We consider the system of equations arising from mantle dynamics introduced by McKenzie (J. Petrology, 1985). In this multi-phase model, the fluid melt velocity obeys Darcy's law while the deformable "solid" matrix is governed by a highly viscous Stokes equation. The system is then coupled through mass conservation and compaction relations. Together these equations form a coupled Darcy-Stokes system on a continuous single-domain mixture of fluid and matrix. The porosity φ, representing the relative volume of fluid melt to the bulk volume, is assumed to be much smaller than one. When coupled with solute transport and thermal evolution in a time-dependent problem, the model transitions dynamically from a non-porous single phase solid to a two-phase porous medium. Such mixture models have an advantage for numerical approximation since the free boundary between the one and two-phase regions need not be determined explicitly. The equations of mantle dynamics apply to a wide range of applications in deep earth physics such as mid-ocean ridges, subduction zones, and hot-spot volcanism, as well as to glacier dynamics and other two-phase flows in porous media. Mid-ocean ridges form when viscous corner flow of the solid mantle focuses fluid toward a central ridge. Melt is believed to migrate upward until it reaches the lithospheric "tent" where it then moves toward the ridge in a high porosity band. Simulation of this physical phenomenon required confidence in numerical methods to handle highly heterogeneous porosity as well as the single-phase to two-phase transition. In this work we present a standard mixed finite element method for the equations of mantle dynamics and investigate its limitations for vanishing porosity. While stable and optimally convergent for porosity bounded away from zero, the stability estimates we obtain suggest, and numerical results show, the method becomes unstable as porosity approaches zero. Moreover, the fluid pressure is no longer a physical variable when the fluid phase disappears and thus is not a good variable for numerical methods. Inspired by the stability estimates of the standard method, we develop a novel stable mixed method with uniqueness and existence of solutions by studying a linear degenerate elliptic sub-problem akin to the Darcy part of the full model: [mathematical equation], where a and b satisfy a(0)=b(0)=0 and are otherwise positive, and the porosity φ ≥ 0 may be zero on a set of positive measure. Using scaled variables and mild assumptions on the regularity of φ, we develop a practical mass-conservative method based on lowest order Raviart-Thomas finite elements. Finally, we adapt the numerical method for the sub-problem to the full system of equations. We show optimal convergence for sufficiently smooth solutions for a compacting column and mid-ocean ridge-like corner flow examples, and investigate accuracy and stability for less regular problems / text
| Identifer | oai:union.ndltd.org:UTEXAS/oai:repositories.lib.utexas.edu:2152/28357 |
| Date | 09 February 2015 |
| Creators | Taicher, Abraham Levy |
| Source Sets | University of Texas |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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