Modeling Geometric State for Fluids in Porous Media: Evolution of the Euler Characteristic

Abstract

Multiphase flow in porous media is strongly influenced by the pore-scale arrangement of fluids. Reservoir-scale constitutive relationships capture these effects in a phenomenological way, relying only on fluid saturation to characterize the macroscopic behavior. Working toward a more rigorous framework, we make use of the fact that the momentary state of such a system is uniquely characterized by the geometry of the pore-scale fluid distribution. We consider how fluids evolve as they undergo topological changes induced by pore-scale displacement events. Changes to the topology of an object are fundamentally discrete events. We describe how discontinuities arise, characterize the possible topological transformations and analyze the associated source terms based on geometric evolution equations. Geometric evolution is shown to be hierarchical in nature, with a topological source term that constrains how a structure can evolve with time. The challenge associated with predicting topological changes is addressed by constructing a universal geometric state function that predicts the possible states based on a non-dimensional relationship with two degrees of freedom. The approach is validated using fluid configurations from both capillary and viscous regimes in ten different porous media with porosity between 0.10 and 0.38. We show that the non-dimensional relationship is independent of both the material type and flow regime. We demonstrate that the state function can be used to predict history-dependent behavior associated with the evolution of the Euler characteristic during two-fluid flow.

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Acknowledgements

J.M. and Z.L. thank Equinor ASA for funding parts of this research through a post-doc project. An award of computer time was provided by the Department of Energy Summit Early Science program. This research also used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725. T.R. acknowledges Equinor ASA for Granting permission to publish this paper.

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McClure, J.E., Ramstad, T., Li, Z. et al. Modeling Geometric State for Fluids in Porous Media: Evolution of the Euler Characteristic. Transp Porous Med 133, 229–250 (2020). https://doi.org/10.1007/s11242-020-01420-1

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Keywords

  • Euler characteristic
  • Topology
  • Integral geometry
  • Minkowski functionals
  • Multiphase flow