A Stochastic Particle-Based Framework for Multiphase Flows in Fractured Porous Media

Abstract

Summary An accurate prediction of transport in sub-surface formations is challenging for Eulerian schemes, especially if advection dominates. Lagrangian particle-tracking schemes, not being hindered by numerical dispersion, are compelling alternatives and have increased relevance in the modelling of non-linear transport, e.g., saturation transport in a multi-phase setting [1]. In this work, we have focused on the development of a particle-tracking scheme for multiphase flows in fractured media with a permeable matrix. To this end, we adopted an Embedded Discrete Fracture Model where fractures are treated as lower dimensional manifolds [2]. The fracture-matrix interfaces are not resolved at the sub-grid level, thereby necessitating an alternative modeling strategy for inter-continuum interactions. In [3], we presented a stochastic particle-tracking scheme for advective solute transport in single-phase flow. A particle’s continuum state is modeled as a two-state Markov chain with transition probabilities for inter-continuum particle transfer. The probabilities are pathline-specific and scale with the particle’s modeled travel time through the grid cell. In a first step, we aim to improve the efficiency of the above-mentioned scheme, especially for the scenarios involving high contrast in matrix-fracture pore-volumes. Then, the scheme is extended to model saturation evolution in two-phase immiscible flows. Following the work of [1], saturation is modeled as a statistical quantity and is estimated with two distinct particle ensembles, i.e., one for each phase. In the absence of dispersive effects, e.g., in flows with high Péclet numbers and without capillary pressure differences, saturation discontinuities are a typical feature. Therefore, near the fronts, inaccurate estimates and instabilities may result due to finite-sized particle ensembles in cells of the computational domain. To render such inaccuracies and instabilities insignificant, an adaptive diffusivity is added to the system, which selectively acts near the front and attains negligible values away from it. To quantify the diffusivity, a Smagorinsky-type [4, 5] model is proposed, which scales with the magnitude of the saturation gradient. In the pursuit of efficient yet accurate large-scale modeling, the proposed Lagrangian scheme can be extended to connect dynamic and locally unresolved sub-grid processes, e.g., phase dissolution, with the macroscopically observed effects.