Abstract
Natural discontinuities are the major concern when considering mass transport in engineered barrier systems and their host rocks. Numerical simulations of highly fractured geological formations are limited because of the contradiction between results accuracy and computational costs. To alleviate such a contradiction, this study proposes an improved fracture continuum method to simulate the radioactive spreading in a complex 3-D fracture system. With Boolean operations and the unified pipe-network method, discontinuities are mapped on structured subdomains and standardized to equivalent paths. Moreover, adaptive mesh refinement is utilized to ease the complexity further. We verify the accuracy of this method in two metric cases, and results show that perfect agreement is achieved with analytical solutions. This method demonstrates its applicability to the simulation of a nuclear leak in a repository for high-level radioactive waste. Effects of the maximum refinement level are discussed. For a room-scale problem, flow rates and mass fluxes on boundary surface converge to stable values when the maximum refinement level is larger than 4. An extensive case with complex fracture networks is modeled and compared with the conventional finite-difference method. The proposed method is capable of conducting robust results with significantly lower computational complexity and negligible errors by avoiding ill-conditioned mesh elements.
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Abbreviations
- AMR:
-
Adaptive mesh refinement
- EDFM:
-
Embedded discrete-fracture modeling
- HLW:
-
High-level radioactive waste
- pEDFM:
-
Projection-based embedded discrete-fracture modeling
- UPM:
-
Unified pipe-network method
- \( b \) :
-
Fracture hydraulic aperture
- \( c \) :
-
Solute concentration
- \( \varvec{k} \) :
-
Permeability tensor
- \( k_{i\tau } \) :
-
Permeability of pipe \( P_{i\tau } \)
- \( l \) :
-
Length of simulated domain size
- \( \varvec{n}_{\varvec{i}} ,\varvec{n}_{\phi } \) :
-
Unit vector of grid surface represented by node \( N_{i} \), \( N_{\phi } \)
- \( \Delta n_{i\tau } \) :
-
Length of pipe \( P_{i\tau } \)
- \( p \) :
-
Total hydraulic pressure
- \( p_{i} ,p_{j} , \ldots ,p_{q} , p_{\tau } ,p_{\phi } \) :
-
Total hydraulic pressure at the corresponding node
- \( q_{i} \) :
-
Net flow rate out of controlled volume of node \( N_{i} \)
- \( r \) :
-
Ratio of grid sizes in two consecutive refinement levels
- \( s_{i\tau } \) :
-
Cross area of the pipe \( P_{i\tau } \)
- \( t \) :
-
Time variable
- \( \varvec{v} \) :
-
Velocity of fluid
- \( \varvec{v}_{\varvec{i}} ,\varvec{v}_{\phi } \) :
-
Velocity of fluid at node \( N_{i} \), \( N_{\phi } \)
- \( w \) :
-
Width of rectangular fracture
- \( \Delta x_{R} \) :
-
Size of grids at mesh-refinement level \( R \)
- \( A_{i} ,A_{\phi } \) :
-
Area of grid surface represented by node \( N_{i} \), \( N_{\phi } \)
- \( D \) :
-
Hydraulic dispersion coefficient
- \( D^{*} \) :
-
Molecular diffusion coefficient in free water
- \( F_{i} \) :
-
Set of all nodes representing surfaces of fine grids sharing a nonconforming interface with the node \( N_{i} \)
- \( H \) :
-
Total hydraulic head
- \( J_{{R_{d} }} \) :
-
Mass flux of solute
- \( M \) :
-
Solute emission on all boundaries
- \( M1,M2, M3, M4 \) :
-
Solute emission on the corresponding boundary
- \( N_{i} ,N_{j} , \ldots ,N_{q} ,N_{\tau } ,N_{\phi } \) :
-
Face-centered node of the corresponding grid surface
- \( P_{ij} ,P_{jm} , \ldots ,P_{pq} \) :
-
Interconnected pipe between corresponding nodes
- \( Q \) :
-
Total flow rate on boundary
- \( R \) :
-
Mesh-refinement level
- \( R_{d} \) :
-
Maximum mesh-refinement level
- \( R_{t} \) :
-
Retardation coefficient
- \( S \) :
-
Source or sink term
- \( T_{i} \) :
-
Set of all nodes connecting to the node \( N_{i} \)
- \( \alpha_{\text{L}} \) :
-
Dispersivity in the longitudinal direction
- \( \theta \) :
-
Dip angle of fracture
- \( \phi \) :
-
Strike angle of fracture
- \( \lambda \) :
-
Decay constant of radioactive material
- \( \mu \) :
-
Dynamic viscosity of fluid
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Acknowledgements
This research was supported by the Research Training Program (RTP) scheme at UWA. The authors wish to thank the National Natural Science Foundation of China (NSFC) for their financial support (Nos. 51778029 and 51627812).
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Ma, G., Li, T., Wang, Y. et al. A Semi-Continuum Model for Numerical Simulations of Mass Transport in 3-D Fractured Rock Masses. Rock Mech Rock Eng 53, 985–1004 (2020). https://doi.org/10.1007/s00603-019-01950-1
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DOI: https://doi.org/10.1007/s00603-019-01950-1