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We consider the interaction between a free flowing fluid and a porous medium flow, where the free flowing fluid is described using the time dependent Stokes equations, and the porous medium flow is described using Darcy’s law in the primal formulation. To solve this problem numerically, we use a diffuse interface approach, where the weak form of the coupled problem is written on an extended domain which contains both Stokes and Darcy regions. This is achieved using a phase-field function which equals one in the Stokes region and zero in the Darcy region, and smoothly transitions between these two values on a diffuse region of width (ϵ) around the Stokes-Darcy interface. We prove convergence of the diffuse interface formulation to the standard, sharp interface formulation, and derive rates of convergence. This is performed by deriving a priori error estimates for discretizations of the diffuse interface method, and by analyzing the modeling error of the diffuse interface approach at the continuous level. The convergence rates are also shown computationally in a numerical example.
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A Three‐Field Formulation for Two‐Phase Flow in Geodynamic Modeling: Toward the Zero‐Porosity Limit
Abstract Two‐phase flow, a system where Stokes flow and Darcy flow are coupled, is of great importance in the Earth's interior, such as in subduction zones, mid‐ocean ridges, and hotspots. However, it remains challenging to solve the two‐phase equations accurately in the zero‐porosity limit, for example, when melt is fully frozen below solidus temperature. Here we propose a new three‐field formulation of the two‐phase system, with solid velocity (vs), total pressure (Pt), and fluid pressure (Pf) as unknowns, and present a robust finite‐element implementation, which can be used to solve problems in which domains of both zero porosity and non‐zero porosity are present. The reformulated equations include regularization to avoid singularities and exactly recover to the standard single‐phase incompressible Stokes problem at zero porosity. We verify the correctness of our implementation using the method of manufactured solutions and analytic solutions and demonstrate that we can obtain the expected convergence rates in both space and time. Example experiments, such as self‐compaction, falling block, and mid‐ocean ridge spreading show that this formulation can robustly resolve zero‐ and non‐zero‐porosity domains simultaneously, and can be used for a large range of applications in various geodynamic settings.
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- Award ID(s):
- 2121568
- PAR ID:
- 10484097
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Journal of Geophysical Research: Solid Earth
- Volume:
- 129
- Issue:
- 1
- ISSN:
- 2169-9313
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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