Stochastic Fluid Travel Times in Heterogeneous Porous Media
Publication: Journal of Hydraulic Engineering
Volume 120, Issue 2
Abstract
An analytic approach is developed for quantifying the distribution of fluid passage times in a heterogeneous porous medium. The basic methodology employed utilizes a diffusion theory description for the displacement of a purely advected fluid subject to a random field of hydraulic conductivity. Within this framework, a governing model is formed using the backward form of the statistical Kolmogorov equation, which yields the inverse Gaussian distribution as a solution to the fluid passage time problem. In proposing the methodology, a rationale is presented for quantifying the associated model parameters using a simple application of the mean and variance to Darcy's law, with subsequent comparisons being made to previous results obtained for perturbation solutions of the associated stochastic partial differential equations. In addition, the validity of the model is discussed within the bounds of a Markovian description for ground‐water flow under a continuum‐based modeling framework. Here, it is argued that acceptably accurate results may be achieved for a statistical Peclet number in excess of 70.
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References
1.
Arnold, L. (1974). Stochastic differential equations: Theory and application. John Wiley & Sons, New York, N.Y.
2.
Bhattacharya, R. N., Gupta, V. K., and Sposito, G. (1976a). “A Markovian stochastic basis for the transport of water through unsaturated soil.” Soil Sci. Soc. Am. J., 40(2), 465–467.
3.
Bhattacharya, R. N., Gupta, V. K., and Sposito, G. (1976b). “On the stochastic foundations of the theory of water flow through unsaturated soil.” Water Resour. Res., 12(3), 503–512.
4.
Cox, D. R., and Miller, H. D. (1965). The theory of stochastic processes. Chapman and Hall, New York, N.Y.
5.
Cushman, J. H. (1987). “Development of stochastic partial differential equations for subsurface hydrology.” Stochastic Hydrol. Hydr., 1(2), 241–262.
6.
Cvetkovic, V. D., Dagan, G., and Shapiro, A. M. (1991). “An exact solution of solute transport by one‐dimensional random velocity fields.” Stochastic Hydrol. Hydr., 5(1), 45–54.
7.
Cvetkovic, V. D., and Shapiro, A. M. (1989). “Solute advection in stratified formations.” Water Resour. Res., 25(6), 1283–1289.
8.
Cvetkovic, V. D., Shapiro, A. M., and Dagan, G. (1992). “A solute flux approach to transport in heterogeneous formations. 2. Uncertainty analysis.” Water Resour. Res., 28(5), 1377–1388.
9.
Dagan, G. (1989). Flow and transport in porous formations. Springer‐Verlag, New York, N.Y.
10.
Dagan, G. (1984). “Solute transport in heterogeneous porous formations.” J. Fluid Mech., 145, 151–177.
11.
Dagan, G., Cvetkovic, V., and Shapiro, A. (1992). “A solute flux approach to transport in heterogeneous formations. 1. The general framework.” Water Resour. Res., 28(5), 1369–1376.
12.
Dagan, G., and Neuman, S. P. (1991). “Nonasymptotic behavior of a common Eulerian approximation of transport in random velocity fields.” Water Resour. Res., 27(12), 3249–3256.
13.
Dagan, G., and Nguyen, V. (1989). “A comparison of travel time and concentration approaches to modeling transport by groundwater.” J. Contam. Hydrol., 4(1), 79–91.
14.
Einstein, A. (1956). Investigations on the theory of the Brownian movement. Dover Publishing, Inc., New York, N.Y.
15.
Folks, J. L., and Chhikara, R. S. (1978). The inverse Gaussian distribution and its statistical application—A review.” J. Royal Statistical Soc., B40(3), 263–289.
16.
Fox, R. F. (1986). “Functional calculus approach to stochastic differential equations.” Physical Review A, 33(1), 467–478.
17.
Gardiner, C. W. (1985). Handbook of stochastic methods for physics, chemistry, and the natural sciences. 2nd Ed., Springer‐Verlag, New York, N.Y.
18.
Gelhar, L. W., and Axness, C. L. (1983). “Three‐dimensional stochastic analysis of macrodispersion in aquifers.” Water Resour. Res., 19(1), 161–180.
19.
Gihman, I. I., and Skorohod, A. V. (1972). Stochastic differential equations. Springer‐Verlag, New York, N.Y.
20.
Gupta, V. K., Bhattacharya, R. N., and Sposito, G. (1981). A molecular approach to the foundations of the theory of solute transport in porous media.” J. Hydrol., 50(1/3), 355–370.
21.
Hathhorn, W. E. (1990). “Diffusion theory and the fluid passage time problem in a porous media,” PhD dissertation, The University of Texas, Austin, Texas.
22.
Jury, W. A. (1982). “Simulation of solute transport using a transfer function model.” Water Resour. Res., 18(2), 363–368.
23.
Jury, W. A., Sposito, G., and White, R. E. (1986). “A transfer function model of solute transport through soil. 1. Fundamental concepts.” Water Resour. Res., 22(2), 243–247.
24.
Knight, J. H., and Smiles, D. E. (1977). “The application of the Markovian hypothesis to the theory of soil water movement: A criticism.” Soil Sci. Soc. Am. J., 41(4), 827–828.
25.
Knighton, R. E., and Wagenet, R. J. (1987). “Simulation of solute transport using a continuous time Markov process. 1. Theory and steady state application.” Water Resour. Res., 23(10), 1911–1916.
26.
Laroussi, C., and De Backer, L. (1977). “Markovian approach to soil water movement.” Soil Sci. Soc. Am. J., 41(4), 829–830.
27.
Lin, Y. T., and Tierney, M. S. (1986). “Preliminary estimates of groundwater travel time and radionuclide transport at the Yucca Mountain repository site.” SAND85‐2701, Sandia Nat. Lab., Albuquerque, N.M.
28.
Massmann, J., and Freeze, R. A. (1987). “Groundwater contamination from waste management sites: The interaction between risk‐based engineering design and regulatory policy 1 methodology.” Water Resour. Res., 23(2), 351–367.
29.
Moran, P. A. P. (1968). An introduction to probability theory. Oxford Univ. Press, New York, N.Y.
30.
Nelson, R. W. (1978). Evaluating the environmental consequences of groundwater contamination. 1. An overview of contaminant arrival distributions as general evaluation requirements.” Water Resour. Res., 14(3), 409–415.
31.
Rao, P. V., Portier, K. M., and Rao, P. S. C. (1981). “A stochastic approach for describing convective‐dispersive solute transport in saturated porous media.” Water Resour. Res., 17(4), 963–968.
32.
Roy, L. K., and Wasan, M. T. (1968). “The first‐passage time distribution of Brownian motion with positive drift.” Math. Biosci., 3, 191–204.
33.
Simmons, C. S. (1982). “A stochastic‐convective transport representation of dispersion in one‐dimensional porous media systems.” Water Resour. Res., 18(4), 1193–1214.
34.
Shapiro, A. M., and Cvetkovic, V. D. (1988). “Stochastic analysis of solute arrival time in heterogeneous porous media.” Water Resour. Res., 24(10), 1711–1718.
35.
Sposito, G., Jury, W. A., and Gupta, V. K. (1986a). “Fundamental problems in the stochastic convection‐dispersion model of solute transport in aquifers and field soils.” Water Resour. Res., 22(1), 77–88.
36.
Sposito, G., White, R. E., Darrah, P. R., and Jury, W. A. (1986b). “A transfer function model of solute transport through soil 3. The convective‐dispersion equation.” Water Resour. Res., 22(2), 255–262.
37.
Tweedie, M. C. K. (1957a). “Statistical properties of inverse Gaussian distributions. I.” Ann. Math. Stats., 28(2), 362–377.
38.
Tweedie, M. C. K. (1957b). “Statistical properties of inverse Gaussian distributions. II.” Ann. Math. Stats., 28(3), 696–705.
39.
White, R. E., Dyson, J. S., Haigh, R. A., Jury, W. A., and Sposito, G. (1986). “A transfer function model of solute transport through soil. 2. Illustrative example.” Water Resour. Res., 22(4), 248–254.
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Published In
Journal of Hydraulic Engineering
Volume 120 • Issue 2 • February 1994
Pages: 134 - 146
Copyright
Copyright © 1994 American Society of Civil Engineers.
History
Received: May 11, 1992
Published online: Feb 1, 1994
Published in print: Feb 1994
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