An empirical probabilistic approach for constraining the uncertainty of long-term solute transport predictions in fractured rock using in situ tracer experiments

Abstract

Although radionuclide tracer tests have been carried out for over 30  years, the role of tracer tests in radioactive waste repository performance assessment (PA) has been questioned due to the differences between the time scales for tracer tests and PA. The possibility of using in situ tracer tests to constrain PA time scale (over 10,000  years) solute transport has been demonstrated using a systematic “microstructural model” approach to define advective and retentive materials. A series of simulations were conducted of the “TRUE-1” sorbing solute transport experiments in a well-characterized block of fractured granite. These experiments were then used to constrain the uncertainty of long-term transport on the same pathways, using a gradient several orders of magnitude smaller. The comparison of uncertainty of this long-term transport, with and without this conditioning step, provided a measure of the ability of tracer tests to reduce PA time-scale uncertainty. Although this approach for quantifying uncertainty reduction is somewhat empirical, it does indicate the potential usefulness of tracer experiments in reducing the uncertainty of the key PA time-scale transport parameters such as the flow wetted surface, provided that immobile zone properties such as sorption (Kd), porosity and diffusivity are available.

Résumé

Bien que des traçages au moyen de radionucléides sont réalisés depuis plus de 30 ans, on s’interroge sur le rôle des traçages pour l’évaluation des performances (EP) des sites d’enfouissement des déchets radioactifs du fait des différences d’échelle de temps entre les traçages et l’EP. La possibilité d’utiliser des traçages in situ pour contraindre le transport de soluté à l’échelle de temps de l’EP (plus de 10 000 ans) a été démontrée au moyen d’une approche systématique de type « modèle microstructurel » pour définir les matériaux advectifs et rétentifs. Une série de simulations des expériences « TRUE-1 » de transport de soluté lié dans un bloc de granite bien caractérisé ont été conduites. Ces expériences ont alors été utilisées pour contraindre l’incertitude du transport à long terme selon les mêmes trajectoires, en utilisant un gradient plusieurs ordres de grandeurs plus petit. La comparaison de l’incertitude sur le transport à long terme avec ou sans cette étape de conditionnement fournit une mesure de la capacité des traçages à réduire l’incertitude pour les échelles de temps des EP. Bien que cette approche de la quantification de la réduction de l’incertitude soit assez empirique, elle pointe l’utilité potentielle des expériences de traçage pour la réduction de l’incertitude sur les paramètres clefs du transport à l’échelle de temps des EP, paramètres tels que la surface mouillée (flow-wetted surface), dans la mesure où les propriétés de la zone immobile, telles que sorption (Kd), porosité et diffusivité, sont connues.

Resumen

Aunque las pruebas de trazadores radionucleidos se llevaron a cabo por más de 30 años, el rol de las pruebas de trazadores en la evaluación de rendimiento (PA) de repositorios de residuos radiactivos ha sido cuestionado debido a las diferencias entre las escalas temporales para las pruebas de trazadores y PA. La posibilidad de usar pruebas de trazadores in situ para restringir la escala temporal para el transporte de soluto en la PA (por encima de 10,000 años) ha sido demostrada usando un enfoque sistemático de modelo microestructural para definir los materiales advectivos y retentivos. Una serie de simulaciones fueron realizadas con los experimentos de transporte de soluto del “TRUE-1” en un bloque de granito fracturado bien caracterizado. Estos experimentos fueron utilizados luego para restringir las incertidumbres del transporte a largo plazo con los mismos lineamientos, usando gradientes varios órdenes de magnitud menor. La comparación de incertidumbres de este transporte a largo plazo con y sin estas etapas condicionantes proporciona una medida de la capacidad de las pruebas de trazadores para reducir la incertidumbre de la escala temporal en la PA. Aunque este enfoque para cuantificar la reducción de incertidumbre es un tanto empírico, indica la utilidad potencial de los experimentos con trazadores en la reducción de incertidumbre de los parámetros claves de transporte en la escala temporal de PA, tales como la superficie mojada de flujo, suponiendo que las propiedades de las zonas inmóviles tales como porción (Kd), porosidad y difusividad estén disponibles.

摘要

虽然放射性核素示踪试验已运用了30多年, 其在放射性废物处置库性能评价 (PA) 中的作用却因示踪试验和PA之间时间尺度的差异而受到质疑。利用原位示踪试验约束PA时间尺度 (超过10000年) 的溶质运移的可能性, 已通过一种系统的定义平流和阻滞材料的“微结构模型”方法进行论证. 对在某研究充分的裂隙花岗岩体中进行的“TRUE-1”吸附溶质运移试验开展了一系列模拟。这些试验随后被用于约束相同路径上长期运移的不确定性, 所用梯度小若干个数量级. 对有无该调节步骤的长期运移的不确定性之间的比较, 为示踪试验降低PA时间尺度不确定性的能力提供了量度. 尽管这种量化不确定性降低的方法在某种程度上带有经验性, 但它确实表明了在不流动层性质如吸附系数 (Kd) 、孔隙度和扩散系数已知的情况下, 示踪试验在降低PA时间尺度上关键运移参数 (如水流润湿表面) 不确定性中的潜在有效性.

Resumo

Apesar dos testes de traçadores com radionuclídeos estarem a ser aplicados desde há mais de 30 anos, o papel dos testes de traçadores na avaliação da performance (PA – Performance Assessment) em repositórios de lixos radioactivos tem sido questionada, devido às diferenças entre as escalas de tempo para os testes de traçadores e para a PA. A possibilidade de usar testes de traçadores in situ para constranger a escala do tempo do transporte de solutos na PA (acima de 10,000 anos) tem sido demonstrada usando uma aproximação sistemática de “modelo microestrutural” para definir materiais advectivos e materiais retentores. Foram realizadas uma série de simulações de ensaios de absorção em transporte de solutos “TRUE-1”, num bloco de granito fracturado bem caracterizado. Estas experiências foram então utilizadas para constranger a incerteza do transporte de longo curso nos mesmos circuitos hidráulicos, usando um gradiente várias ordens de magnitude inferior. A comparação da incerteza deste transporte de longo curso, com e sem este condicionalismo, forneceu uma medida da capacidade dos testes de traçadores para reduzirem a incerteza da escala temporal da PA. Apesar desta aproximação para quantificar a redução da incerteza ser um tanto empírica, permite mostrar a utilidade potencial das experiências de traçadores na redução da incerteza dos parâmetros chave da escala temporal da PA, tais como o fluxo na superfície húmida, desde que sejam conhecidas propriedades fixas desta zona, tais como a sorção (Kd), a porosidade e a difusividade.

Appendix: Goldsim implementation of transport in fracture networks with matrix diffusion

The GoldSim solute transport equation is based upon the advection diffusion equation, assuming steady-state flow and a second-order approach to describe the diffusive mass transfer of a solute between the groundwater in a pipe and the multiple immobile porosity zones attached to it. The advective-dispersive transport of solute species n in a pipe network is given by:

$$\begin{array}{*{20}c} {A\left[ {R_{\text{n}} \left( \ell \right)\frac{{\partial C_{\text{n}} }}{{\partial t}} + q\left( \ell \right)\frac{{\partial C_{\text{n}} }}{{\partial \ell }} - \frac{\partial }{{\partial \ell }}D_{{\text{ln}}} \left( \ell \right)\frac{{\partial C_{\text{n}} }}{{\partial \ell }} + R_{\text{n}} \left( \ell \right)\lambda _{\text{n}} C_{\text{n}} - R_{{\text{n}} - 1} \left( \ell \right)\lambda _{{\text{n}} - 1} C_{{\text{n}} - 1} } \right]} \\ { \pm \sum\limits_{\ell \prime } {\mathop M\limits^ \bullet \delta \left( {\ell - \ell \prime } \right)} + \sum\limits_{^{\ell *} } {Q\left( {C_{\text{n}} - C_{\text{n}}^ * } \right)\delta \left( {\ell - \ell ^ * } \right)} \; + \sum\limits_{{\text{im}} = 1}^{IM} {P_{{\text{im}}} \theta _{{\text{im}}} D_{{\text{im}}} } \frac{{\partial C_{\text{n}}^{im} }}{{\partial w}}\left| {_{w = 0} = 0} \right.} \\ \end{array} $$

(1)

where:

n :

Nuclide index [-]

im:

Immobile zone class number (note: if desired im can equal 0) [-]

IM($\ell$):

Total number of immobile zones attached to pipe $\ell$ [-]

A($\ell$):

Pipe cross-sectional area [L²]

R _n($\ell$):

Retardation factor [-]

q ($\ell$):

Specific discharge (≡ pipe velocity v) [L/T]

$D_{\ell _{\text{n}} } \left( \ell \right)$ :

Dispersion coefficient = $\alpha v + D_{\text{n}}^o$ [L²/T]

α :

Pipe longitudinal dispersivity [L],

$D_{\text{n}}^o$ :

Free-solution diffusion coefficient [L²/T]

λ _n :

Decay constant [1/T]

$\mathop M\limits^ -$ (t):

Internal solute mass source/sink [M/T]

Q :

External fluid source/sink [L³/T]

δ $\left( {\ell - \ell \prime } \right)$ :

Dirac delta [1/L]

δ $\left( {\ell - \ell ^ * } \right)$ :

Dirac delta [1/L]

P _im :

Block surface area per unit length of matrix (equivalent to the effective perimeter of immobile zone “im”) [L]

D _im :

Matrix effective diffusion coefficient [L²/T]

θ _im :

Immobile zone porosity for immobile zone “im”

C_n :

Pipe concentration [M/L³]

$C_{\text{n}}^ *$ :

Concentration of injectate in external fluid source [M/L³]

$C_{\text{n}}^{im}$ :

Immobile zone concentration [M/L³]

$\ell$ :

Distance along interconnected pipe network [L]

$\ell \prime$ :

Location of solute mass source/sink [L]

$\ell ^ *$ :

Location of external fluid source/sink [L]

w :

Distance perpendicular to plane of fracture [L]

t :

Time [T]

GoldSim solves this equation using a Laplace Transform Galerkin method (Sudicky and McLaren 1992).

Boundary conditions may be either of the Dirichlet-type where the input concentration history of each species is a specified function of time, or of the Cauchy-type where the advective input mass flux can be prescribed as a function of time at the origin of a pipe on the boundary of the domain. Mathematically, these boundary conditions are described by:

$${\text{Dirichlet:}}\,C_{\text{n}} = C_{\text{n}}^o \left( t \right)\,{\text{on}}\,\Gamma $$

(2)

$${\text{Cauchy:}}\,{\text{A}}\left( \ell \right)q\left( \ell \right)C_{\text{n}}^o \left( t \right) = A\left( \ell \right)\left[ {q\,\left( \ell \right)C_{\text{n}} \left( {\ell ,t} \right) - D_{\ell _{\text{n}} } \left( \ell \right)\frac{{\partial C_{\text{n}} }}{{\partial \ell }}} \right]{\text{on}}\,\Gamma $$

(3)

where $C_{\text{n}}^o$(t) is the specified concentration for species n. LTG also allows the concentration or flux rate (e.g. mol/year) to be specified at an interior point.

In order to represent the diffusive exchange of solute mass between the pipes and any on the (im) immobile zones attached to them, LTG uses a second-order approach described by:

$$\begin{array}{*{20}l} {\theta _{{\text{im}}} \left( {im,\ell } \right)R_{\text{n}}^{{\text{im}}} \left( {im.\ell } \right)\frac{{\partial C_{\text{n}}^{{\text{im}}} }}{{\partial t}} - \frac{\partial }{{\partial w}}\theta _{{\text{im}}} \left( {im,\ell } \right)D_{{\text{im}}} \frac{{\partial C_{\text{n}}^{{\text{im}}} }}{{\partial w}}} \hfill \\ { + \theta _{{\text{im}}} \left( {im,\ell } \right)R_{\text{n}}^{{\text{im}}} \left( {im,\ell } \right)\lambda _{\text{n}} C_{\text{n}}^{{\text{im}}} - \theta _{{\text{im}}} \left( {im,\ell } \right)R_{{\text{n - 1}}}^{{\text{im}}} \left( {im,\ell } \right)\lambda _{{\text{n - 1}}} C_{{\text{n - 1}}}^{{\text{im}}} = 0} \hfill \\ \end{array} $$

(4)

where:

θ _im $\left( {{\text{im}},\ell } \right)$ :

Immobile zone porosity for immobile zone “im” attached to pipe “$\ell$” [-]

$R_n^{im}$ $\left( {{\text{im}},\ell } \right)$ :

Immobile zone retardation factor for immobile zone “im” attached to pipe “$\ell$” [-]

$C_n^{im}$ :

Concentration in matrix [M/L³]

D _im :

Matrix effective diffusion coefficient [L²/T]

$D_n^0$ τ

$D_n^o$ :

Free-solution diffusion coefficient [L²/T]

τ :

Tortuosity [-]

If a particular immobile zone is fluid-filled such as within an immobile water zone attached to a pipe within a fracture plane, then the immobile zone porosity, θ _im, would equal 1.0.