Numerical modeling of two-phase flow for solitary porosity waves and transport of trace elements within melt
Authors:   , Dohmen Janik 
Published: June 2023

Melting inside earth is a common phenomenon and can be observed in many different regions where melt travels through the mantle and crust to eventually reach the surface where it crystallizes to build large volcanic provinces, whole stratigraphic layers of flood basalts, or even the oceanic crust. Often, melt reaching the surface is a good source of information. It can be used to achieve a better understanding about processes taking place in deeper regions inside the mantle and it is therefore essential to fundamentally understand melting and melt percolation processes. In order to achieve a deeper understanding, the aim of this thesis is to investigate processes that are connected to melting by using numerical models. The physical model used is a so called two-phase flow model which describes the ability of melt to percolate through a viscously deforming, partially molten matrix. A famous feature of two-phase flow are solitary porosity waves, which are waves of locally higher porosity ascending through a partially molten background, keeping its shape constant, driven by decompaction and compaction of the solid matrix in front and behind the wave. The viscosity law for shear- and volume viscosity was strongly simplified in most previous studies that modeled solitary waves. Often the porosity dependency is underestimated or its influence on the volume viscosity is even neglected, leading to too high viscosities. In this work more realistic laws are used that strongly decrease for small melt fractions. Those laws are incorporated into a 2D Finite Difference mantle convection code with two-phase flow to study the ascent of solitary porosity waves. The model results show that an initial Gaussian-shaped wave rapidly evolves into a solitary wave with a certain amplitude, traveling upwards with constant velocity. Even though strongly weaker viscosities are used, the effect on dispersion curves and wave shape are only minor as long as the background porosity is rather small. The results are still in agreement to semi-analytical solutions which neglect shear stresses in the melt segregation equation. Higher background porosities and wave amplitudes lead to significant decrease in phase velocity and wave width, as the viscosity is strongly effected. However, the models show that solitary waves are still a possible mechanism for more realistic matrix viscosities. While the ascending of porosity waves are mostly described by the movement of fluid melt, partially molten regions inside Earth trigger upwelling of both, solid and fluid phases, which can be called diapirism. While diapirs can have a wide variety of wavelengths, porosity waves are restricted to a few times the compaction length. The size of a melt perturbation in terms of compaction length therefore describes whether material is transported by diapirism or porosity waves. In this thesis we study the transition from diapiric rise to solitary porosity waves by systematically changing the size of a porosity perturbation from 1.8 to 120 times the compaction length. In case of a perturbation of the size of a few times the compaction length a single porosity wave will emerge, either with a positive or negative vertical matrix flux and if melt is not allowed to move relative to the matrix a diapir will emerge. In between these physical end members a regime can be observed where the partially molten perturbation will split up into numerous solitary waves, whose phase velocity is low compared to the Stokes velocity and the swarm of solitary waves will ascend jointly as a diapir, slowly elongating due to a higher amplitude main solitary wave. Solitary waves will always emerge from a melt perturbation as long as two-phase flow is enabled, but the time for a solitary wave to emerge increases non-linearly with the perturbation radius in terms of compaction length. In nature, in many cases this time might be too long for solitary waves to emerge. Another important feature when it comes to two-phase flow is the transport of trace elements in melt. Incompatible elements prefer to go into the melt, which eventually enriches the area where it crystallizes again. In order to model this redistribution, the code FDCON was extended to allow for fully consistent transport of elements in melt, including melting, freezing and re-equilibration with time. A 2D model, a simple representation of a volcanic back arc, is set up to investigate the behavior of trace elements. The influence of retention number and re-equilibration time is examined. Lava-lamp like convection can be observed in the lower part of the model, producing melt, that eventually leads to enrichment in trace elements in the upper high-viscous layer. The total enrichment in this layer approaches an asymptotic value and a 0D model is introduced to recreate this behavior.