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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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Modeling liquid transport in the Earth's mantle as two-phase flow: effect of an enforced positive porosity on liquid flow and mass conservation
Abstract. Fluid and melt transport in the solid mantle can be modeled as a two-phase flow in which the liquid flow is resisted by the compaction of the viscously deforming solid mantle. Given the wide impact of liquid transport on the geodynamical and geochemical evolution of the Earth, the so-called “compaction equations” are increasingly being incorporated into geodynamical modeling studies. When implementing these equations, it is common to use a regularization technique to handle the porosity singularity in the dry mantle. Moreover, it is also common to enforce a positive porosity (liquid fraction) to avoid unphysical negative values of porosity. However, the effects of this “capped” porosity on the liquid flow and mass conservation have not been quantitatively evaluated. Here, we investigate these effects using a series of 1- and 2-dimensional numerical models implemented using the commercial finite-element package COMSOL Multiphysics®. The results of benchmarking experiments against a semi-analytical solution for 1- and 2-D solitary waves illustrate the successful implementation of the compaction equations. We show that the solutions are accurate when the element size is smaller than half of the compaction length. Furthermore, in time-evolving experiments where the solid is stationary (immobile), we show that the mass balance errors are similarly low for both the capped and uncapped (i.e., allowing negative porosity) experiments. When Couette flow, convective flow, or subduction corner flow of the solid mantle is assumed, the capped porosity leads to overestimations of the mass of liquid in the model domain and the mass flux of liquid across the model boundaries, resulting in intrinsic errors in mass conservation even if a high mesh resolution is used. Despite the errors in mass balance, however, the distributions of the positive porosity and peaks (largest positive liquid fractions) in both the uncapped and capped experiments are similar. Hence, the capping of porosity in the compaction equations can be reasonably used to assess the main pathways and first-order distribution of fluids and melts in the mantle.
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- Award ID(s):
- 2246804
- PAR ID:
- 10509475
- Publisher / Repository:
- Copernicus Publications
- Date Published:
- Journal Name:
- Solid Earth
- Volume:
- 15
- Issue:
- 1
- ISSN:
- 1869-9529
- Page Range / eLocation ID:
- 23 to 38
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.5194/se-15-23-2024
