Transport in Porous Media

, Volume 95, Issue 1, pp 213–238 | Cite as

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

Article

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 (τ1) for the relaxation of the effective macroscale parameters (i.e., when t ≤ τ1); (2) a time local, two-equation model (2eq). This model can be adopted when (t) is significantly larger than (τ1) (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 (τ2) 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.

Keywords

Porous media Homogenization Volume averaging Dispersion Spatial moments 

List of Symbols

Variables

bij

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

ci

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

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

Superficial spatial average of ci (mol m−3)

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

Intrinsic spatial average of ci (mol m−3)

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

Weighted spatial average concentration (mol m−3)

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

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

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

Diffusion tensor in the iregion (m2 s −1)

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

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

Dij

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)

Ji

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

L

Characteristic length of the field-scale (m)

i

Characteristic length of the i-region (m)

mni

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)

mn

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 (−)

nij

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

pk

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

Qi(x, t)

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

Qγω

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

R

Radius of the REV, (m)

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

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

Sij

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

ri

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)

vi

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

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

Superficial spatial average of vi (m s−1)

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

Intrinsic spatial average of vi (m s−1)

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

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

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

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

Vij

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 (−)

Vi

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

\({\mathcal{V}}\)

Domain of the averaging volume (−)

V

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

Greeks

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, VγγVωω (m s−1)

ΔD

Dispersion contrast between the γ and ω regions, DγγDωω (m2 s −1)

Φi

iregion volume fraction (−)

\({\varepsilon_{i}}\)

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

ω-region

Second region (−)

τ1

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

τ2

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

μni

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

μnγω

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

μn

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

Subscript

i, j

Indices for γ or ω (−)

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Y. Davit
    • 1
  • B. D. Wood
    • 2
  • G. Debenest
    • 3
    • 4
  • M. Quintard
    • 3
    • 4
  1. 1.University of Oxford, Mathematical InstituteOxfordUK
  2. 2.School of Chemical, Biological and Environmental EngineeringOregon State UniversityCorvallisUSA
  3. 3.Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse)ToulouseFrance
  4. 4.CNRS, IMFTToulouseFrance

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