Transport in Porous Media

, Volume 68, Issue 2, pp 187–218 | Cite as

Upscaling, relaxation and reversibility of dispersive flow in stratified porous media

  • C. W. J. BerentsenEmail author
  • C. P. J. W. van Kruijsdijk
  • M. L. Verlaan
Original Paper


Dispersive tracer released in a unidirectional velocity field belonging to a stratified porous of finite height describes a transition, called relaxation, from a convective dominated behaviour for short times to Fickian behaviour for asymptotic long times. The temporal relaxation state of the tracer is controlled by the transverse mixing term. In most practical applications, the orders of the time and length scales of the relaxation mechanism are such that in an upscaled model of a stratified medium the dispersive flux is in a pre-asymptotic state. Explicit modelling of the relaxation of the dispersive flux in the pre-asymptotic region is required to improve the accuracy. This paper derives a pre-asymptotic one-dimensional upscaled model for the transverse averaged tracer concentration. The model generalises Taylor dispersion (Proc. R. Soc. London 219, 186–203 (1953)) and extends the method of Camacho (Phys. Rev. E 47(2), 1049–1053 (1993a); Phys. Rev. E 48 (1993b)) to dispersion tensors that may vary as function of the transverse direction. In the averaging step, the governing two-dimensional equation is first spectrally decomposed in terms of the eigenfunctions of the transverse mixing term. Next, the resulting modal relaxation equations are combined into an effective relaxation equation for the extended dispersive Taylor flux. Contrary to the one-dimensional Fickian approach, the upscaled model approximates the multi-scale relaxation behaviour as a single scale relaxation process and accounts for the partial reversibility of convective dispersion upon reversal of the flow direction. The upscaled model is evaluated against the original two-dimensional model by means of moment analysis. The longitudinal tracer variance predicted by our model is quantitatively correct in the short and long time limits and is qualitatively correct for intermediate times.


Relaxation Reversibility Taylor dispersion Non-Fickian dispersion Upscaling Tracer flow Stratified flow 


\({\bf B}/{\tilde{\bf B}}/{\bf B}_N\)

[m/s] Modal velocity interaction matrix w.r.t. {eigenfunction base of −∂ y (D T (y)∂ y )/cosine Fourier base/eigenfunction base of \({\tilde {\bf D}}_{T,N}\)}.


[kg/m3] 2D concentration.


[kg/m3]Transverse or height averaged concentration.

\(c_n /\tilde {c}_n /c_{N,n}\)

[kg/m3]n-th modal concentration w.r.t. {eigenfunction base of −∂ y (D T (y)∂ y )/cosine Fourier base/eigenfunction base of \({\tilde {\bf D}}_{T,N}\)}.

\({\bf c}/{\tilde {\bf c}}\)

[kg/m3] Infinite column vector with modal concentrations \(\{c_{n}/\tilde{c}_{n}\} for n\, = 1,{\ldots},\infty\).

\({\tilde {\bf c}}_N/{\bf c}_N\)

[kg/m3] Finite column vector with modal concentrations \(\{\tilde{c}_{N, n} / {c}_{N, n}\}\) for n =  1,..., N.


[m] Total layer height or transverse width.


[m2/s] Effective dispersion coefficient.


[m2/s] Longitudinal dispersion coefficient.


[m2/s] Height or transverse averaged of longitudinal dispersion coefficient.

\({\boldsymbol D}_{L,n}/{\boldsymbol{\tilde D}}_{L,n}\)

[m2/s] n-th spectral mode of longitudinal dispersion coefficient w.r.t {eigenfunction base of −∂ y (D T (y)∂ y )/cosine Fourier base}.

\({\bf D}_{L} /{\tilde {\bf D}}_{L} /{\bf D}_{L,N}\)

[m2/s] Modal interaction matrix resulting from spectral decomposition of D L (y)c (x,y,t) w.r.t. {eigenfunction base of −∂ y (D T (y)∂ y )/ cosine Fourier base/eigenfunction base of \({\tilde {\bf D}}_{T,N}\)}.

\({\bf d}_{L} / {\tilde {\bf d}}_{L}\)

[m2/s] Infinite column vector with spectral modes of longitudinal dispersion coefficients \(\{{\boldsymbol D}_{L,n}/{\boldsymbol{\tilde D}}_{L,n}\}\) for n =  1,...,,∞.


[m2/s] Molecular diffusion coefficient.


[m2/s] Transverse dispersion coefficient.

\({\tilde {\bf D}}_T\)

[m2/s] Matrix resulting from cosine Fourier decomposition of −∂ y (D T (y)∂ y ).

\({\boldsymbol{\tilde D}}_{T,N}\)

[m2/s] \({\boldsymbol{\tilde D}}_{T}\) truncated at N modes \((={\boldsymbol{\tilde D}}_{T}(1\ldots N, 1\ldots N))\).

Eξ ,x,k

k-th non-centred non-normalised spatial moment of quantity ξ


[-] Tortuosity factor.


[-] Unity matrix.

JE,n /JE,T

[kg/m2s] Extended {modal dispersive flux/Taylor flux}.

Mξ ,x,k

k-th non-centred normalised spatial moment of quantity ξ.


[kg] Tracer mass.


[s] {Time/Reversal time}.

\({\bf T}_{\cos,\phi}\)

[-] Transformation matrix from cosine Fourier base to eigenfunction base of −∂ y (D T (y)∂ y ).


[-] Transformation matrix from cosine Fourier base truncated at n-modes to eigen function base of \({\tilde{\bf D}}_{T,N}\).


[m/s] Longitudinal velocity.


[m/s] Height or transverse averaged velocity.

\(v_n /\tilde {v}_n /v_{N,n}\)

(m/s) n-th modal velocity belonging to {eigenfunction base of −∂ y (D T (y)∂ y )/cosine Fourier base/eigenfunction base of \({\tilde {\bf D}}_{T,N}\)}.

\({\bf v}/{\tilde{\bf v}}\)

[m/s] Finite column vector with modal velocities {\({\bf v}_{n}/\tilde{\bf v}_{n}\}\) for n = 1,...,∞.

\({\tilde {\bf v}}_N /{\bf v}_N\)

[m/s] Infinite column vector with model velocities \(\{{\tilde {\bf v}}_n/{\bf v}_{N,n}\}\) for n =  1,...,N.


[m] {Longitudinal/transverse} co-ordinate.


[m] Initial longitudinal position.




[m] {Longitudinal/Transverse} dispersivity.

\(\phi _n (y)\)

[-] n-th eigenfunction of operator −∂ y (D T (y )∂ y ).

λ nN,n

[s−1] n-th eigenvalue of {operator −∂ y (D T (y)∂ y ) / matrix \({\tilde {\bf D}}_{T,N}\)}.


[s−1] (Diagonal) matrix with eigenvalues of {−∂ y (D T (y)∂ y ) / \({\tilde {\bf D}}_{T,N}\)}.




Co-variance or variance.


[m2] Spatial variance belonging to concentration.

\(\Delta \sigma _{\rm uni}\)

[m2] Asymptotic deviation of variance from Fickian behaviour for particle distributions that are initially distributed uniform over the height.


[s] \(=(\lambda _n^{-1} /\lambda _{N,n}^{-1} )\) n-th modal relaxation time.


[s] Relaxation time.


[s] Effective relaxation time.










Truncated at N modes.




With respect to the interaction of the n-th and m-th base functions (n,m >  0).


After flow reversal.




Generalised Telegraph equation.





For asymptotic long times.




[-] W.r.t cosine Fourier base.


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  1. Aris R. (1956) On the dispersion of a solute in a fluid through a tube. Proc. R. Soc. London, Ser. A 235, 67–77Google Scholar
  2. Bear, J.: Dynamics of Fluid in Porous Media. Dover publications, INC, New York, ISBN 0-486-65675-6 (1972)Google Scholar
  3. Berentsen, C.W.J.: Upscaling of flow in porous media from a tracer perspective, PhD-dissertation, Delft University of Technology, Delft, The Netherlands, Delft University Press, ISBN 90-407-2450-4 (2003)Google Scholar
  4. Berentsen, C.W.J., Verlaan, M.L., van Kruijsdijk, C.P.J.W. Upscaling and reversibility of Taylor dispersion in heterogeneous porous media. Phys. Rev. E, 71, 046308-1–046308-16 (2005)Google Scholar
  5. Camacho J. (1993a) Thermodynamics of Taylor dispersion: Constitutive equations. Phys. Rev. E 47(2): 1049–1053CrossRefGoogle Scholar
  6. Camacho J. (1993b) Purely global model for Taylor dispersion. Phys. Rev. E 48(1): 310–321CrossRefGoogle Scholar
  7. De Josselin de Jong, G.: Longitudinal and transverse diffusion in granular deposits. Trans. Am. Geophys. Union 39(1), 67ff (1958)Google Scholar
  8. Gelhar, L.W.: Stochastic subsurface hydrology. Prentice Hall Inc, Simon & Schuster Company, Englewood Cliffs, New Jersey 07632, ISBN 0-13-846767-6 (1993)Google Scholar
  9. Gelhar L.W., Gutjahr A.L., Naff R.L. (1979) Stochastic analysis of macrodispersion in a stratified aquifer. Water Resour. Res. 15(6): 1387–1397Google Scholar
  10. Hiby, J.W.: Longitudinal and transverse mixing during single-phase flow through granular beds. Interaction between Fluids & Particles, pp. 312–325. I Chem E., London (1962)Google Scholar
  11. Lake, L.W., Hirasaki, G.J.: Taylor’s dispersion in stratified porous media. SPEJ (August 1981), 459–468 (1981)Google Scholar
  12. Marle C., Simandoux P., Pacsirsky J., Gaulier C. (1967) Etude du deplacement de fludes miscibles en milieu poreux stratifie. Revue IFP 22, 272–294Google Scholar
  13. Matheron G., de Marsily G. (1980) Is transport in porous media always diffusive? A counterexample. Water Resour. Res. 16(5): 901–917CrossRefGoogle Scholar
  14. Perkins T.K., Johnston O.C. (1963) A review of diffusion and dispersion in porous media. SPEJ, SPE reprints series 8(3): 70Google Scholar
  15. Rigord, P., Calvo, A., Hulin, J.P.: Transition to irreversibility for the dispersion of a tracer in porous media. Phys. Fluids A 2(5), 681ff (1990)Google Scholar
  16. Saffman, P.G.: A theory of dispersion in a porous medium. J. Fluid Mech. 6, 321ff (1959)Google Scholar
  17. Saffman P.G. (1960) Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries. J. Fluid Mech. 7, 194–208CrossRefGoogle Scholar
  18. Taylor G.I. (1953) Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. London, Ser. A 219: 186–203CrossRefGoogle Scholar
  19. Taylor G.I. (1954) Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion. Proc. R. Soc. London, Ser. A 225, 473–477Google Scholar
  20. Verlaan, M.L., Berentsen, C.W.J., van Kruijsdijk, C.P.J.W., Nederlof, E.F.: Convection dispersion and reversibility. European Conference on the Mathematics of Oil Recovery (ECMOR VII), Baveno, Italy (2000)Google Scholar
  21. Verlaan, M.L.: Dispersion in heterogeneous media; Fundamental issues inspired by underground gas storage. PhD Thesis, Delft, The Netherlands, ISBN 90-77017-17-8 (2001)Google Scholar
  22. Zauderer, E.: Partial differential equations of applied mathematics, 2nd edn. Wiley, New-York, ISBN 0-471-61298-7 (1989)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • C. W. J. Berentsen
    • 1
    • 2
    Email author
  • C. P. J. W. van Kruijsdijk
    • 1
    • 3
  • M. L. Verlaan
    • 1
    • 4
  1. 1.Department of GeotechnologyDelft University of TechnologyDelftThe Netherlands
  2. 2.Department of Earth Sciences, Hydrogeology groupUtrecht University3508 TA UtrechtThe Netherlands
  3. 3.Shell Canada Ltd.Calgary Research CentreCalgaryCanada
  4. 4.Improved Oil Recovery and Water ManagementShell International Exploration and Production2288 GS RijswijkThe Netherlands

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