Published online 17 May 2007
Published in Vadose Zone J 6:336-343 (2007)
DOI: 10.2136/vzj2006.0051
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SPECIAL SECTION: SAVANNAH RIVER SITE
An Unsteady Dual Porosity Representation of Tritium Leaching from Buried Concrete Rubble
G. P. Flach*,
K. P. Crapse,
M. A. Phifer,
L. B. Collard and
L. D. Koffman
Savannah River National Laboratory, Savannah River Site, Aiken, SC 29808
* Corresponding author (gregory.flach{at}sml.doe.gov).
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.
Received 29 March 2006.
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ABSTRACT
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Decontamination and decommissioning at the Savannah River Site near Aiken, SC, have produced on-site disposals of low-level solid radioactive waste in the form of concrete rubble. In the case of a former tritium extraction facility, building demolition produced a significant volume of rubble containing tritium. The contaminated debris comprises a heterogeneous mixture of coarse aggregate sizes, shapes, and internal tritium distributions. The rubble was disposed in unlined earthen trenches that were subsequently backfilled and exposed to normal infiltration. To forecast tritium flux to the water table, an unsteady dual-porosity model was developed to describe vadose zone leaching and transport. Tritium release was assumed to be controlled by diffusion within concrete, while advective and diffusive transport occur in the surrounding backfill. Rubble size and shape variations were characterized through a combination of physical measurement and photographic image analysis. For simplicity, the characterization data were reduced to an approximately equivalent distribution of one-dimensional slab thicknesses for representation in the dual-porosity formulation. Tritium flux to the water table from concrete rubble was predicted to be roughly 40% of that from uniformly contaminated soil. The lower flux is a result of slow release to soil pore water and a reduced effective trench conductivity from the presence of impervious concrete. At early times, tritium release from concrete in the lower trench is depressed by downward migration of tritium from overlying material. The pattern reverses at later times, when tritium is largely exhausted in the upper trench but higher residuals occupy the lower trench.
Abbreviations: GIS, geographic information system.
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INTRODUCTION
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Nuclear facility decontamination and decommissioning at the USDOE Savannah River Site near Aiken, SC, have produced on-site disposals of low-level solid radioactive waste in the form of concrete rubble. In the case of a former tritium extraction facility, Building 232-F demolition (Fig. 1a) produced a significant volume of rubble with embedded tritium as tritiated water (HTO). Tritium (3H) is nonsorbing and decays with a half-life of 12.3 yr. The contaminated concrete debris (Fig. 1b) comprises a heterogeneous mixture of coarse aggregate sizes, shapes, and internal tritium distributions. The rubble was disposed in long, shallow, unlined, earthen trenches that were subsequently backfilled with excavated soil and exposed to normal rainfall infiltration. An accurate prediction of tritium flux to the water table was desired to demonstrate compliance with USDOE performance objectives for on-site disposal. To this end, an unsteady dual-porosity model was developed to describe tritium leaching from concrete to soil, and vadose zone transport to the water table. A novel development in the work is a practical technique for representing a highly heterogeneous mixture of arbitrary rubble shapes and sizes with an approximately equivalent distribution of one-dimensional slab geometries.
The dual-porosity concept has been applied to several physical settings, including fractured media, laboratory soil columns, heterogeneous granular aquifers, and in the situation most analogous to the present, aggregated media (Passioura, 1971; Skopp and Warrick, 1974; van Genuchten and Wierenga, 1976, 1977; van Genuchten et al., 1977; Rao et al., 1980; Lafolie and Hayot, 1993a, 1993b; Brusseau et al., 1994; Griffioen et al., 1998). Specific formulations range from "mobile–immobile" regions with first-order mass transfer (Coats and Smith, 1964) to "dual-permeability," in which advection occurs in both regions (Gerke and van Genuchten, 1993). Generalizations of the concept have also been developed (e.g., Haggerty and Gorelick, 1995; Wang et al., 2005a).
For the present application, the low permeability of intact concrete precludes significant advective transport, and an accurate representation of diffusive leaching from concrete rubble was desired. Thus, the "unsteady" dual-porosity model, using the terminology of Birkholzer and Rouve (1994), was chosen. The unsteady formulation (Rasmuson and Neretnieks, 1980; Grisak and Pickens, 1980; Bibby, 1981; Huyakorn et al., 1983; van Genuchten, 1985) fully incorporates Fick's second law, in contrast to approximate first-order mass transfer approaches (e.g., Coats and Smith, 1964). The unsteady dual-porosity model also avoids ambiguity (Griffioen et al., 1998) in the proper mass transfer coefficient setting for the simpler first-order approach and readily accommodates nonuniform initial conditions.
The size and shape of concrete chunks have a strong effect on the time-varying rate of tritium leaching from building rubble. Leaching is relatively slow from large chunks with a low surface area–to–volume ratio. Small concrete pieces and/or a high surface area–to–volume ratio produce a fast release. Rather than explicitly considering every potential geometry in a leaching model, our approach approximated concrete rubble by a distribution of equivalent one-dimensional slab geometries. The size and shape variations of rubble were characterized through a combination of physical measurement and photographic image analysis. Each size classification was simulated separately to show the effect on flux. The individual flux results were then blended in proportion to the thickness distribution to produce a composite flux.
The overall system model comprises a pair of complementary unsaturated flow and solute transport models: (i) a two-dimensional trench cross-section, single-porosity, vadose zone model extending from the ground surface to the water table and (ii) a one-dimensional, dual-porosity, waste zone model focused on the near-field interaction between concrete rubble and surrounding backfill. Effective porosity and conductivity are assigned to the trench waste zone in the vadose zone model, assuming concrete rubble is impermeable. The vadose zone flow model generates the flow field for vadose zone transport and defines boundary conditions for the waste zone flow model. The waste zone flow model simulates flow through soil backfill in the disposal trench. The waste zone transport model defines the diffusive flux of tritium from concrete to soil backfill, which is then used as the source term for larger-scale vadose zone transport model simulations of tritium migration from waste zone soil backfill through underlying undisturbed soil to the water table.
The sections that follow describe the development of the unsteady dual-porosity model for tritium leaching from coarse aggregate, an approach for approximating an arbitrary aggregate shape with a one-dimensional slab, characterization of concrete rubble at the Savannah River Site, vadose zone modeling and the influence of relatively impervious rubble on unsaturated flow through a disposal trench, leaching flux as a function of characteristic aggregate size, and representative water-table flux results compared with tritium-contaminated soil disposal.
Unsteady Dual-Porosity Model
Under humid southeastern U.S. conditions, buried concrete is assumed to be fully saturated for practical purposes. Liquid-phase molecular diffusion is considered to be the primary mechanism for tritium release to surrounding backfilled soil. The effective diffusion coefficient for tritium in concrete is estimated to be 5 x 10–8 cm2 s–1, compared to about 5 x 10–6 cm2 s–1 for soil. Therefore, radioactivity concentration gradients in the backfill can be neglected compared with those in adjoining concrete, and the boundary condition for diffusion in concrete is effectively the backfill tritium concentration. The mathematical formulation assumed for diffusive release from a one-dimensional slab of dimension s is
 | [1] |
where C denotes activity concentration, D is the effective diffusion coefficient for porous medium transport, 
is the first-order coefficient for radioactive decay, and C0(x) is the initial concentration. After time zero, concentration at the two exterior surfaces varies with time according to Cb(t), the tritium concentration in the backfill pore water. The flux leaving concrete rubble is computed as
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where nc is the porosity of concrete, assumed to be 0.18. Equation set [1] is solved using a numerical technique (Anderson et al. [1984, section 4-2.5) summarized by
 | [3] |
with the fully implicit option (
= 1) selected. In Eq. [3], 
t denotes time step, x is the grid spacing, n is the time step number, and j is the grid node.
For the backfilled soil residing between solid pieces of concrete, tritium is assumed to be transported by liquid-phase advection and dispersion through an unsaturated porous medium. The PORFLOW code (Analytic & Computational Research, 2002), version 5.97.0, was selected for numerical flow and transport simulation involving backfilled soil. Unsaturated flow is represented by Richards' equation, and the standard advection–dispersion equation for unsaturated conditions is used to predict solute transport. The PORFLOW User's Manual (Analytic & Computational Research, 2002) provides additional information on mathematical formulations and numerical implementation. Because gradients in the horizontal plane are relatively small, only one-dimensional flow and transport in the vertical direction are modeled. The resolution of the 1D PORFLOW model is 0.3 m (1 ft). Trench disposal at the Savannah River Site produces a 4.9-m (16-ft) thick waste zone, which results in 16 grid cells. Dispersivity is set to zero (as a conservative assumption), so plume spreading occurs through molecular diffusion and numerical dispersion. Additional properties are defined in subsequent sections describing the larger-scale, companion vadose zone model extending from ground surface to the water table.
Under the dual-porosity paradigm, the concrete (immobile water) and backfilled soil (mobile water) regions are assumed to coexist spatially and compose a mathematical dual continuum. The concrete model is coupled to the backfill model through the concentration transient Cb(t). The backfill model is coupled to the concrete model through the flux term Fb(t), accounting for the volume fraction (fc) and equivalent one-dimensional thickness (s) of concrete present. Specifically, the rate of activity (mass) transfer from concrete to backfill per unit total volume is fc/s x Fb(t). Assignment of appropriate values for fc and s for the present application is discussed in subsequent sections. A separate concrete diffusion model is coupled to each grid cell of the backfill advection–dispersion model. The overall coupled waste zone model is solved iteratively by alternating between concrete and backfill simulations and updating the concentration transients using an under-relaxation technique. The loose coupling approach is numerically inefficient but simple to implement and effective in achieving convergence.
Equivalent One-Dimensional Slab Thickness
One method for defining the thickness of an equivalent one-dimensional slab is to preserve the volume (V) and surface area (A) of the three-dimensional object (Fig. 2). The resulting slab thickness (s) is
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where Af = A/2 is the area of one face of the slab. The effectiveness of this approach can be assessed by comparing exact solutions for mass diffusion from a three-dimensional brick geometry and a one-dimensional slab of characteristic dimension defined by Eq. [4]. The following initial value formulation is considered for simplicity:
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The equation set describes diffusion from a brick with dimensions a, b, c and a uniform initial concentration of C0. Concentration is denoted by C, and D is the effective diffusion coefficient for porous medium transport. After time zero, the exterior surfaces are held at a concentration of zero. First-order decay is omitted from Eq. [5] to focus solely on the impact of geometry (one-dimensional vs. three-dimensional) on diffusion. The exact series solution is (Ozisik 1980)
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with the eigenvalues defined as the positive roots of sin(ßma) = sin(
nb) = sin(
pc) = 0.
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FIG. 2. Three-dimensional diffusive leaching from concrete rubble of arbitrary size and shape approximated by one-dimensional diffusion from a slab.
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For a brick with dimensions a, b, c, Eq. [4] defining equivalent one-dimensional slab thickness evaluates as
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and the exact series solution for a one-dimensional slab is
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with eigenvalues defined by sin(ßma) = 0 (Ozisik 1980).
The fraction of initial mass residing in the three-dimensional brick at time t can be derived by integrating Eq. [6] over the volume, with the following result:
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Similarly, the mass fraction remaining in the one-dimensional slab is
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Figure 3 displays the relative residual mass curves for a block of unit dimensions (1 x 1 x 1 unitless) and a diffusion coefficient of 1 (nondimensional parameter case), and the corresponding one-dimensional slab. For early times, the two solutions agree exactly, because the primary control on flux is surface area. At later times, as the diffusion front penetrates the two geometries, shape also affects the flux. Cumulative flux from the one-dimensional slab is significantly larger compared with the three-dimensional block. As the three-dimensional block is flattened while holding volume constant, the agreement becomes better, as indicated by the example results for a brick of dimensions 
x 2 x 2, also shown in Fig. 3.
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FIG. 3. Relative residual mass for diffusion from three-dimensional (3D) bricks of two shapes and corresponding one-dimensional (1D) approximations based on volume and area preservation.
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Improved agreement between the three-dimensional and one-dimensional simulations can be achieved by empirically adjusting the thickness of the one-dimensional slab while holding volume constant. Results such as those depicted in Fig. 3 indicate that the more "three-dimensional" the shape from a diffusion perspective, the less accurate the approximating one-dimensional slab. One measure of the "three-dimensional" nature of an object is the volume–to–surface area ratio. For a rectangular geometry, the ratio is maximized when the three sides are equal; that is, the brick is a cube. Similarly, for elliptical shapes, the surface area is minimized for the special case of a sphere. These special cases produce the maximum value of s in Eq. [4] for a fixed volume, as can be seen by noting that V/A = s/2. For both a cube and sphere,
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where d is either the side length for the cube or the diameter of the sphere, and smax is the largest possible value for a fixed volume. For a cube,
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and for a sphere,
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For an arbitrary three-dimensional shape, the ratio s/smax is an indication of the three-dimensional nature of the object and ranges between 0 and 1. An empirical adjustment factor to s, producing significantly improved agreement between the one-dimensional and three-dimensional results, is given by
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and plotted in Fig. 4. The revised slab thickness becomes
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The empirical factor f was developed in this study by fitting the chosen functional form to calibration data. The calibration data consisted of optimal multipliers for a range of s/smax values. The "round" numerical values in Eq. [14] have no particular theoretical basis, and other functional forms could be chosen to accomplish the same effect. Figure 5 shows the comparison of the one-dimensional and three-dimensional results based on revised slab dimension. In addition to the 1 x 1 x 1 and 1/4 x 2 x 2 brick dimensions, results for a brick of dimensions 1/2 x 
2 x 2 are shown. The agreement is observed to be much improved, and adequate for the purpose of this study.
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FIG. 4. Empirical factor f for adjusting equivalent slab thickness s based on preserving volume and surface area. Defined by Eq. [14]: f = 1 + (2 – 1)(s/smax)2. (3D = three-dimensional; 1D = one-dimensional.)
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FIG. 5. Relative residual mass for diffusion from three-dimensional (3D) bricks of three shapes, and corresponding one-dimensional (1D) approximations based on optimal slab thickness given by Eq. [15].
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Concrete Rubble Size Distribution
The size and shape variations of concrete rubble from Building 232-F were estimated from comparable building demolition sites, through a combination of physical measurement and photographic image analysis. For each concrete chunk analyzed, an approximately equivalent one-dimensional slab thickness was derived from Eq. [4] and Eq. [11–15], from measurements of volume V and surface area A. Size distributions were defined in terms of a geometric series of discrete characteristic dimensions (Table 1) by grouping thicknesses within certain size ranges. Table 1 shows the estimated size distribution of rubble from Building 232-F demolition.
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TABLE 1. Characteristic one-dimensional slab thicknesses simulated by the unsteady dual-porosity model and the volume fraction of each size class associated with Savannah River Site, Building 232-F rubble.
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An example of photographic image analysis to determine V and A is depicted in Fig. 6. The raw photograph (Fig. 6a) was first translated into a two-dimensional map of larger individual chunks of concrete, clusters of smaller pieces, and void–air space, using geographic information system (GIS) software (ArcView 3.2.a, Environmental Systems Research Institute, Redlands, CA) (Fig. 6b). The two-dimensional area and perimeter of each polygon representing an individual concrete piece were computed by the GIS software. An approximating ellipse with the same area and perimeter as the polygon was then generated, followed by a three-dimensional ellipsoid, assuming that the third axis was the average of the two ellipse axes. Finally, the three axes were used to compute three-dimensional volume and surface area for the approximating ellipsoid.
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FIG. 6. Concrete rubble from a demolition site comparable to Building 232-F. (A) Outlined region of interest. (B) Translation into geographic information system (GIS) polygons.
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Vadose Zone Flow
The larger-scale vadose model was developed to investigate tritium migration beyond the trench waste zone (e.g., flux to water table), and provide appropriate boundary conditions for the dual-porosity waste zone leaching model described earlier. The two models share the same waste zone material properties.
Figure 7 depicts the vadose zone flow model extent, grid resolution, material zones, and boundary conditions. Nominal dimensions for a disposal trench are 6.1 m (20 ft) wide and 6.1 m (20 ft) deep. Buried waste occupies the bottom 4.9 m (16 ft). The water table averages 14 m (45 ft) below grade, or 7.6 m below trench bottom. Facility practice is to excavate trenches in groups of five with a 3.0 m (10 ft) separation between individual trenches. The numerical model focuses on an outside trench among a group of five. No flow was assumed to occur across a vertical line midway between trenches (due to an approximately symmetrical flow pattern) and along the opposite far-field boundary. Average annual infiltration at the disposal facility is estimated to be 30 cm yr–1 from HELP model (USEPA, 1994a, 1994b) simulations, and this setting is uniformly applied over the top boundary. The water table coincides with the bottom boundary, where pressure head is set to zero.
Undisturbed soil is treated as a uniform clayey sand with a porosity of 0.42, a hydraulic conductivity of 1 x 10–4 cm s–1, and soil characteristic curves as shown in Fig. 8. The disturbed soil backfill above the waste zone is estimated to have a porosity of 0.456 and hydraulic conductivity of 2 x 10–4 cm s–1. Soil curves for backfilled soil are assumed to be the same as for undisturbed soil. In the vadose zone model, the waste zone is treated as a homogenous material with effective properties reflecting the presence of impervious rubble embedded in backfilled soil. The soil curves shown in Fig. 8 are assumed to apply to the waste zone. Effective porosity and conductivity depend on the volume fraction of concrete present.
The volume of concrete in Building 232-F has been estimated at 1031 m3, and disposal records indicate that about 1584 m3 of rubble was buried in trenches. The volume increase is presumed to reflect bulking of the material. The data imply a volume fraction of solid concrete within bulked rubble of 65%. This percentage is consistent with the bulk density of shale rip-rap (1682 kg m–3 [105 lb ft–3]) and the rock density of shale (2803 kg m–3 [175 lb ft–3]), which imply a solid fraction of 60% for the bulked material. Disposal operations personnel estimated that 80 to 85% of the trench volume would typically be filled with rubble. Thus, the overall fraction of solid concrete within the waste zone is estimated to be 0.825 x 0.65 = 0.54. Backfilled soil is assumed to occupy the remaining fraction of 0.46, after some initial settling. Assuming a tortuosity of 2 for macroscopic flow around chunks of concrete, the effective porosity and conductivity of backfilled rubble are approximately 0.46 x 0.456 = 0.21 and (0.46/2) x (2 x 10–4) = 4.6 x 10–5 cm s–1, respectively. The tortuosity value is based on the data of Wang et al. (2005b), who measured tortuosity for rock samples in the size range of 3.8 to 4.5 cm in diameter and 8.9 to 10.5 cm in length.
Impervious concrete produces a lower effective hydraulic conductivity for the composite waste zone, which impacts flow within and surrounding the trench. The result is lower Darcy velocities both within and beneath the trench. However, impermeable material also reduces the effective porosity for advective transport, which increases pore velocity in the trench relative to Darcy velocity. The overall impact on travel time is unclear. To examine these phenomena, trenches backfilled with soil alone (Fig. 9a) and a combination of soil and concrete rubble (Fig. 9b) were simulated. The simulated path lines shown in the figures include 1-yr time markers based on pore velocity. Note that impervious rubble increases pore velocity and reduces travel time through the trench, despite a lower Darcy velocity within the waste zone, but diverts moisture around the trench and increases travel time beneath the trench. The net result is nearly the same travel time from the ground surface to the water table for the two scenarios.
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FIG. 9. Simulated saturation and path lines for disposal trenches containing (A) soil only and (B) concrete rubble backfilled with soil. One-year time markers are shown on the path lines, which were computed from pore velocity.
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Vadose Zone Transport
Vadose zone flow model results are used to set boundary conditions for the waste zone flow model. The coupled dual-porosity waste zone transport model is then run to generate a foot-by-foot leaching source term for the larger-scale vadose zone transport model, which uses the grid and flow field described in the previous section. Dispersivity is set to zero (as in the waste zone model), so plume spreading occurs through molecular diffusion and numerical dispersion.
Figure 10 illustrates normalized flux (Ci yr–1 per Ci disposed), leaving the bottom of the burial trench as a function of characteristic concrete thickness, and a uniform initial distribution of tritium. Figures 11 and 12 show how tritium flux and concentration vary with elevation in the trench, for rubble with an 8-cm characteristic dimension. Backfill concentration increases with depth (Fig. 12) as tritium is carried downward from higher layers, thus depressing the early tritium release with depth (Fig. 11). At later times, the bottom layers release tritium at a higher rate than upper layers.
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FIG. 10. Simulated flux of tritium released at the bottom of a disposal trench as a function of one-dimensional slab thickness.
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FIG. 11. Flux as a function of elevation in the disposal trench for an 8-cm slab thickness and uniform initial tritium distribution.
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FIG. 12. Soil backfill concentration as a function of elevation in the disposal trench for an 8-cm slab thickness and uniform initial tritium distribution.
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Figure 13 illustrates flux to the water table for tritium-contaminated soil alone in the trench, and mixed soil and concrete rubble with each of the characteristic dimensions defined in Table 1 plus a 0-cm asymptote. The latter is generated directly from the vadose transport model by setting the effective porosity for the waste zone to 0.408 = (0.49 x 0.21 + 0.18)/0.49, which preserves the combined water content of the backfill and concrete for soil saturation averaging about 0.49 (Fig. 9b). The implicit assumption is that pore waters in backfill and rubble are in local equilibrium, which would be the case for very thin concrete slabs (or high diffusion coefficient). Concrete rubble in disposal trenches hinders release of embedded tritium and reduces Darcy velocity, compared with tritium contaminated soil, thus significantly reducing flux to the water table.
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FIG. 13. Simulated flux to the water table from tritium-contaminated soil and concrete rubble of various characteristic one-dimensional slab dimensions.
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The contaminated soil and 0-cm asymptotic curves highlight the impact of impervious concrete on vadose zone flow (Fig. 9), as both scenarios involves instantaneous release of tritium to mobile water, albeit with somewhat different porosities (0.456 vs. 0.408, respectively). Lower effective hydraulic conductivity and Darcy velocity for the 0-cm asympototic case more than compensate for a slightly lower effective porosity. The result is a significantly lower water-table flux, even though activity (mass) transfer from the concrete to backfill is unhindered. Additional declines in peak flux are observed with increasing concrete thickness.
For the concrete size distribution defined in Table 1, a composite flux curve for Building 232-F rubble was generated by blending the type curves shown in Fig. 13 and similar curves for a nonuniform initial tritium distribution. Predemolition characterization of Building 232-F indicated a uniform distribution of tritium in most concrete but a nonuniform distribution in the stack (Fig. 1a). For the building stack, activity was confined to a depth of 7.6 cm (3 in) from the interior wall exposed to tritium gas. In the waste zone model, tritium was assumed to reside within the first 7.6 cm (3 in) adjoining only one side of the surrogate one-dimensional slab. The resulting peak water table flux is predicted to be approximately 40% of that for contaminated soil alone in a trench (not plotted).
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Conclusions
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Concrete rubble disposal in earthen trenches at the Savannah River Site presents a unique opportunity to apply the unsteady dual-porosity transport formulation in an unconventional setting. The contaminated concrete debris comprises a heterogeneous mixture of coarse aggregate sizes, shapes, and internal tritium distributions. Approximating the waste with a distribution of equivalent one-dimensional slab thicknesses is effective in capturing diffusional flux variations with concrete chuck size, shape, and internal source distribution. The unsteady dual-porosity formulation facilitated more accurate forecast of tritium releases from burial trenches compared with prior assumptions of instantaneous release. Specifically, the peak flux from concrete rubble is predicted to be roughly 40% of that from tritium-contaminated soil. The lower flux is a result of slow release of tritium from concrete and a reduced effective hydraulic conductivity of the trench due to the presence of impervious concrete. Downward migration of tritium released in upper portions of a disposal trench was observed to initially depress leaching in lower portions, due to elevated concentrations in backfilled soil. Tritium initially held up in the lower trench later produces elevated releases compared with overlying material.
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ACKNOWLEDGMENTS
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We are sincerely grateful to Washington Savannah River Company LLC and the USDOE for making this work possible under Contract No. DE-AC09-96SR18500 and for permission to publish our findings.
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ABSTRACT
INTRODUCTION
