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Correspondence Between One- and Two-Equation Models for Solute Transport in Two-Region Heterogeneous Porous Media

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Abstract

In this work, we study the transient behavior of homogenized models for solute transport in two-region porous media. We focus on the following three models: (1) a time non-local, two-equation model (2eq-nlt). This model does not rely on time constraints and, therefore, is particularly useful in the short-time regime, when the timescale of interest (t) is smaller than the characteristic time (τ 1) for the relaxation of the effective macroscale parameters (i.e., when t ≤ τ 1); (2) a time local, two-equation model (2eq). This model can be adopted when (t) is significantly larger than (τ 1) (i.e., when \({t\gg\tau_{1}}\)); and (3) a one-equation, time-asymptotic formulation (1eq ∞). This model can be adopted when (t) is significantly larger than the timescale (τ 2) associated with exchange processes between the two regions (i.e., when \({t\gg\tau_{2}}\)). In order to obtain insight into this transient behavior, we combine a theoretical approach based on the analysis of spatial moments with numerical and analytical results in several simple cases. The main result of this paper is to show that there is only a weak asymptotic convergence of the solution of (2eq) towards the solution of (1eq ∞) in terms of standardized moments but, interestingly, not in terms of centered moments. The physical interpretation of this result is that deviations from the Fickian situation persist in the limit of long times but that the spreading of the solute is eventually dominating these higher order effects.

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Abbreviations

b ij :

Closure mapping vector in the i-region associated with \({{\nabla}\langle c_{j}\rangle^{j} ({\rm m})}\)

c i :

Pointwise solute concentration in the i-region (mol m−3)

\({\langle c_{i}\rangle}\) :

Superficial spatial average of c i (mol m−3)

\({\langle c_{i}\rangle^{i}}\) :

Intrinsic spatial average of c i (mol m−3)

\({\langle c\rangle^{\gamma\omega}}\) :

Weighted spatial average concentration (mol m−3)

\({\tilde{c}_{i}}\) :

Solute concentration standard deviation in the i−region (mol m−3)

\({\boldsymbol{{D}}_{i}}\) :

Diffusion tensor in the i−region (m2 s −1)

\({\boldsymbol{{D}}_{ij}}\) :

Dispersion tensor in the two-equation models associated with \({\partial_{t}\langle c_{i}\rangle^{i}}\) and \({\Delta\langle c_{j}\rangle^{j}\; ({m}^{2} {s}^{-1})}\)

D ij :

Dispersion coefficient in the 1-D two-equation models associated with \({\partial_{t}\langle c_{i}\rangle^{i}}\) and \({\Delta\langle c_{j}\rangle^{j}}\) (m2 s −1)

\({\boldsymbol{{D}}^{\infty}}\) :

Dispersion tensor of the one-equation time-asymptotic model (m2 s −1)

D ∞ :

Dispersion coefficient of the 1-D one-equation time-asymptotic model (m2 s−1)

exp:

Exponentially decaying terms (−)

h :

Transient effective mass exchange kernel (s−1)

h ∞ :

Effective mass exchange coefficient (s−1)

\({\tilde{{\bf j}}_{i}}\) :

Deviation of the total mass flux for region i (mol m−2 s−1)

J i :

Average of the total mass flux for region i (mol m−2 s−1)

L :

Characteristic length of the field-scale (m)

ℓ i :

Characteristic length of the i-region (m)

m i n :

nth-order centered moment associated with \({\langle c_{i}\rangle^{i}}\) for the two-equation model (mn mol)

\({m_{n}^{\gamma\omega}}\) :

nth-order centered moment associated with \({\langle c\rangle^{\gamma\omega}}\) for the two-equation model (mn mol)

m ∞ n :

nth-order centered moment associated with \({\langle c\rangle^{\gamma\omega}}\) for the one-equation asymptotic model (mn mol)

\({m_{n}^{\gamma\omega}}\) :

nth-order standardized moment associated with \({\langle c\rangle^{\gamma\omega}}\) for the two-equation model (−)

\({M_{n}^{\infty}}\) :

nth-order standardized moment associated with \({\langle c\rangle^{\gamma\omega}}\) for the one-equation asymptotic model (−)

n ij :

Normal unit vector pointing from the i-region towards the j-region (−)

p k :

Three lattice vectors that are needed to describe the 3-D spatial periodicity (m)

Q i(x, t):

Macroscopic source term in the i-region (mol m−3 s −1)

Q γω :

Weighted macroscopic source term (mol m−3 s −1)

R :

Radius of the REV, (m)

\({\mathcal{S}_{ij}}\) :

Boundary between the i-region and the j-region (−)

S ij :

Area associated with \({\mathcal{S}_{ij} ({\rm {m}}^{2})}\)

r i :

Closure parameter in the i-region associated with \({\langle c_{\gamma}\rangle^{\gamma}-\langle c_{\omega}\rangle^{\omega}\; (-)}\)

t :

Time (s)

t’:

Non-dimensionalized time (−)

T :

Period of the oscillations (s)

v i :

Velocity at the microscale in the i-region (m s−1)

\({\langle{\bf v}_{i}\rangle}\) :

Superficial spatial average of v i (m s−1)

\({\langle{\bf v}_{i}\rangle^{i}}\) :

Intrinsic spatial average of v i (m s−1)

\({\langle{v}_{i}\rangle^{i}}\) :

Norm of the intrinsic spatial average of v i (m s−1)

\({\tilde{{\bf v}}_{i}}\) :

Velocity standard deviation in the i-region (m s−1)

V ij :

Effective velocity in the two-equation models associated with \({\partial_{t}\langle c_{i}\rangle^{i}}\) and \({\nabla\langle c_{j} \rangle^{j}\; ({m} {s}^{-1})}\)

V ∞ :

Effective velocity of the one-equation time-asymptotic model (m s−1)

\({\mathcal{V}_{i}}\) :

Domain of the averaging volume that is identified with the i-region (−)

V i :

Volume of the domain \({\mathcal{V}_{i} ({\rm {m}}^{3})}\)

\({\mathcal{V}}\) :

Domain of the averaging volume (−)

V :

Volume of the domain \({\mathcal{V}\; ({m}^{3})}\)

α:

Weighted mass transfer coefficient, \({h_{\infty}\left(\frac{1}{\Phi_{\gamma}\varepsilon_{\gamma}} +\frac{1}{\Phi_{\omega}\varepsilon_{\omega}}\right) ({\rm {s}}^{-1})}\)

\({\boldsymbol{\beta}_{1}^{*}}\) and \({\boldsymbol{\beta}_{2}^{*}}\) :

Source terms in the closure problems (m s−1)

γ-region:

First region (−)

ΔV :

Velocity contrast between the γ and ω regions, V γγ−V ωω (m s−1)

ΔD :

Dispersion contrast between the γ and ω regions, D γγ−D ωω (m2 s −1)

Φ i :

i−region volume fraction (−)

\({\varepsilon_{i}}\) :

Darcy-scale fluid fraction (porosity) within the i-region (−)

ω-region:

Second region (−)

τ 1 :

Characteristic time for the relaxation of the two-equation model effective parameters (s)

τ 2 :

Characteristic time for the transition towards the one-equation asymptotic regime (s)

μ i n :

nth-order raw moment associated with the \({\langle c_{i}\rangle^{i}}\) for the two-equation model (mn mol)

μ γω n :

nth-order raw moment associated with the \({\langle c\rangle^{\gamma\omega}}\) for the two-equation model (mn mol)

μ ∞ n :

nth-order raw moment associated with the \({\langle c\rangle^{\gamma\omega}}\) for the one-equation asymptotic model (mn mol)

i, j :

Indices for γ or ω (−)

References

  • Ahmadi A., Quintard M., Whitaker S.: Transport in chemically and mechanically heterogeneous porous media, V, Two-equation model for solute transport with adsorption. Adv. Water Resour. 22, 59–86 (1998)

    Article  Google Scholar 

  • Aris R.: Dispersion of linear kinematic waves. Proc. Roy. Soc. Ser. A 245, 268–277 (1958)

    Article  Google Scholar 

  • Bensoussan A., Lions J., Papanicolau G.: Asymptotic Analysis for Periodic Structures. North-Holland Publishing Company, Amsterdam (1978)

    Google Scholar 

  • Brenner H.: Dispersion resulting from flow through spatially periodic porous media. Proc. Roy. Soc. Lond. Ser. A 297(1430), 81–133 (1980)

    Google Scholar 

  • Carslaw H., Jaeger J.: Conduction of Heat in Solids. Clarendon Press, Oxford (1946)

    Google Scholar 

  • Chastanet J., Wood B.: The mass transfer process in a two-region medium. Water Resour. Res. 44, W05413 (2008)

    Article  Google Scholar 

  • Cherblanc F., Ahmadi A., Quintard M.: Two-medium description of dispersion in heterogeneous porous media: Calculation of macroscopic properties. Water Resour. Res. 39(6), SBH6–1 (2003)

    Article  Google Scholar 

  • Cherblanc F., Ahmadi A., Quintard M.: Two-domain description of solute transport in heterogeneous porous media: Comparison between theoretical predictions and numerical experiments. Adv. Water Resour. 30(5), 1127–1143 (2007)

    Article  Google Scholar 

  • Chiogna G., Cirpka O., Grathwohl P., Rolle M.: Relevance of local compound-specific transverse dispersion for conservative and reactive mixing in heterogeneous porous media. Water Resour. Res. 47, W07540 (2011)

    Article  Google Scholar 

  • Coats K., Smith B.: Dead end pore volume and dispersion in porous media. Soc. Petrol. Eng. J. 4, 73–84 (1964)

    Google Scholar 

  • Cushman J.: The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles. Kluwer Academic Press, Norwell (1997)

    Google Scholar 

  • Cushman J., Ginn T.: non-local dispersion in media with continuously evolving scales of heterogeneity. Transp. Porous Media 13(1), 123–138 (1993)

    Article  Google Scholar 

  • Cushman J., Hu B.: A resumé of non-local transport theories. Stoch. Hydrol. Hydraul. 9(2), 105–116 (1995)

    Article  Google Scholar 

  • Cushman J., Hu B., Deng F.: non-local reactive transport with physical and chemical heterogenity: Localization errors. Water Resour. Res. 31(9), 2219–2237 (1995)

    Article  Google Scholar 

  • Cushman J.H., Bennethum L.S., Hu B.X.: A primer on upscaling tools for porous media. Adv. Water Resour. 25(8-12), 1043–1067 (2002)

    Article  Google Scholar 

  • Dagan G.: Flow and Transport in Porous Formations. Springer, New York (1989)

    Book  Google Scholar 

  • Davarzani H., Marcoux M., Quintard M.: Theoretical predictions of the effective thermodiffusion coefficients in porous media. Int. J. Heat Mass Transfer 53, 1514–1528 (2010)

    Article  Google Scholar 

  • Davit Y., Quintard M.: Comment on “Frequency-dependent dispersion in porous media”. Phys. Rev. E 86, 013201 (2012)

    Article  Google Scholar 

  • Davit Y., Quintard M., Debenest G.: Equivalence between volume averaging and moments matching techniques for mass transport models in porous media. Int. J. Heat Mass Transfer 53, 4985–4993 (2010)

    Article  Google Scholar 

  • Govindaraju R., Bhabani D.: Moment Analysis for Subsurface Hydrologic Applications (Water Science and Technology Library). Springer, New York (2007)

    Google Scholar 

  • Gray W., Leijnse A., Kolar R., Blain C.: Mathematical Tools for Changing Spatial Scales in the Analysis of Physical Systems. CRC Press, Boca Raton (1993)

    Google Scholar 

  • Haggerty R., Gorelick S.: Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity. Water Resour. Res. 31(10), 2383–2400 (1995)

    Google Scholar 

  • Haggerty R., Harvey C., Freiherrvon Schwerin C., Meigs L.: What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results. Water Resour. Res. 40, W01510 (2004)

    Article  Google Scholar 

  • Haggerty R., McKenna S., Meigs L.: On the late-time behavior of tracer test breakthrough curves. Water Resour. Res. 36(12), 3467–3479 (2000)

    Article  Google Scholar 

  • Howes F., Whitaker S.: The spatial averaging theorem revisited. Chem. Eng. Sci. 40(8), 1387–1392 (1985)

    Article  Google Scholar 

  • Kfoury M., Ababou R., Noetinger B., Quintard M.: Matrix-fracture exchange in a fractured porous medium: stochastic upscaling. Comptes Rendus Mécanique 332(8), 679–686 (2004)

    Google Scholar 

  • Kfoury M., Ababou R., Noetinger B., Quintard M.: Upscaling fractured heterogeneous media: Permeability and mass exchange coefficient. J. Appl. Mech. Trans. ASME 73(1), 41–46 (2006)

    Article  Google Scholar 

  • Koch D., Brady J.: A non-local description of advection–diffusion with application to dispersion in porous media. J. Fluid Mech. Dig. Arch. 180, 387–403 (1987)

    Article  Google Scholar 

  • Koch J., Brady J.: Anomalous diffusion in heterogeneous porous media. Phys. Fluids 31, 965–973 (1988)

    Article  Google Scholar 

  • Landereau P., Noetinger B., Quintard M.: Quasi-steady two-equation models for diffusive transport in fractured porous media: large-scale properties for densely fractured systems. Adv. Water Resour. 24, 863–876 (2001)

    Article  Google Scholar 

  • Le Borgne T., Dentz M., Bolster D., Carrera J., de Dreuzya J., Davya P.: Non-fickian mixing: Temporal evolution of the scalar dissipation rate in heterogeneous porous media. Adv. Water Resour. 33, 1468–1475 (2010)

    Article  Google Scholar 

  • Luo J., Cirpka O., Dentz M., Carrera J.: Temporal moments for transport with mass transfer described by an arbitrary memory function in heterogeneous media. Water Resour. Res. 44, W01502 (2008)

    Article  Google Scholar 

  • Moyne C.: Two-equation model for a diffusive process in porous media using the volume averaging method with an unsteady-state closure. Adv. Water Resour. 20(2-3), 63–76 (1997)

    Article  Google Scholar 

  • Moyne C., Didierjean S., Amaral Souto H., da Silveira O.: Thermal dispersion in porous media: one-equation model. Int. J. Heat Mass Transfer 43, 3853–3867 (2000)

    Article  Google Scholar 

  • Neuman S.: Eulerian-lagrangian theory of transport in space-time nonstationary velocity fields: Exact non-local formalism by conditional moments and weak approximation. Water Resour. Res. 29, 633–645 (1993)

    Article  Google Scholar 

  • Parker J., Valocchi A.: Constraints on the validity of equilibrium and first-order kinetic transport models in structured soils. Water Resour. Res. 22(3), 399–407 (1986)

    Article  Google Scholar 

  • Quintard M., Cherblanc F., Whitaker S.: Dispersion in heterogeneous porous media: One-equation non-equilibrium model. Transp. Porous Media 44(1), 181–203 (2001)

    Article  Google Scholar 

  • Quintard M., Whitaker S.: Transport in chemically and mechanically heterogeneous porous media III; Large-scale mechanical equilibrium and the regional form of Darcy’s law. Adv. Water Resour. 21(7), 617–629 (1998)

    Article  Google Scholar 

  • Souadnia A., Didierjean S., Moyne C.: Transient dispersion in porous media: A comparison between exact and approximate solutions in a case study. Transp. Porous Media 47, 245–277 (2002)

    Article  Google Scholar 

  • Stagnitti F., Allinson G., Morita M., Nishikawa M., Il H., Hirata T.: Temporal moments analysis of preferential solute transport in soils. Environ. Model. Assess. 5, 229–236 (2000)

    Article  Google Scholar 

  • Valdes-Parada F., Alvarez- Ramirez J.: Frequency-dependent dispersion in porous media. Phys. Rev. E 84, 031201 (2011)

    Article  Google Scholar 

  • Whitaker S.: Introduction to Fluid Mechanics. R.E. Krieger, Malabar (1981)

    Google Scholar 

  • Whitaker S.: The Method of Volume Averaging. Kluwer Academic Publishers, Dordrecht (1999)

    Google Scholar 

  • Wood B.: The role of scaling laws in upscaling. Adv. Water Resour. 32, 723–736 (2009)

    Article  Google Scholar 

  • Wood, B., Valdès-Parada, F.: Volume averaging: local and non-local closures using a green’s function approach. Adv. Water Resour. (2012). http://dx.doi.org/10.1016/j.advwatres.2012.06.008

  • Xu J., Hu B.: Eulerian spatial moments for solute transport in three-dimensional heterogeneous, dual-permeability media. Stoch. Environ. Res. 18, 47–60 (2004)

    Article  Google Scholar 

  • Young D.F., Ball W.: Column experimental design requirements for estimating model parameters from temporal moments under nonequilibrium conditions. Adv. Water Resour. 23, 449–460 (2000)

    Article  Google Scholar 

  • Zanotti F., Carbonell R.: Development of transport equations for multiphase system-1: General development for two phase system. Chem. Eng. Sci. 39(2), 263–278 (1984)

    Article  Google Scholar 

  • Zhang Y.: Moments for tempered fractional advection-diffusion equations. J. Stat. Phys. 139, 915–939 (2010)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. University of Oxford, Mathematical Institute, 24–29 St. Giles’, Oxford, OX1 3LB, UK

    Y. Davit

  2. School of Chemical, Biological and Environmental Engineering, Oregon State University, Corvallis, OR, 97331, USA

    B. D. Wood

  3. Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, 31400, Toulouse, France

    G. Debenest & M. Quintard

  4. CNRS, IMFT, 31400, Toulouse, France

    G. Debenest & M. Quintard

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  1. Y. Davit
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  2. B. D. Wood
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  3. G. Debenest
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  4. M. Quintard
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Correspondence to Y. Davit.

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Davit, Y., Wood, B.D., Debenest, G. et al. Correspondence Between One- and Two-Equation Models for Solute Transport in Two-Region Heterogeneous Porous Media. Transp Porous Med 95, 213–238 (2012). https://doi.org/10.1007/s11242-012-0040-y

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