Adaptive Multi-Scale Pore Network Method for Two-Phase Flow in Porous Media

Abstract:

Dynamic pore network simulators are important tools in studying macroscopic quantities in two-phase flow through porous media. However, these simulators have a time complexity of order ${\mathrm{N²}}^{ }$for $N$ pore bodies, which limits their usage to small domains. Quasi-static pore network simulators, which assume capillary dominated flow, are more efficient with a time complexity of order $Nlog(N)$, but are unable to capture phenomena caused by viscous effects such as viscous fingering and stable displacement.

It has been experimentally observed that, in several flow scenarios, capillary forces are dominant at the pore scale and viscous forces at larger scales. In order to take advantage of this behaviour and to reduce the time complexity of existing dynamic pore network simulators, we propose a multi-scale pore-network method for two phase flow. In our solution algorithm, the pore network is first divided into smaller subnetworks. The algorithm to advance the fluid interfaces within each subnetwork consists of three steps: 1) The saturation rate of each subnetwork is obtained by solving a two-phase meso-scale mass balance equation over the domain of subnetworks. Here, a multi-point flux scheme is used. 2) Depending on the local capillary number computed in the subnetwork, either an invasion percolation algorithm or a dynamic network algorithm is used to locally advance the fluid-fluid interfaces within each subnetwork until a new saturation value is matched. 3) The transmissibilities for the meso-scale equation are updated based on the updated fluid configurations in each subnetwork. For this purpose the methodoloy of the existing multi-scale finite volume (MSFV) method is employed.

An important feature of the multi-scale pore-network method is that it maintains consistency of both fluid occupancy and fluxes at subnetwork interfaces. Viscous effects such as viscous fingering (see figure) can be captured at a decreased computational cost compared to dynamic pore network simulators. In addition, it is straightforward to parallelize the solution algorithm. Validation studies are presented and compared to results obtained with an existing dynamic pore network model. 

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