Abstract
Mathematical models are developed for two-dimensional transient transport of colloids, and cotransport of contaminant/colloids in a fracture-rock matrix system with spatially variable fracture aperture. The aperture in the fracture plane is considered as a lognormally distributed random variable with spatial fluctuations described by an exponential autocovariance function. Colloids are envisioned to irreversibly deposit onto fracture surfaces without penetrating the rock matrix; whereas, the contaminant is assumed to decay, sorb onto fracture surfaces and onto colloidal particles, as well as to diffuse into the rock matrix. The governing stochastic transport equations are solved numerically for each realization of the aperture fluctuations by a fully implicit finite difference scheme. Emphasis is given on the effects of variable aperture on colloid and colloid-facilitated contaminant transport. Simulated breakthrough curves of ensemble averages of several realizations show enhanced colloid transport and more pronounced fingering when colloids are subject to size exclusion from regions of small aperture size. Moreover, it is shown that an increase in the fracture aperture fluctuations leads to faster transport and increases dispersion. For the case of contaminant/colloids cotransport it is shown, for the conditions considered in this work, that colloids enhance contaminant mobility and increase contaminant dispersion.
Similar content being viewed by others
Abbreviations
- b :
-
fracture aperture, L
- c :
-
contaminant concentration in the fracture, M/L3
- c m :
-
contaminant concentration in the rock matrix, M/L3
- c o :
-
source contaminant concentration, M/L3
- c * :
-
contaminant concentration adsorbed onto fracture surfaces, M/L2
- c * m :
-
contaminant concentration adsorbed inside the rock matrix, M/M
- d p :
-
colloidal particle diameter, L
- D :
-
hydrodynamic dispersion coefficient dyadic, L2/t
- D :
-
Brownian diffusion coefficient for colloids and molecular diffusion coefficient for contaminants, L2/t
- D m :
-
effective diffusion coefficient in the rock matrix, L2/t
- h :
-
total head potential in the fracture, L
- K f :
-
partition coefficient for contaminant sorption onto fracture surfaces, L
- K m :
-
contaminant partition coefficient in the rock matrix, L3/M
- K n :
-
partition coefficient for contaminant sorption onto suspended colloids, L
- K n* :
-
partition coefficient for contaminant sorption onto deposited colloids, L3/M
- ℓ x :
-
fracture length in thex-direction, L
- ℓ y :
-
fracture length in they-direction, L
- n :
-
colloid concentration in the liquid phase, M/L3
- n o :
-
source colloid concentration, M/L3
- n * :
-
colloid concentration adsorbed onto fracture surfaces, M/L2
- n *max :
-
maximum deposited colloid concentration on fracture surfaces, M/L2
- N * :
-
number of deposited colloidal particles per unit surface area of the fracture, 1/L2
- N *max :
-
maximum number of deposited colloidal particles per unit surface area of the fracture, 1/L2
- q * :
-
diffusive mass flux into the rock matrix, M/L2t
- R :
-
retardation factor in the fracture
- R m :
-
retardation factor in the rock matrix
- s :
-
contaminant concentration adsorbed on colloids in the liquid phase, M/M
- s o :
-
source solid-phase contaminant concentration onto suspended colloids, M/M
- s * :
-
contaminant concentration adsorbed on deposited colloids, M/M
- t :
-
time, t
- U :
-
interstitial velocity vector, L/t
- x :
-
coordinate along the fracture length, L
- y :
-
coordinate along the fracture width, L
- z :
-
coordinate perpendicular to the fracture plane, L
- α :
-
area blocked by a deposited colloidal particle, L2
- α L :
-
longitudinal dispersivity, L
- α T :
-
transversal dispersivity, L
- γ :
-
fluid specific weight, M/L2t2
- ε :
-
fraction of the fracture surface physically covered by colloids
- gz :
-
dummy integration variable
- θ :
-
porosity of the rock matrix
- κ :
-
colloid deposition coefficient, L
- λ :
-
first-order decay coefficient, 1/t
- Μ :
-
fluid dynamic viscosity, M/Lt
- ξ :
-
defined in (18)
- ρ b :
-
bulk density of the rock matrix, M/L3
- ρ p :
-
colloidal particle density, M/L3
- σ :
-
standard deviation of the lognormally distributed fluctuations of the fracture aperture
References
Abdel-Salam, A. and Chrysikopoulos, C. V.: 1994, Analytical solutions for one-dimensional colloid transport in saturated fractures,Advances in Water Resour. 17(5), 283–296.
Abelin, H.: 1986, Migration in a single fracture: An in situ experiment in a natural fracture, Ph.D. dissertation, Dep. of Chem. Eng., R. Inst, of Tech., Stockholm, 170 pp.
Bear, J. and Verruijt, A.: 1987,Modeling Groundwater Flow and Pollution, D. Reidel, Dordrecht.
Bianchi, L. and Snow, D.: 1968, Permeability crystalline rock interpretated from measured orientations and apertures of fractures,Ann. Arid Zone 8(2), 231–245.
Bourke, P. J., Dunance, E. M., Heath, M. J. and Hodgkinson, D. D.: 1985, Fracture hydrology relevant to radionuclide transport, AERE Rep. 11414, Atomic Energy Res. Estab., Harwell, United Kingdom.
Bourke, P. J.: 1987, Channeling of flow through fractures in rock,Proceedings, GEOVAL87 Symposium, Swed. Nucl. Power Inst., Stockholm, pp. 167–177.
Bowen, B. D. and Epstein, N.: 1979, Fine particle deposition in smooth parallel-plate channels,J. Colloid and Interface Sci. 72(1), 81–97.
Bradbury, M. H. and Green, A.: 1986, Investigations into the factors influencing long range matrix diffusion rates and pore space accessibility at depth in granite,J. Hydrol. 89, 123–139.
Buddemeier, R. W. and Hunt, J. R.: 1988, Transport of colloidal contaminants in groundwater radionuclide migration at the Nevada test site,Appl. Geochem. 3, 535–548.
Champ, D. R. and Schroeter, J.: 1988, Bacterial transport in fractured rock: A field-scale tracer test at the Chalk River Nuclear Laboratories,Water Sci. Technol. 20(11/12), 81–87.
Enfield, C. G. and Bengtsson, G.: 1988, Macromolecular transport of hydrophobic contaminants in aqueous environments,Ground Water 26(4) 64–70.
Enfield, C. G., Bengtsson, G. and Lindqvist, R.: 1989, Influence of Macromolecules on chemical transport,Environ. Sci. Technol. 23(10) 1278–1286.
Gale, J. E.: 1982, The effects of fracture type (induced versus natural) on the stress-fracture closure-fracture permeability relationships, inProceedings at 23rd Symposium on Rock Mechanics, Univ. Calif., Berkeley, pp. 290–298.
Gale, J. E., Rouleau, A. and Atkinson, L. C.: 1985, Hydraulic properties of fractures,Proceedings, Int. Assoc. of Hydrogeologists, Memoirs, Tucson Congress17 pp. 1–11.
Grisak, G. E. and Pickens, J. F.: 1981, An analytical solution for solute transport through fractured media with matrix diffusion,J. Hydrol. 52, 47–57.
Hakami, E. and Barton, N.: 1990, Aperture measurements and flow experiments using transparent replicas of rock joints, in N. Barton and O. Stephansson (eds),Rock Joints, A. A. Balkema, Rotterdam, pp. 383–390.
Haldeman, W. R., Chuang, Y., Rasmussen, T. C. and Evans, D. D.: 1991, Laboratory analysis of fluid flow and solute transport through a fracture embedded in porous tuff,Water Resour. Res. 27(1), 53–65.
Harvey, R. W., George, L. H., Smith, R. L. and LeBlanc, D. R.: 1989, Transport of fluorescent microshpere and indigenous bacteria through a sandy aquifer: Results of natural- and forced-gradient tracer experiment,Environ. Sci. Technol. 23, 51–56.
Harvey, R. W. and Garabedian, S. P.: 1991, Use of colloid filtration theory in modeling movement of bacteria through a contaminated sandy aquifer,Environ. Sci. Technol. 25, 178–185.
Herzig, J. P., Leclerc, D. M. and Le Goff, P.: 1970, Flow of suspension through porous media: Application to deep filtration,Ind. Eng. Chem. 62(5), 9–35.
Higgo, J. J. W., Williams, G. M., Harrison, I., Warwick, P., Gardiner, M. P. and Longworth, G.: 1993, Colloid transport in a glacial sand aquifer. Laboratory and field studies,Colloids and Surfaces A 73, 179–200.
Huyakorn, P. S. and Pinder, G. F.: 1983,Computational Methods in Subsurface Flow, Academic Press, New York.
Johns, R. A. and Roberts, P. V.: 1991, A solute transport model for channelized flow in a fracture,Water Resour. Res. 27(8), 1797–1808.
KBS-3, 1983, Final storage of spent nuclear fuel, 4, technical report, Swedish Nuclear Fuel Supply, 124 pp.
Kinzelbach, W.: 1986,Developments in Water Science, Groundwater Modelling, Elsevier, Amsterdam.
Krishnamoorthy, T. M., Nair, R. N. and Sarma, T. P.: 1992, Migration of radionuclides from a granite repository,Water Resour. Res. 28(7), 1927–1934.
Lapidus, L. and Pinder, G.: 1982,Numerical Solution of Partial Differential Equations in Science and Engineering, Wiley, New York.
McCarthy, J. F. and Zachara, J. M.: 1989, Subsurface transport of contaminants,Environ. Sci. Technol.,23(5), 496–502.
Moreno, L., Tsang, Y. W., Tsang, C. F., Hale, F. V. and Neretnieks, I.: 1988, Flow and tracer transport in a single fracture: A stochastic model and its relation to some field observations,Water Resour. Res. 24, 2033–2048.
Moulin, V. and Ouzounian, G.: 1992, Role of colloids and humic substances in the transport of radio-elements through the geosphere,Appl. Geochem. Suppl. Issue (1), 179–186.
Neretnieks, I., Eriksen, T. and Tahtinen, P.: 1982, Tracer movement in a single fissure in granitic rock: Some experimental results and their interpretation,Water Resour. Res. 18, 849–858.
Neretnieks, I.: 1983, A note on fracture flow dispersion mechanisms in the ground,Water Resour. Res. 23, 561–570.
Neretnieks, I.: 1985, Transport in fractured rocks,Proceedings, Int. Assoc. of Hydrogeologists, Memoirs, Tucson Congress 17, pp. 301–318.
Neuzil, C. E. and Tracy, J. V.: 1981, Flow through fractures,Water Resour. Res. 17, 191–199.
Novakowski, K. S., Evans, G. V., Lever, D. A. and Raven, K.: 1985, A field example of measuring hydrodynamic dispersion in a single fracture,Water Resour. Res. 21, 1165–1174.
Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P.: 1992,Numerical Recipes in Fortran: The Art of Scientific Computing, Cambridge University Press.
Puls, R. W., Paul, C. J. and Clark, D. A.: 1993, Surface chemical effects on colloid stability and transport through natural porous media,Colloids and Surfaces A 73, 287–300.
Pyrak, L. R., Myer, L. R. and Cook, N. G. W.: 1985, Determination of fracture void geometry and contact area at different effective stress,Eos Trans. AGU Abstract,66(46), 903.
Raven, K. G., Novakowski, K. S. and Lapeciv, P. A.: 1988, Interpretation of field tracer tests of a single fracture using a transient solute storage model,Water Resour. Res. 24, 2019–2032.
Sakthivadivel, R.: 1969, Clogging of a granular porous medium by sediment, Hydraul. Eng. Lab., Univ. of Calif., Berkeley, Rep. HEL 15-17, 106 pp.
Schrauf, T. W. and Evans, D. D.: 1986, Laboratory studies of gas flow through a single natural fracture,Water Resour. Res. 22, 1038–1050.
Shapiro, A. M. and Nicholas, J. R.: 1989, Estimating the statistical properties of fracture aperture using field scale hydraulic and tracer tests: Theory and application,Water Resour. Res. 25, 817–828.
Skagius, K. and Neretnieks, I.: 1986, Porosities and diffusivities of some nonsorbing species in crystalline rocks,Water Resour. Res. 22(3), 389–398.
Sposito, G.: 1984,The Surface Chemistry of Soils, Oxford University Press, New York.
Song, L. and Elimelech, M.: 1993, Dynamics of colloid deposition in porous media: Modeling the role of retained particles,Colloids and Surfaces A 73, 49–63.
Streltsova, T. D.: 1988,Well Testing in Heterogeneous Formations, Wiley, New York.
Strikwerda, J. C.: 1989,Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks/Cole.
Tang, D. H., Frind, E. O. and Sudicky, E. A.: 1981, Contaminant transport in fractured porous media: Analytical solution for a single fracture,Water Resour. Res. 17(3), 555–564.
Toran, L. and Palumbo, A. V.: 1992, Colloid transport through fractured and unfractured laboratory sand columns,J. Contam. Hydrol. 9, 289–303.
Tsang, Y. W. and Tsang, C. F.: 1987, Channel model of flow through fractured media,Water Resour. Res. 23, 467–479.
Williams, S. A. and El-Kadi, A. I.: 1986, COVAR: A computer program for generating two-dimensional fields of autocorrelated parameters by matrix decomposition,Int. Groundwater Model. Cent., Holcomb Res. Inst., Colorado School of Mines, Golden, CO.
Witherspoon, P. A., Wang, J. S. Y. Iwai, K. and Gale, J. E.: 1980, Validity of cubic law for fluid flow in a deformable rock fracture,Water Resour. Res. 16(6), 1016–1024.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Abdel-Salam, A., Chrysikopoulos, C.V. Modeling of colloid and colloid-facilitated contaminant transport in a two-dimensional fracture with spatially variable aperture. Transp Porous Med 20, 197–221 (1995). https://doi.org/10.1007/BF01073172
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01073172