1887

Abstract

Summary

We present the recently developed Double Control Volume Finite Element Method (DCVFEM) in combination with dynamic mesh adaptivity in parallel computing to simulate immiscible viscous fingering in two- and three-dimensions.

Immiscible viscous fingering may occur during the waterflooding of oil reservoirs, resulting in early breakthrough and poor areal sweep. Similarly to miscible fingering it is triggered by small-scale permeability heterogeneity while it is controlled by the mobility ratio of the fluid and the level of transverse dispersion / capillary pressure. Up to this day, most viscous fingering studies have focussed on the miscible problem since immiscible fingering is significantly more challenging. It requires numerical simulations capable to capture the interaction of the shock front with the capillary pressure, which is a saturation dependent dispersion term. That leads to models with very fine mesh in order to minimise numerical diffusion, resulting in computationally intensive simulations.

In this study, we apply the dynamic mesh adaptive DCVFEM in parallel computing to simulate immiscible viscous fingering with capillary pressure. Parallelisation is achieved by using the MPI libraries. Dynamic mesh adaptivity is achieved by mapping of data between meshes. The governing multiphase flow equations are discretised using double control volumes on tetrahedral finite elements. The discontinuous representation for pressure and velocity allows the use of small control volumes, yielding higher resolution of the saturation field.

We demonstrate convergence of fingers using our parallel numerical method in 2d and 3d, on fixed and adaptive meshes, quantifying the speed-up due to parallelisation and mesh adaptivity and the achieved accuracy. Dynamic mesh adaptivity allows resolution to be automatically employed where it is required to resolve the fingers with lower resolution elsewhere, enabling capture of complex non-linearity such as tip-splitting. We achieve convergence with less than 10k elements, approximately 5 times fewer elements than are needed for the converged fixed mesh solution, consequently the computational cost is also significantly reduced.

Initial growth rates as a function of wavenumber, viscosity ratio, relative permeability and capillary pressure are compared with the literature. We demonstrate that the structure of the mesh plays a key role in simulation of fingering as it can itself trigger the instability, control the dominant wavelength of the fingers and their growth rate. We show that it is important to characterise the amount of transverse to longitudinal numerical diffusion in a general unstructured mesh in order to ensure that the correct fingering pattern is simulated.

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2019-04-08
2020-04-09
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