Abstract
A simple theoretical model is described for deriving a 1-dimensional equation for the spreading of a tracer in a steady flow at the field scale. The originality of the model is to use a stochastic appoach not in the 3-dimensional space but in the 1-D space of the stream tubes. The simplicity of calculation comes from the local relationship between permeability and velocity in a 1-D flow. The spreading of a tracer front is due to local variations in the cross-sectional area of the stream tubes, which induces randomness in travel time. The derived transport equation is averaged in the main flow direction. It differs from the standard dispersion equation. The roles of time and space variables are exchanged. This result can be explained by using the statistical theory of Continuous Time Random Walk instead of a standard Random Walk. However, the two equations are very close, since their solutions have the same first and second moments. Dispersivity is found to be equal to the product of the correlation length by the variance of the logarithm of permeability, a result similar to Gelhar's macrodispersion.
Similar content being viewed by others
Abbreviations
- A :
-
total cross-section area of the sample
- C :
-
(resident) concentration of tracer
- D,D * :
-
dispersion coefficient
- F :
-
flux of tracer
- G :
-
probability distribution function for permeability in the stream-tube segments
- I :
-
tracer intensity (mass crossing a surface per unit time)
- K :
-
permeability
- L :
-
length of the medium
- M :
-
number of stream tubes in the medium
- N :
-
number of segments along a stream tube
- P :
-
pressure
- Q :
-
total flow rate in the sample
- a :
-
length of an elementary stream-tube segment
- g :
-
probability distribution function for permeability in the space
- i, j :
-
indices, tube numbers
- q :
-
flow rate in each stream tube
- s :
-
variable cross-section area of a stream tube
- t, t′ :
-
time
- u :
-
front velocity
- x :
-
space variable in the flow direction
- ε :
-
small local variation in time
- α, α t :
-
longitudinal, transverse dispersivity
- Φ :
-
porosity of the porous medium
- λ :
-
correlation length in the permeability field
- Μ :
-
viscosity of the fluid
- Τ :
-
time for filling an elementary stream tube segment
- σ :
-
standard deviation of a stochastic variable
- γ:
-
probability distribution of arrival times (Gaussian)
References
Araktingi, U. G. and Orr, Jr, F. M., 1988, Viscous fingering in heterogeneous porous media, SPE 18095, Proc. 63rd annual conference of the Soc. of Petrol Eng., Houston.
Arya, A., Hewett, T. A., Larson, R., and Lake, L. W., 1985, Dispersion and reservoir heterogeneity, SPE 14364, Proc. 60th annual conference of the Soc. of Petrol. Eng., Las Vegas.
Avrain, T., PhD thesis, 1993, Influence des hétérogénéités et du contraste de viscosité sur les déplacements de fluides miscibles en milieu poreux: expériences et modélisation, University Paris VI (in French).
Bear, J., 1988,Dynamics of Fluids in Porous Media, Dover, New York.
Bouchaud, J.-P. and Georges, A., 1990, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,Physics Reports 195, 128–293.
Dagan, G., 1988, Time-dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers,Water Res. Res. 24, 1491–1500.
Feder, J. 1988,Fractals, Plenum Press, New York.
Fogler, H. S., 1992,Segments of Chemical Reaction Engineering, Prentice-Hall International Series in the Physical and Chemical Engineering Sciences, Prentice-Hall, New Jersey.
Fried, J. J. and Combarnous, M., 1971, Dispersion in porous media, Adv. Hydrosci.7, 169–282.
Gelhar, L. W. and Axness, C. L., 1983, Three-dimensional stochastic analysis of macrodispersion in aquifers,Water Res. Res. 19, 161–180.
Hewett, T. A. and Behrens, R. A., 1990, Considerations affecting the scaling of displacements in heterogeneous permeability distributions, SPE 20739, proceedings of the 65th annual conference of the Soc. of Petrol. Eng., New Orleans.
Jury, W. A. and Uttermann, J., 1992, Solute transport through layered soil profiles: zero and perfect travel time correlation models,Transport in Porous Media 8, 277–297.
Klafter, J., Blumen, A., and Shlesinger, M. F., 1987, Stochastic pathway to anomalous diffusion,Phys. Rev. A. 35, 3081–3083.
Lenormand, R. and Wang, B., 1995, A streamtube model for miscible flow. Part 2: macrodispersion in porous media with long-range correlations,Transport in Porous Media 18, 263–282, this issue.
Neuman, S. P., Winter, C. L., and Newman, C. M., 1987, Stochastic theory of field-scale fickian dispersion in anisotropic porous media,Water Res. Res. 23, 1453–466.
Perkins, T. K. and Johnston, O. C., 1963, A review of dispersion in porous media,Soc. Petrol Eng. J., 70–84.
Renard, G., 1990, A 2D Reservoir streamtube EOR model with periodical automatic regeneration of streamlines,In Situ 14, 175–200.
Rinaldo, A., Maran, A. and Rigon, R., 1991, Geomorphological dispersion,Water Res. Res. 27, 513–525.
Sahimi, M. and Imdakm, A. O., 1988, The effect of morphological disorder on hydrodynamic dispersion in flow through porous media,J. Phys. A: Math. Gen. 21, 3833–3870.
Shapiro, A. M. and Cvetkovic, V. D., 1988, Stochastic analysis of solute arrival time in heterogeneous porous media,Water Res. Res. 24, 1711–1718.
Sposito, G., Jury, W. A. and Gupta, V. K., 1986, Fundamental problems in the stochastic convection-dispersion model of solute transport in aquifers and field soils,Water Res. Res. 22, 77–88.
Van Kampen, 1981,Stochastic Processes in Physics and Chemistry, Elsevier Science Publishers, Amsterdam.
Warren, J. E., 1964, Macroscopic dispersion,Soc. Petrol. Eng. J., 215–230.
Yortsos, Y. C., 1991, A theoretical analysis of vertical flow equilibrium, SPE 22612, Proc. 66th annual conference of the Soc. of Petrol Eng., Dallas.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lenormand, R. A stream tube model for miscible flow. Transp Porous Med 18, 245–261 (1995). https://doi.org/10.1007/BF00616934
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00616934