 57 (4), 453–476.Google Scholar
Greenkorn, R. A. & Kessler, D. P. 1969 Dispersion in heterogeneous nonuniform anisotropic porous media. Ind. Engng Chem. 61 (9), 14–32.CrossRefGoogle Scholar
Janković, I., Fiori, A. & Dagan, G. 2003 Flow and transport in highly heterogeneous formations: 3. Numerical simulations and comparison with theoretical results. Water Resour. Res. 39 (9), 1721.CrossRefGoogle Scholar
Kampman, N., Bickle, M. J., Maskell, A., Chapman, H. J., Evans, J. P., Purser, G., Zhou, Z., Schaller, M. F., Gattacceca, J. C., Bertier, P., et al. . 2014 Drilling and sampling a natural CO$_2$ reservoir: implications for fluid flow and CO$_2$-fluid–rock reactions during CO$_2$ migration through the overburden. Chem. Geol. 369, 51–82.CrossRefGoogle Scholar
Koch, D. L. & Brady, J. F. 1988 Anomalous diffusion in heterogeneous porous media. Phys. Fluids 31 (5), 965–973.CrossRefGoogle Scholar
Law, J. 1944 A statistical approach to the interstitial heterogeneity of sand reservoirs. Trans. AIME 155 (01), 202–222.CrossRefGoogle Scholar
Lessoff, S. C. & Dagan, G. 2001 Solute transport in heterogeneous formations of bimodal conductivity distribution: 2. Applications. Water Resour. Res. 37 (3), 473–480.CrossRefGoogle Scholar
Levenspiel, O. & Smith, W. K. 1957 Notes on the diffusion-type model for the longitudinal mixing of fluids in flow. Chem. Engng Sci. 6 (4–5), 227–235.CrossRefGoogle Scholar
Matheron, G. & De Marsily, G. 1980 Is transport in porous media always diffusive? A counterexample. Water Resour. Res. 16 (5), 901–917.CrossRefGoogle Scholar
Phillips, O. M. 2009 Geological Fluid Dynamics: Sub-Surface Flow and Reactions. Cambridge University Press.CrossRefGoogle Scholar
Rabinovich, A., Dagan, G. & Miloh, T. 2012 Boundary effects on effective conductivity of random heterogeneous media with spherical inclusions. Phys. Rev. E 86 (4), 046601.CrossRefGoogle ScholarPubMed
Scott, D. W. 1979 On optimal and data-based histograms. Biometrika 66 (3), 605–610.CrossRefGoogle Scholar
Srzic, V., Cvetkovic, V., Andricevic, R. & Gotovac, H. 2013 Impact of aquifer heterogeneity structure and local-scale dispersion on solute concentration uncertainty. Water Resour. Res. 49 (6), 3712–3728.CrossRefGoogle Scholar
Sturges, H. A. 1926 The choice of a class interval. J. Am. Stat. Assoc. 21 (153), 65–66.CrossRefGoogle Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219 (1137), 186–203.Google Scholar
Woods, A. W. 2015 Flow in Porous Rocks. Cambridge University Press.CrossRefGoogle Scholar
Zavala-Sanchez, V., Dentz, M. & Sanchez-Vila, X. 2009 Characterization of mixing and spreading in a bounded stratified medium. Adv. Water Resour. 32 (5), 635–648.CrossRefGoogle Scholar