Abstract
This article compares for the first time, local longitudinal and transverse dispersion coefficients obtained by homogenization with experimental data of dispersion coefficients in porous media, using the correct porosity dependence. It is shown that the longitudinal dispersion coefficient can be reasonably represented by a simple periodic unit cell (PUC), which consists of a single sphere in a cube. We present a slightly modified and simplified approach to derive the homogenized equations, which emphasizes physical aspects of homogenization. Subsequently, we give full dimensional expressions for the dispersion tensor based on a comparison with the convective dispersion equation used for contaminant transport, inclusive the correct dependence on porosity. For the PUC of choice, the dispersion relations are identical to the relations obtained for periodic media. We show that commercial finite element software can be readily used to compute longitudinal and transverse dispersion coefficients in 2D and 3D. The 3D results are for the first time obtained at relevant Peclet numbers. There is good agreement for longitudinal dispersion. The computed transverse dispersion coefficients for a single sphere in a cube are much too low. The effect of adsorption on the dispersion coefficient is also studied. Adsorption does not affect the transverse dispersion coefficient. However, adsorption enhances the longitudinal dispersion coefficient in agreement with an analysis of homogenization applied to Taylor dispersion discussed in the literature.
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Abbreviations
- c (x, y, z, t):
-
Tracer concentration
- c a :
-
Equilibrium surface concentration
- c s :
-
Absorbed concentration (Eq. 2.6)
- c (n) :
-
See Eq. 2.7
- D :
-
Longitudinal or transverse component of D
- D m :
-
Effective molecular diffusion tensor
- D d :
-
Hydrodynamic dispersion tensor
- D 0 :
-
Molecular diffusion coefficient
- \({{D}_{xx}^{\rm d},{D}_{xx}^{\rm m}}\) :
-
Longitudinal component of D d and D m
- \({{D}_{yy}^{\rm d},{D}_{yy}^{\rm m}}\) :
-
Transverse component of D d and D m
- D :
-
D m + D d
- K :
-
Distribution coefficient
- L :
-
Characteristic macroscopic length
- ℓ :
-
Characteristic length of the PUC
- N :
-
Number of dimensions
- n :
-
Outward unit normal
- PUC:
-
Periodic unit cell
- p :
-
Pressure
- Pe :
-
Peclet number
- \({{\langle Q\rangle}}\) :
-
Average of quantity Q
- Q R, Q D :
-
Reference and dimensionless quantity
- R :
-
Retardation
- r b :
-
Big scale (global) coordinate
- r s :
-
Small scale (local) coordinate
- t :
-
Time
- u :
-
Darcy velocity
- u inj :
-
Darcy injection velocity
- \({\overline{{\bf u}}}\) :
-
Average Darcy velocity vector
- v :
-
Fluid velocity
- \({\overline{{\bf v}}}\) :
-
Average interstitial velocity vector
- v R :
-
Reference velocity
- Γ:
-
Grain boundary
- δ :
-
Thickness of sorption layer
- \({{\partial\Omega}}\) :
-
Outer boundary of PUC
- \({{\varepsilon}}\) :
-
Scaling parameter \({\ell/L\ll 1}\)
- μ :
-
Viscosity
- \({{\varphi}}\) :
-
Porosity
- \({\overrightarrow{\mathbf{\chi}}}\) :
-
\({{c}^{(1)}{=\overrightarrow{\mathbf{\chi}}\cdot}\mathbf{grad}_{\rm b}}\) c (0)
- \({{\chi}_{x}}\) :
-
x-Component of \({\overrightarrow{\chi}}\)
- Ω:
-
Total domain of PUC
- Ωl :
-
Fluid domain in the PUC
- Ωm :
-
Grain domain in the PUC
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Acknowledgments
This article is the result of the Master Thesis work of Aiske Rijnks. In an early stage, we had a three-day discussion with Marjan Smit, Andrea Cortis, Ruud Schotting, and Jacob Bear leading to an inventory of the physical assumptions underlying homogenization. We thank Hamidreza Salimi for many useful suggestions. We would like to acknowledge Fred Vermolen for valuable suggestions on the numerical implementation. Furthermore, we thank Sorin Pop for contributing to the discussion on the physical perspective of homogenization, and for reading this manuscript. We thank Florian Kleinendorst (COMSOL) for his suggestion to use the appropriate Neumann condition in the periodic boundary conditions. An unexpected meeting with Pierre Tardif d’Hamonville during a Marie-Curie programme (GRASP) sponsored workshop on CO2 sequestration led to enlightening discussions. Finally, countless excellent comments of the referees and audiences in oral presentations have greatly contributed to this article. The study described here was supported by The Netherlands Organization for Scientific Research (NWO) for the project “Solubility/mobility of arsenic under changing redox conditions” (2001–2004).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bruining, H., Darwish, M. & Rijnks, A. Computation of the Longitudinal and Transverse Dispersion Coefficient in an Adsorbing Porous Medium Using Homogenization. Transp Porous Med 91, 833–859 (2012). https://doi.org/10.1007/s11242-011-9875-x
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DOI: https://doi.org/10.1007/s11242-011-9875-x