In this work we present a new numerical method for solving coupled 1D-3D flow models, which can be used, for example, to model the flow in a well and the surrounding reservoir. Assuming 1D Poiseuille flow in the well, we use Stirling’s law of filtration to couple it to the 3D flow model used for the reservoir. The presence of a line source in the governing equations of the reservoir is known to cause a logarithmic type singularity in the solution around the well. For this reason, the solution is difficult to approximate numerically. We therefore introduce a decomposition technique where the solution is split into an explicitly known logarithmic term capturing the singularity and a well-behaved correction term. After a decoupling, the coupled 1D-3D model can then be reformulated as a fixed point iteration scheme iterating over the 1D pressure in the well and the 3D correction term for the reservoir. The iteration scheme can be implemented using both Galerkin and mixed finite element methods, the former of which leads to mass conservative solutions. The advantage of the decoupling and reformulation is twofold: Firstly, it recasts the model into a system for which the discretization schemes and solution methods are readily available. Secondly, it recovers optimal convergence rates without needing to perform a mesh-refinement around the well.


Article metrics loading...

Loading full text...

Full text loading...


  1. Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M. and Wells, G.
    [2015] The FEniCS Project Version 1.5.Archive of Numerical Software, 3(100).
    [Google Scholar]
  2. Cattaneo, L. and Zunino, P.
    [2013] Computational models for coupling tissue perfusion and microcirculation.MOX–Report, 25.
    [Google Scholar]
  3. [2014] A computational model of drug delivery through microcirculation to compare different tumor treatments. International Journal for Numerical Methods in Biomedical Engineering, 30(11), 1347–1371.
    [Google Scholar]
  4. D’Angelo, C.
    [2007] Multiscale modelling of metabolism and transport phenomena in living tissues, 183.
    [Google Scholar]
  5. [2012] Finite Element Approximation of Elliptic Problems with Dirac Measure Terms in Weighted Spaces: Applications to One- and Three-dimensional Coupled Problems. SIAM Journal on Numerical Analysis, 50(1), 194–215.
    [Google Scholar]
  6. D’Angelo, C. and Quarteroni, A.
    [2008] On the coupling of 1D and 3D diffusion-reaction equations: application to tissue perfusion problems. Mathematical Models and Methods in Applied Sciences, 18(08), 1481–1504.
    [Google Scholar]
  7. Grinberg, L., Cheever, E., Anor, T., Madsen, J.R. and Karniadakis, G.E.
    [2011] Modeling Blood Flow Circulation in Intracranial Arterial Networks: A Comparative 3D/1D Simulation Study. Annals of Biomedical Engineering, 39(1), 297–309.
    [Google Scholar]
  8. Kuchta, M., Nordaas, M., Verschaeve, J.C.G., Mortensen, M. and Mardal, K.A.
    [2016] Preconditioners for Saddle Point Systems with Trace Constraints Coupling 2D and 1D Domains. SIAM Journal on Scientific Computing, 38(6), B962–B987.
    [Google Scholar]
  9. Nabil, M. and Zunino, P.
    [2016] A computational study of cancer hyperthermia based on vascular magnetic nanoconstructs.Royal Society Open Science, 3(9).
    [Google Scholar]
  10. Peaceman, D.W.
    [1978] Interpretation of Well-Block Pressures in Numerical Reservoir Simulation.Society of Petroleum Engineers, 18(03).
    [Google Scholar]
  11. Reichold, J., Stampanoni, M., Keller, A.L., Buck, A., Jenny, P. and Weber, B.
    [2009] Vascular Graph Model to Simulate the Cerebral Blood Flow in Realistic Vascular Networks.Journal of Cerebral Blood Flow & Metabolism, 29(8), 1429–1443. PMID: 19436317.
    [Google Scholar]

Data & Media loading...

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error