The Journal of Engineering Geology (지질공학)

The Korean Society of Engineering Geology (대한지질공학회)
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Variations of Longitudinal Moments for a Contaminant Transport in Physically and Chemically Heterogeneous Media

물리.화학적 불균질 특성을 지닌 매질 내 오염운 이동시 보이는 종적률 변화

  • Published : 2009.03.31
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Abstract

Two dimensional Monte-Carlo simulations of a non-reactive solute plume in isotropic porous media which are physically and chemically heterogeneous are conducted to determine the variations of moment. Retardation factors of 1, 2 and 5 are given to ascertain how the second moments are changed as adsorption increased. Retarded longitudinal second spatial moment, ${Z_{11}}^{'R}(t',l')$, increased during the transport process and as the dimensionless lengths of line plume source, $l_2'$, increased. ${Z_{11}}^{'R}(t',l')$ decreased as the retardation factors increased, and the simulated moments fit well to the first-order analytical results. Retarded longitudinal plume centroid variance, ${Z_{11}}^{'R}(t',l')$, decreased as the dimensionless lengths of line plume source, $l_2'$, increased and as the retardation factor increased. The result indicates that the uncertainty about the plume center decreased, and the ergodic condition for the second spatial moments is far from reaching. Simulated longitudinal one particle displacement covariance, ${Z_{11}}^{'R}(t')$, well consistent with the first-order analytical results for the three degrees of retardation factors of 1, 2 and 5 respectively. It is, consequently, concluded that the retarded longitudinal second moments could be produced by stochastic simulation, and that the first-order analytical results definitely provides very close values of the longitudinal retarded moments.

물리 화학적 불균질 특성을 가진 매질 중 K 임의장과 $K_d$ 임의장이 정(+)의 상관관계를 가지는 경우, 매질 내 이송되는 오염운이 보이는 적률변화를 알아보기 위해 등방매질 내 이송되는 비반응성 오염운에 대한 2차원 몬테카를로 시뮬레이션이 시행되었다. 흡착성이 증가함에 따라 변화하는 이차적률들을 관찰하기 위해 1, 2, 5의 지연요소가 설정되었다. 지연된 종이차공간적률, ${Z_{11}}^{'R}(t',l')$은 오염운의 이송이 진행됨에 따라 증가하며 초기 오염운의 크기가 증가함에 따라 증가한다. 또한 지연요소가 증가함에 따라 감소하며, 일차분석해에 의한 이론적 결과와 비교적 잘 일치함을 보인다. 지연된 종오염운중심분산, ${Z_{11}}^{'R}(t',l')$은 초기오염원의 크기, $l_2'$가 증가할수록 감소하며 설정된 시뮬레이션에 의해서는 아직 에르고딕 이송상태에 도달하지 않았음을 지시해준다. 지연요소가 증가함에 따라 현저히 감소하여 오염운 중심점에 대한 불확실성이 감소함을 보인다. 지연된 단일입자종이송분산, ${X_{11}}^{'R}(t')$은 세 가지 다른 등급의 지연요소에 대해서 모두 일차근사법에 의한 이론적 결과와 비교적 잘 일치한다. 따라서 종방향의 지연된 이차적률들은 추계론적 시뮬레이션에 의해 산출될 수 있으며, 일차근사법에 의한 분석해는 종방향의 지연된 적률들에 대해 상당히 정확한 근사값을 제공하고 있음을 결론지을 수 있다.

Keywords

References

  1. 서병민, 2005, 불균질도가 높은 대수층 내에서의 비에르고딕 용질이동에 관한 수치시뮬레이션, 지질공학회지, 15(3), 245-255
  2. Bellin, A., Salandin, P., and Rinaldo, A., 1992, Simulation of dispersion in heterogeneous porous formations:statistics, first order theories, convergence of computations, Water Resour. Res., 28(9), 2211-2227 https://doi.org/10.1029/92WR00578
  3. Barry, D., Coves, A.J., and Sposito, G., 1988, On the dagan model of solute transport in ground-water application to the Borden site. Water Resour. Res.,24(10), 1805-1817 https://doi.org/10.1029/WR024i010p01805
  4. Burr, D.T., Sudicky, E.A., and Naff, R.L., 1994, Nonreactive and reactive solute transport in three-dimensional heterogeneous porous media: Mean displacement, plume spreading, and uncertainty, Water Resour. Res., 30(3), 791-815 https://doi.org/10.1029/93WR02946
  5. Chin, D.A. and Wang, T., 1992, An investigation of the validity of first order stochastic dispersion theories in isotropic porous media, Water Resour. Res., 28(6), 1531-1542 https://doi.org/10.1029/92WR00666
  6. Cushman, J.H., Hu, B.X., and Ginn, T.R., 1994, Nonequilibrium statistical mechanics of preasymptotic dispersion, J. Stat. Phys., 75, 859-878 https://doi.org/10.1007/BF02186747
  7. Cvetkovic, V. and Dagan, G., 1994, Transport of Kinetically sorbing solute by steady random velocity in heterogeneous porous formations, J. Fluid Mech.,265, 189-215 https://doi.org/10.1017/S0022112094000807
  8. Dagan, G., 1988, Time-dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers, Water Resour. Res., 24(9), 1491-1500 https://doi.org/10.1029/WR024i009p01491
  9. Dagan, G., 1990, Transport in heterogeneous porous formations: Spatial moments, ergodicity, and effective dispersion, Water Resour. Res., 26(6), 1281-1290 https://doi.org/10.1029/WR026i006p01281
  10. Dagan, G., 1991, Dispersion of a passive solute in nonergodic transport by steady velocity fields in heterogeneous formations, J. Fluid Mech., 233, 197-210 https://doi.org/10.1017/S0022112091000459
  11. Dentz, M., Kinzelbach, H., Attinger, S., and Kinzelbach, W., 2002, Temporal behavior of a solute cloud in a heterogeneous porous medium, 3. Numerical simulations, Water Resour. Res., 38, 23-1-13
  12. Espinoza, C. and Valocchi, A.J., 1997, Stochastic analysis of one-dimensional transport of kinetically adsorbing solutes in chemically heterogeneous aquifers, Water Resour. Res., 33(11), 2429-2445 https://doi.org/10.1029/97WR02169
  13. Fiori, A. and Belline, A., 1999, Non-ergodic transport of kinetically sorbing solutes, J. Contam. Hydrol., 40, 201-219 https://doi.org/10.1016/S0169-7722(99)00054-6
  14. Federico, V.D. and Zhang, Y.K., 1999, Solute transport in heterogeneous porous media with long-range correlations, Water Resour. Res., 35(10), 3185-3192 https://doi.org/10.1029/1999WR900021
  15. Garabedian, S.P., Leblanc, D.R., Gelhar, L.W., and Celia, M.A., 1991, Large-scale natural gradient tracer test in sand and gravel, Cape Code, Massachusetts. 2. Analysis of Spatial moments for a non-reactive tracer, Water Resour. Res., 27(5), 911-924 https://doi.org/10.1029/91WR00242
  16. Hassan, A., Cushman, J.H., and Delleur, J.W., 1998, A Monte Carlo assessment of eulerian flow and transport perturbation models, Water Resour. Res., 34, 1143-1163 https://doi.org/10.1029/98WR00011
  17. Hassan, A., Andricevic, R. and Cvetkovic, V., 2002, Evaluation of analytical solute discharge moments using numerical modeling in absolute and relative dispersion frameworks, Water Resour. Res., 38, 1-1-8
  18. Hubbard, S., Chen, J., Peterson, J., Majer, E.L., Willianms, K.H., Swift, D.J., Mailloux, B., and Rubin, Y., 2001, Hydrogeological characterization of the South Oyster vacterial transport site suing geophysical data, Water Resour. Res., 37 10), 2431-2456 https://doi.org/10.1029/2001WR000279
  19. Killey, R.W.D. and Moltyaner, G.L., 1988, Twin lake tracer tests: setting methodology, and hydraulic conductivity distribution, Water Resour. Res., 24(10),1585-1612 https://doi.org/10.1029/WR024i010p01585
  20. Kitanidis, P.K., 1988, Prediction by the method of moments of transport in a heterogeneous formation, Jour. Hydrology, 102(1-4), 453-473 https://doi.org/10.1016/0022-1694(88)90111-4
  21. Mackay, D.M., Freyberg, D.L., Roberts, P.V., and Cherry, J.A., 1986, A natural gradient experiment in a sand aquifer, 1. Approach and overview of plume movement, Water Resour. Res., 22, 2017-2030 https://doi.org/10.1029/WR022i013p02017
  22. Rajaram, H. and Gelhar, L.W., 1993, Plume scale-dependent dispersion in heterogeneous aquifer, 1. Lagrangian analysis in a stratified aquifer, Water Resour. Res., 29(9), 3249-3260 https://doi.org/10.1029/93WR01069
  23. Robin, M.J.L., Gutjahr, A.L., Sudicky, E.A., and Wilson, J.L., 1993, Cross-correlated random field generator with direct Fourier transform method, Water Resour. Res., 29(7), 2385-2397 https://doi.org/10.1029/93WR00386
  24. Rubin, Y., 1990, Stochastic modeling of macrodispersion in heterogeneous porous media, Water Resour. Res., 26(1), 133-141 https://doi.org/10.1029/WR026i001p00133
  25. Selroos, J.O., 1995, Temporal moments for non-ergodic solute transport in heterogeneous aquifers, Water Resour. Res., 31(7), 1705-1712 https://doi.org/10.1029/95WR00945
  26. Zhang, D. and Neuman, S.P., 1995, Eulerian-Lagrangian analysis of transport conditioned on hydraulic data, 3. Spatial moments, travel time distribution, mass flow rate, and cumulative release across a compliance surface, Water Resour. Res., 31(1), 65-75 https://doi.org/10.1029/94WR02236
  27. Zhang, Y.K., Zhang, D., and Lin, J., 1996, Non-ergodic solute transport in three-dimensional heterogeneous isotropic aquifers, Water Resour. Res., 32(9), 2955-2963 https://doi.org/10.1029/96WR01467
  28. Zhang, Y.K. and Zhang, D., 1997, Time-dependent dispersion of non-ergodic plumes in two-dimensional heterogeneous porous media, Journal of Hydrologic Engineering, 2(2), 91-94 https://doi.org/10.1061/(ASCE)1084-0699(1997)2:2(91)
  29. Zhang, Y.K. and Federico, V.D., 1998, Solute transport in three-dimensional heterogeneous media with a Gaussian covariance of log hydraulic conductivity, Water Resour. Res., 34(8), 1929-1934 https://doi.org/10.1029/98WR01142
  30. Zhang, Y.K. and Lin, J., 1998, Numerical simulations of transport of non-ergodic plumes in heterogeneous aquifers, Stochastic Hydrology and Hydraulics, 12(2),117-140 https://doi.org/10.1007/s004770050013
  31. Zhang, Y.K. and Federico, D.V., 2000, Nonergodic solute transport in heterogeneous porous media: Influence of multiscale structure, in Zhang, D., and Winter, C.L., eds., Theory, Modeling, and Field Investigation in Hydrogology: A Special Volume in Honor of Shlomo P. Neuman's 60th Birthday: Boulder, Colorado, Geological Sociery of America Special Paper, 348, 61-72
  32. Zhang, Y.K., 2003, Non-ergodic solute transport in physically and chemically heterogeneous porous media, Water Resour. Res., 39(7), 1197 https://doi.org/10.1029/2003WR002116
  33. Zhang, Y.K. and Seo, B., 2004, Numerical simulations of non-ergodic solute transport in three-dimensional heterogeneous porous media, Stochastic Environmental Research and Risk Assessment, 18, 205-215 https://doi.org/10.1007/s00477-004-0178-4