Research Project B03

Heterogeneous multi-scale methods for two-phase flow in dynamically fracturing porous media


Data sets published by project B03 can be found on the DaRUS website

Publications in scientific journals

  1. Conference papers

    1. Gander, M. J., Lunowa, S. B., & Rohde, C. (2022). Consistent and Asymptotic-Preserving Finite-Volume Robin Transmission Conditions for Singularly Perturbed Elliptic Equations. In S. C. Brenner, E. Chung, A. Klawonn, F. Kwok, J. Xu, & J. Zou (Eds.), Domain Decomposition Methods in Science and Engineering XXVI (pp. 443--450). Springer International Publishing.
    2. Burbulla, S., & Rohde, C. (2020). A Fully Conforming Finite Volume Approach to Two-Phase Flow in Fractured Porous Media. In R. Klöfkorn, E. Keilegavlen, F. A. Radu, & J. Fuhrmann (Eds.), Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples (pp. 547--555). Springer International Publishing.
  2. (Journal-) Articles

    1. Mel’nyk, T., & Rohde, C. (2024). Asymptotic expansion for convection-dominated transport in a thin graph-like junction. Analysis and Applications, 22(05), Article 05.
    2. Mel’nyk, T. A., & Rohde, C. (2024). Asymptotic approximations for semilinear parabolic convection-dominated transport problems in thin graph-like networks. Journal of Mathematical Analysis and Applications, 529(1), Article 1.
    3. Hörl, M., & Rohde, C. (2024). Rigorous derivation of discrete fracture models for Darcy flow in the limit of vanishing aperture. Networks and Heterogeneous Media, 19(1), Article 1.
    4. Mel’nyk, T., & Rohde, C. (2024). Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: Strong boundary interactions. Asymptotic Analysis, 137(1–2), Article 1–2.
    5. Burbulla, S., Hörl, M., & Rohde, C. (2023). Flow in Porous Media with Fractures of Varying Aperture. SIAM Journal on Scientific Computing, 45(4), Article 4.
    6. Burbulla, S., Formaggia, L., Rohde, C., & Scotti, A. (2023). Modeling fracture propagation in poro-elastic media combining phase-field and discrete fracture models. Computer Methods in Applied Mechanics and Engineering, 403, 115699.
    7. Gander, M. J., Lunowa, S. B., & Rohde, C. (2023). Non-Overlapping Schwarz Waveform-Relaxation for Nonlinear Advection-Diffusion Equations. SIAM Journal on Scientific Computing, 45(1), Article 1.
    8. Burbulla, S., Dedner, A., Hörl, M., & Rohde, C. (2022). Dune-MMesh: The Dune Grid Module for Moving Interfaces. Journal of Open Source Software, 7(74), Article 74.
    9. Seus, D., Radu, F. A., & Rohde, C. (2022). Towards hybrid two-phase modelling using linear domain decomposition. Numerical Methods for Partial Differential Equations.
    10. Burbulla, S., & Rohde, C. (2022). A finite-volume moving-mesh method for two-phase flow in fracturing porous media. Journal of Computational Physics, 111031.
    11. Berre, I., Boon, W. M., Flemisch, B., Fumagalli, A., Gläser, D., Keilegavlen, E., Scotti, A., Stefansson, I., Tatomir, A., Brenner, K., Burbulla, S., Devloo, P., Duran, O., Favino, M., Hennicker, J., Lee, I.-H., Lipnikov, K., Masson, R., Mosthaf, K., … Zulian, P. (2021). Verification benchmarks for single-phase flow in three-dimensional fractured porous media. Advances in Water Resources, 147, 103759.
    12. Koch, T., Gläser, D., Weishaupt, K., Ackermann, S., Beck, M., Becker, B., Burbulla, S., Class, H., Coltman, E., Emmert, S., Fetzer, T., Grüninger, C., Heck, K., Hommel, J., Kurz, T., Lipp, M., Mohammadi, F., Scherrer, S., Schneider, M., … Flemisch, B. (2020). DuMux 3 – an open-source simulator for solving flow and transport problems in porous media with a focus on model coupling. Computers & Mathematics with Applications.
    13. Köppel, M., Martin, V., Jaffré, J., & Roberts, J. E. (2019). A Lagrange multiplier method for a discrete fracture model for flow in porous media. Computational Geosciences, 23(2), Article 2.
    14. Seus, D., Mitra, K., Pop, I. S., Radu, F. A., & Rohde, C. (2018). A linear domain decomposition method for partially saturated flow in porous media. Computer Methods in Applied Mechanics and Engineering, 333, 331--355.


About this Project

The local geometry of single fractures as well as the global structure of entire fracture networks in fluid-filled porous media can change significantly in time, depending in an intricate way on the likewise inter-related flow field in the pore/fracture space and on the mechanical displacement field in the solid skeleton.

We consider incompressible two-phase flow in suchlike dynamically changing porous media on a continuum-mechanical REV scale of cm-length. Under the assumption that the mean fracture aperture is separated from the REV scale as well as from the pore scale, the fractures are kept up to the REV scale as time-dependent manifolds of co-dimension one. Thus we apply a discrete fracture network approach which enables us to equip the fracture network with its own lower-dimensional flow models and to account for arbitrary interfacial dynamics.

The major goal of this project is to develop models and associated numerical discretisations for increasing geometric and fluid-mechanical complexity. In particular we are interested in the case where the propagation of the fracture network is driven by the fluid pressure field and by stress-induced deformations of the porous medium. For the case of propagating and possibly bifurcating fractures, the network becomes part of the unknowns and the whole problem a (kind of) free boundary value problem. The growth of the fracture depends on microscopic pore scale physics such that the overall dynamics requires a multi-scale ansatz.


We have developed models and discretisation methods for evolving networks of fractures. Having the same type of two-phase flow dynamics in the fractures and the bulk porous medium we have been able to perform simulations for infiltration fronts in two and three space dimensions.

Pressure field for single-phase flow in a fractured 3D porous medium.

Our major new contribution is a numerical method that accounts for fracture propagation. The Riemann-solver based finite-volume method have been generalised to cover the case of propagating fracture evolution still using a conforming mesh.  For artificially given geometrical evolution an existing moving-mesh concept has been further developed to provide at any discrete time-step a mesh which displays the (discrete) fracture as connected mesh facets. We have used the re-meshing routines together with Dumux and have simulated two-phase flow scenarios with given propagation, bifurcation and crossing of fractures.

All code has been developed within the DUNE framework, and a grid manager based on our implementation dune-mmesh has been added to the CRC~1313 platform Dumux. In order to provide sustainable software, we also have created python packages of DUNE and uploaded them to the Python Package Index.

Infiltration of a wetting-fluid front from the left domain boundary into a propagating fracture network with bifurcating pattern.

In cooperation with Project B01 a heterogeneous multi-scale method has been developed to track the propagation of fractures by a phase-field approach as the microscopic model. An implementation of poroelasticity on the basis of an enhanced finite-element discretisation is available within Dumux and has been coupled to our model implementation. The phase-field model uses the force fields to determine the small-scale evolution of the fracture. It is only solved on a small domain around the fracture tip by the FEM approach developed in project B01.

To explore the range of validity of this approach the implemented software framework is currently used for a comparative study with micro-fluidic experiments in B04.

Numerical results for the multi-scale method. The pressure field of the discrete fracture model (left) is coupled to a phase-field model (right) that governs the fracture tip movement in a small region around it. The grey shadow indicates bulk porous medium with higher Griffith critical energy.

Future Work

The first major objective of the project in the future is to develop mathematical models and corresponding numerical methods that apply to highly diversified, time-dependent fracture geometries and to heterogeneous flow regimes, i.e., the flow model in fracture and bulk porous medium might differ.

We will advance the models to cover spatially and temporally varying fracture geometries driven by, and interacting with the flow and mechanical displacement fields. In the fractures we will employ aperture-averaged models that account for e.g. high-permeability or capillarity-driven flow regimes. The heterogeneous multi-scale method will be generalised towards time-dependent networks with changing topology due to fracture interaction and branching.

The second objective is to validate the new models and discretisation methods by appropriate benchmarking with D03, based on the simulation results on experimental data from Project B05.

From the mathematical point-of-view we aim at providing analytical results for the new aperture-averaged models coupled to the strongly non-linear mixed-type systems in the fully-dimensional bulk domains.

The major numerical advancement will be the complete transfer of the implementation to a higher-order DG setting.

International cooperation

University of Bergen

Together with the Mercator Fellow  Florin Radu we provide convergence statements for a domain-decomposition approach based on  the usage of the L - scheme.

For further information please contact

This image shows Christian Rohde

Christian Rohde

Prof. Dr. rer. nat.

Deputy Spokesperson, Principal Investigator, Research Projects B03 and C02, Project MGK

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