A new criterion for determining the representative elementary volume of 1 translucent porous media and inner contaminant

12 Representative elementary volume (REV) is essential to measure and quantify the effective 13 parameters of a complex heterogeneous medium. To overcome the limitations of the existing 14 REV estimation criteria, a new REV estimation criterion (χ i ) based on dimensionless range 15 and gradient calculation is proposed in this study to estimate REV of a translucent material 16 based on light transmission techniques. Three sandbox experiments are performed to 17 estimate REVs of porosity, density, tortuosity and perchloroethylene (PCE) plume using 18 multiple REV estimation criteria. In comparison with χ i , previous REV estimation criteria 19 based on the coefficient of variation ( 𝐶 𝑉𝑖 ), the entropy dimension ( DI i ) and the relative 20 gradient error ( 𝜀 𝑔𝑖 ) are tested in REV quantification of translucent silica and inner PCE 21 plume to achieve their corresponding effects. Results suggest that new criterion (χ i ) can 22 effectively identify the REV in the materials, whereas the coefficient of variation and 23 entropy dimension ( i D I )are not effective. The relative gradient error can make the 24 REV plateau obvious, while random fluctuations make the REV plateau uneasy to 25 identify accurately. Therefore, the new criterion is appropriate for REV estimation of the 26 translucent materials and inner contaminant. Models are built based on Gaussian equation 27 to simulate the distribution of REVs for media properties, which frequency of REV is dense 28 in the middle and sparse on both sides. REV estimation of PCE plume indicates high level 29 of porosity lead to large value of mean and standard deviation for REVs of PCE saturation 30 (S o ) and PCE-water interfacial area (A OW ). Fitted equations are derived from distribution of 31 REVs for PCE plume related to d m (distances from mass center to considered point) and d I 32 (distances from injection position to considered point). Moreover, relationships between

translucent materials of a given thickness, the emergent light intensity after the absorptive 124 and interfacial losses can be expressed as (Niemet and Selker, 2001;Bob et al., 2008;Wu 125 et al., 2017): where I0 is the original light intensity; C is a constant of correction for light emission 128 and light observation; τj is the transmittance when light penetrate from phase i to i+1; 129 αi is the absorption coefficient when light penetrate in phase i; di is the length of light 130 penetration path in phase i. 131 To derive the porosity, the 2D translucent porous medium should be only saturated 132 by water. Consequently, the emergent light intensity can be expressed as:  If we arbitrarily select an infinitesimal element, its area Ao approaches zero 139 (Ao→0) from the 2D translucent porous media (Fig. 1b), and suppose the infinitesimal 140 element with thickness LT containing solid particles and pores that can be regarded as 141 https://doi.org/10.5194/hess-2020-91 Preprint. Discussion started: 20 May 2020 c Author(s) 2020. CC BY 4.0 License.
where do is the median diameter of pores; θ is porosity. . β and γ can be determined from 149 experimental data, then porosity can be obtained.

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The density and tortuosity are derived as (Wu et al., 2018): where ρ is the density of translucent porous media; ρw is the density of water; ρs is the 154 density of solid particles; τ is tortuosity .

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The saturation of dense nonaqueous phase liquid (DNAPL) was quantified by light 156 transmission technique based on light pass through translucent materials (Niemet and I ln I ln where So is the saturation of DNAPL; Is is the light intensity after light penetration  Hsies, 2000). In the range of REV, the value of one associated property will meet 169 the condition:  Table 2.

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To evaluate the REV of porosity, the coefficient of variation where i is the cuboid window ( Fig. 1b)  sub-grids. When number of sub-grids (N) is less than 10, a correction is utilized to 185 replace Eq. (10). According to Nordahl and Ringrose (2008), is defined 186 as homogeneous and can be used as criterion to identify the REV scale.

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Similarly, for porosity of translucent silica, entropy dimension ( i DI ) (Table 2) is 188 utilized for REV analysis and estimation (Martínez et al., 2007), which is defined as: where, Lε is the scale of sub-grid; "≈" indicates the asymptotic equivalence as  To achieve the REV for multiple system variables, such as porosity, moisture 208 saturation and air-water interfacial areas in an unsaturated porous medium, a criterion 209 named the relative gradient error (Table 2) was applied (Costanza-Robinson et al., 2011): where i g ε is relative gradient error; ΔL is the measured cuboid window size increment 212 length for REV estimation. Usually, i g ε less than 0.2 (Costanza-Robinson et al., 2011) is 213 utilized to identify a REV sizes.

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A new criterion based on the required condition of REV is proposed to estimate 215 the REV range for the translucent silica in this study:  which can be identified as REV plateau in region II (Fig. 1c). The relative gradient error