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SRNL-STI-2011-00583 27 February 2012
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Ground Water Technical Commentary
1
Effective Porosity Implies Effective Bulk Density in Sorbing Solute Transport
2
G. P. Flach
3
Savannah River National Laboratory
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The concept of an effective porosity is widely used in solute transport modeling to account for
5
the presence of a fraction of the medium that effectively does not influence solute migration,
6
apart from taking up space. This non-participating volume or ineffective porosity plays the
7
same role as the gas phase in single-phase liquid unsaturated transport: it increases pore
8
velocity, which is useful towards reproducing observed solute travel times. The prevalent use of
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the effective porosity concept is reflected by its prominent inclusion in popular texts, e.g., de
10
Marsily (1986), Fetter (1988, 1993) and Zheng and Bennett (2002).
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The purpose of this commentary is to point out that proper application of the concept for
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sorbing solutes requires more than simply reducing porosity while leaving other material
13
properties unchanged. More specifically, effective porosity implies the corresponding need for
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an effective bulk density in a conventional single-porosity model. The reason is that the
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designated non-participating volume is composed of both solid and fluid phases, both of which
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must be neglected for consistency. Said another way, if solute does not enter the ineffective
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porosity then it also cannot contact the adjoining solid. Conceptually neglecting the fluid
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portion of the non-participating volume leads to a lower (effective) porosity. Likewise,
19
discarding the solid portion of the non-participating volume inherently leads to a lower or
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effective bulk density. In the author's experience, practitioners virtually never adjust bulk
21
density when adopting the effective porosity approach.
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Effective bulk density is easily derived in terms of assumed effective porosity. The following
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exercise assumes that the participating and non-participating volumes have the same pore
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scale porosity and solid density, but that is not required. Let V = total volume, V
f
= fluid volume,
25
φ = V
f
/V = porosity, M
s
= solid mass, ρ
s
= M
s
/(V-V
f
) = solid density, ρ
b
= M
s
/V = (1- φ)ρ
s
= bulk
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density, V
p
= participating (mobile) volume, and f
p
= V
p
/V = participating fraction. Then the
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effective (participating, mobile) porosity is defined by
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(1)
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where V
fp
is the fluid volume within the participating volume. Similarly the effective bulk
30
density is defined by
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(2)
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SRNL-STI-2011-00583 27 February 2012
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where M
sp
is the solid mass within the participating volume. Combining Equations 1 and 2
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produces
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(3)
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One can also define an effective solid density, which is useful for modeling software that takes
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(or requires) solid density as input. Using Equations 2 and 3 the result is
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(4)
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We next examine the impact of alternative density assignments on solute retardation. To
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generate example values, we consider the following specific settings representative of a
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sedimentary aquifer at the Savannah River Site: φ = 0.40,
eff
= 0.25 (Flach et al. 2004), and ρ
b
=
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1.6 g/cm
3
. The sorption coefficient (K
d
) is arbitrarily assumed to be 1.0 cm
3
/g.
42
As one intuitively anticipates, Equation 3 preserves retardation between the total (R) and
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effective porosity (R
eff
) systems
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(5)
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In contrast, if the unaltered original bulk density is used with an effective porosity in forward
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model predictions, then retardation is biased high
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(6)
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The bias is larger still if the unaltered solid density is coupled with an effective porosity
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(7)
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If experimental retardation data are fit using a single porosity model with variable effective
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porosity but bulk density fixed at the total porosity value, then the apparent sorption
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coefficient will be biased low because the analysis assumes excess solid is present
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(8)
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Thus the direction of the bias differs for inverse modeling versus forward simulations. Biases
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are zero when effective porosity is equal to total porosity, and increase with increasing non-
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zero ineffective porosity. These modeling biases can be eliminated by adopting an effective bulk
57
density using Equation 3.
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SRNL-STI-2011-00583 27 February 2012
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References
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de Marsily, G. 1986. Quantitative Hydrogeology: Groundwater Hydrology for Engineers.
60
Orlando: Academic Press, 440 p.
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Fetter, C. W. 1988. Applied Hydrogeology. Second edition. New York: Macmillan Publishing
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Company, 592 p.
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Fetter, C. W. 1993. Contaminant Hydrogeology. New York: Macmillan Publishing Company, 458
64
p.
65
Flach, G.P., S.A. Crisman, and F.J. Molz III. 2004. Comparison of single-domain and dual-domain
66
subsurface transport models. Ground Water 42, no. 6: 815–828.
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Zheng, C., and G.D. Bennett. 2002. Applied Contaminant Transport Modeling. Second edition.
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New York: John Wiley and Sons, 621 p.
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