Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-18T12:43:23.205Z Has data issue: false hasContentIssue false
  • English
  • Français

Solute transport in heterogeneous porous formations

Published online by Cambridge University Press:  20 April 2006

Gedeon Dagan
Gedeon Dagan
Affiliation:
Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel
Get access Rights & Permissions [Opens in a new window]

Abstract

Solute transport in porous formations is governed by the large-scale heterogeneity of hydraulic conductivity. The two typical lengthscales are the local one (of the order of metres) and the regional one (of the order of kilometres). The formation is modelled as a random fixed structure, to reflect the uncertainty of the space distribution of conductivity, which has a lognormal probability distribution function. A first-order perturbation approximation, valid for small log-conductivity variance, is used in order to derive closed-form expressions of the Eulerian velocity covariances for uniform average flow. The concentration expectation value is determined by using a similar approximation, and it satisfies a diffusion equation with time-dependent apparent dispersion coefficients. The longitudinal coefficients tend to constant values in both two- and three-dimensional flows only after the solute body has travelled a few tens of conductivity integral scales. This may be an exceedingly large distance in many applications for which the transient stage prevails. Comparison of theoretical results with recent field experimental data is quite satisfactory.

The variance of the space-averaged concentration over a volume V may be quite large unless the lengthscale of the initial solute body or of V is large compared with the conductivity integral scale. This condition is bound to be obeyed for transport at the local scale, in which case the concentration may be assumed to satisfy the ergodic hypothesis. This is not generally the case at the regional scale, and the solute concentration is subjected to large uncertainty. The usefulness of the prediction of the concentration expectation value is then quite limited and the dispersion coefficients become meaningless.

In the second part of the study, the influence of knowledge of the conductivity and head at a set of points upon transport is examined. The statistical moments of the velocity and concentration fields are computed for a subensemble of formations and for conditional probability distribution functions of conductivity and head, with measured values kept fixed at the set of measurement points. For conditional statistics the velocity is not stationary, and its mean and variance vary throughout the space, even if its unconditional mean and variance are constant. The main aim of the analysis is to examine the reduction of concentration coefficient of variation, i.e. of its uncertainty, by conditioning. It is shown that measurements of transmissivity on a grid of points can be effective in reducing concentration variance, provided that the distance between the points is smaller than two conductivity integral scales. Head conditioning has a lesser effect upon variance reduction.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 145 , August 1984 , pp. 151 - 177
Copyright
© 1984 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beran, M. J. 1968 Statistical Continuum Theories. Interscience.
Buyevich, Yu. A., Leonov, A. J. & Safrai, V. M. 1969 Variations in filtration velocity due to random large-scale fluctuations of porosity. J. Fluid Mech. 37, 371–381.Google Scholar
Clifton, P. M. & Neuman, S. P. 1982 Effects of kriging and inverse modeling on conditional simulation of the Avra Valley aquifer in Southern Arizona. Water Resources Res. 18, 1215–1237.Google Scholar
Dagan, G. 1982a Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 1. Conditional simulation and the direct problem. Water Resources Res. 18, 813–833.Google Scholar
Dagan, G. 1982b Stochastic modeling of groundwater flow by unconditional and conditional by probabilities, 2. The solute transport. Water Resources Res. 18, 835–848.Google Scholar
Delhomme, J. P. 1979 Spatial variability and uncertainty in groundwater flow parameters: a geostatistical approach. Water Resources Res. 15, 269–280.Google Scholar
Devary, J. L. & Doctor, P. G. 1982 Pore velocity uncertainties estimation. Water Resources Res. 18, 1157–1164.Google Scholar
Freeze, R. A. 1975 A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media. Water Resources Res. 11, 725–741.Google Scholar
Fried, J. J. & Combarnous, M. A. 1971 Dispersion in porous media. Adv. Hydrosci. 7, 169–282.Google Scholar
Gelhar, L. J. & Axness, C. L. 1983 Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resources Res. 19, 161–180.Google Scholar
Gelhar, L. W., Gtjtjahr, A. L. & Naff, R. L. 1979 Stochastic analysis of macrodispersion in a stratified aquifer. Water Resources Res. 15, 1387–1397.Google Scholar
Journel, A. G. & Huijbregts, C. J. 1978 Mining Geostatistics. Academic Press.
Lallemand-Barres, P. & Peaudecerf, P. 1978 Recherche des relations entre la valeur de la dispersivité macroscopique d'un milieu aquifere, ses autres caracteristiques et les conditions de mesure. Etude bibliographique. Bull. BRGM., section III no. 4, 277–284.Google Scholar
Mathèron, G. & de Marsily, G. 1980 Is transport in porous media always diffusive? A counterexample. Water Resources Res. 16, 901–917.Google Scholar
Mood, A. M. F. & Graybill, F. A. 1963 Introdudionto the Theory of Statistics, 2nd edn. McGraw-Hill..
Phythian, R. 1975 Dispersion by random velocity fields. J. Fluid Mech. 67, 145–153.Google Scholar
Simmons, C. S. 1982 A stochastic-convective ensemble method for representing dispersive transport in groundwater. EPRI Palo-Alto, Calif., Tech. Rep. CS-2558 RP1406-1.Google Scholar
Smith, L. & Schwartz, F. W. 1980 Mass transport, 1. Stochastic analysis of macro-dispersion. Water Resources Res. 16, 303–313.Google Scholar
Sudicky, E. A., Cherry, J. A. & Frind, E. O. 1980 Hydrologic studies of a sandy aquifer at an abandoned land fill, Part 4. A natural gradient tracer test. Rep. under Contract OSU79-00195, Dept Earth Sci. Univ. Waterloo.Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Land. Math. Soc. (2) 20, 196–212.Google Scholar