Abstract
In the development of oil and gas reservoirs, the transport of miscible fluids in porous rocks is a key issue for oil and gas recovery. The simplified unidirectional flow model is employed to investigate the effects of the viscosity ratio on dispersion in semi-infinite homogenous media. The viscosity is supposed to be unsteady due to changing component concentration over time. In cases of both a viscosity ratio larger and less than 1, the pollutant concentration and flow velocity are computed at different initial conditions and viscosity ratios. The analytical solutions are then developed by introducing new variables and transforming the equation governing advection–diffusion equation in semi-infinite homogeneous media with a continuous source. A comparison of the numerical solution with the analytical solution revealed a similarity over 98%, highlighting the usability of the analytical solution. If the viscosity ratio is larger than 1, flow velocity declines exponentially and concentration attenuates with transporting time. In addition, the timescale plays a significant role and the effects become more prominent in long-term transport. In the case of a viscosity ratio less than 1, both the timescale and viscosity ratio variables have little influence on the changing speed of the concentration profile. This work helps to predict the position and the time required to reach the harmless pollutant concentration when monitoring fracturing fluids transportation into groundwater system and would be especially useful in designing and interpreting laboratory experiments studying the miscible flow.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Al-Niami ANS, Rushton KR (1977) Analysis of flow against dispersion in porous media. J Hydrol 33:87–97. https://doi.org/10.1016/0022-1694(77)90100-7
Aral MM, Tang Y (1992) Flow against dispersion in two-dimensional regions. J Hydrol 142:261–277. https://doi.org/10.1016/0022-1694(92)90243-O
Banks RB, Ali J (1964) Dispersion and adsorption in porous media flow. J Hydraul Div 90:13–31
Banks RB, Jerasate S (1962) Dispersion in unsteady porous media flow. J Hydraul Div 88:1–21
Ebach EH, White R (1958) Mixing of fluids flowing through beds of packed solids. J Am Inst Chem Eng 4:161–164. https://doi.org/10.1002/aic.690040209
Guvanasen V, Volker RE (1983) Experimental investigations of unconfined aquifer pollution from recharge basins. Water Resour Res 19:707–717. https://doi.org/10.1029/WR019i003p00707
Harleman DRF, Rumer RR (1963) Longitudinal and lateral dispersion in an isotropic porous medium. J Fluid Mech 16:385–394. https://doi.org/10.1017/S0022112063000847
Jackson RB, Pearson BR, Osborn SG, Warner NR, Vengosh A (2011) Research and policy recommendations for hydraulic fracturing and shale gas extraction. Center on Global Change, Duke University, Durham
Jaiswal DK, Kumar A, Yadav RR (2011) Analytical solution to one-dimensional advection–diffusion equation with temporally dependent coefficient. J Water Resour Prot 3:76–84. https://doi.org/10.1016/j.jhydrol.2009.11.008
Koch M, Zhang G (1991) Numerical simulation of the migration of density dependent contaminant plumes, SCRI. Report to the Florida Dept. of Environmental Regulation, Florida state University, Tallahassee, FL US
Koch M, Zhang G (1992) Numerical simulation of the effects of variable density in a contaminant plume. Groundwater 30:731–742. https://doi.org/10.1111/j.1745-6584.1992.tb01559.x/full
Lai SH, Jurinak JJ (1971) Numerical approximation of cation exchange in miscible displacement through soil columns. Soil Sci Soc Am Proc 35:894–899. https://doi.org/10.2136/sssaj1971.03615995003500060017x
Leij FJ, Toride N, Genucheten V (1993) Analytical solutions for non-equilibrium solute transport in three-dimensional porous media. J Hydrol 151:193–228. https://doi.org/10.1016/0022-1694(93)90236-3
Marino MA (1974) Distribution of contaminants in porous media flow. Water Resour Res 10:1013–1018. https://doi.org/10.1029/WR010i005p01013
Marshal TJ, Holmes JW, Rose CW (1996) Soil physics, 3rd edn. Cambridge University Press, Cambridge
Nunge RJ, Lin TS, Grill WN (1972) Laminar dispersion in curved tubes and channels. J Fluid Mech 52:363–383. https://doi.org/10.1017/S0022112072001247
Ogata A (1970) Theory of dispersion in granular media. US Geological Survey Professional Papers, 411-1
Ogata A, Banks RB (1961) A solution of the differential equation of longitudinal dispersion in porous media. US Geological Survey Professional Papers, 411-A
Rumer RR (1962) Longitudinal dispersion in steady and unsteady flow. J Hydraul Div 88(4):147–172
Scheidegger AE (1957) The physics of flow through porous media. University of Toronto Press, Toronto
Scheidegger AE (1961) General theory of dispersion in porous media. J Geophys Res 66:3273–3278. https://doi.org/10.1029/JZ066i010p03273
Serrano SE (1995) Forecasting scale-dependent dispersion from spills in heterogeneous aquifers. J Hydrol 169:151–169. https://doi.org/10.1016/0022-1694(94)02663-V
Singh MK, Mahato NK, Singh P (2008) Longitudinal dispersion with time-dependent source concentration in semi-infinite aquifer. J Earth Syst Sci 117:945–949. https://doi.org/10.1007/s12040-008-0079-x
Taylor G (1953) Dispersion of soluble matter in solvent flowing slowly through a tube. Proc R Soc A 219:186–203. https://doi.org/10.1098/rspa.1953.0139
Turner GA (1958) The flow-structure in packed beds: a theoretical investigation utilizing frequency response. Chem Eng Sci 7:156–165. https://doi.org/10.1016/0009-2509(58)80022-6
Yadav RR, Jaiswal DK, Yadav HK, Gulrana (2011) Temporally dependent dispersion through semi-infinite homogeneous porous media: an analytical solution. Int J Res Rev Appl Sci 6:158–164
Acknowledgements
This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and China Scholarship Council for sponsoring as a visiting student (No. 2009644010) in University of Calgary.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sang, Q., Chen, P., Dong, M. et al. The effect of viscosity ratio on the dispersal of fracturing fluids into groundwater system. Environ Earth Sci 77, 307 (2018). https://doi.org/10.1007/s12665-018-7460-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12665-018-7460-0