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Simulation of Solute Transport Through Fractured Rock: A Higher-Order Accurate Finite-Element Finite-Volume Method Permitting Large Time Steps

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Abstract

Discrete-fracture and rock matrix (DFM) modelling necessitates a physically realistic discretisation of the large aspect ratio fractures and the dissected material domains. Using unstructured spatially adaptively refined finite-element meshes, we find that the fastest flow often occurs in the smallest elements. Flow velocity and element size vary over many orders of magnitude, disqualifying global Courant number (CFL)-dependent transport schemes because too many time steps would be necessary to investigate displacements of interest. Here, we present a higher-order accurate implicit pressure–(semi)-implicit transport scheme for the advection–diffusion equation that overcomes this CFL limitation for DFM models. Using operator splitting, we solve the pressure and the transport equations on finite-element, node-centred finite-volume meshes, respectively, using algebraic multigrid methods. We apply this approach to field data-based DFM models where the fracture flow velocity and mesh refinement is 2–4 orders of magnitude greater than that of the matrix. For a global CFL of ≤10,000, this implies sub-CFL, second-order accurate behaviour in the matrix, and super-CFL, at least first-order accurate, transports in fast-flowing fractures. Their greater refinement, however, largely offsets this numerical dispersion, promoting a highly accurate overall solution. Numerical and fracture-related mechanical dispersions are compared in the realistic DFM models using second-order accurate runs as reference cases. With a CFL histogram, we establish target error criteria for CFL overstepping. This analysis indicates that for extreme fracture heterogeneity, only a few transport steps can be sufficient to analyse macro-dispersion. This makes our implicit method attractive for quick analysis of transport properties on multiple realisations of DFM models.

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Authors and Affiliations

  1. Department of Mineral Resources and Petroleum Engineering, Montan University of Leoben, Max Tendler Str. 4, 8700, Leoben, Austria

    Stephan K. Matthäi

  2. Department of Earth Science and Engineering, Imperial College London, South Kensington Campus, Exhibition Road, London, SW7 2AZ, UK

    Hamidreza M. Nick & Christopher Pain

  3. Institute of Fluid Mechanics and Environmental Physics in Civil Engineering, Appelstrae 9A, 30167, Hannover, Germany

    Insa Neuweiler

Authors
  1. Stephan K. Matthäi
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  2. Hamidreza M. Nick
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  3. Christopher Pain
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  4. Insa Neuweiler
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Correspondence to Hamidreza M. Nick.

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Matthäi, S.K., Nick, H.M., Pain, C. et al. Simulation of Solute Transport Through Fractured Rock: A Higher-Order Accurate Finite-Element Finite-Volume Method Permitting Large Time Steps. Transp Porous Med 83, 289–318 (2010). https://doi.org/10.1007/s11242-009-9440-z

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  • DOI: https://doi.org/10.1007/s11242-009-9440-z

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