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In partially molten regions inside the earth melt buoyancy may trigger upwelling of both solid and fluid phases, i.e. diapirism. If the melt is allowed to move separately with respect to the matrix, melt perturbations may evolve into solitary porosity waves. While diapirs may form on a wide range of scales, porosity waves are restricted to sizes of a few times the compaction length. Thus, the size of a partially molten perturbation controls whether a diapir or a porosity wave will emerge. We study the transition from diapiric rise to solitary porosity waves by solving the two-phase flow equations of conservation of mass and momentum in 2D with porosity dependent matrix viscosity. We systematically vary the initial size of a porosity perturbation from 1 to 100 times the compaction length. If the perturbation is much larger than a regular solitary wave, its Stokes velocity is large and therefore faster than the segregating melt. Consequently, the fluid is not able to form a porosity wave and a diapir emerges. For small perturbations solitary waves emerge, either with a positive or negative vertical matrix velocity inside. In between the diapir and solitary wave regimes we observe a third regime of solitary wave induced focusing of melt. In these cases, diapirism is dominant but the fluid is still fast enough to locally build up small solitary waves which rise slightly faster than the diapir and form finger like structures at the front of the diapir. In our numerical simulations the width of these fingers is controlled by the compaction length or the grid size, whichever is larger. In cases where the compaction length becomes similar to or smaller than the grid size the finger-like leading solitary porosity waves are no more properly resolved, and too big and too fast waves may be the result. Therefore, one should be careful in large scale two-phase flow modelling with melt focusing especially when compaction length and grid size are of similar order.
In partially molten regions inside the Earth, melt buoyancy may trigger upwelling of both solid and fluid phases, i.e., diapirism. If the melt is allowed to move separately with respect to the matrix, melt perturbations may evolve into solitary porosity waves. While diapirs may form on a wide range of scales, porosity waves are restricted to sizes of a few times the compaction length. Thus, the size of a partially molten perturbation in terms of compaction length controls whether material is dominantly transported by porosity waves or by diapirism. We study the transition from diapiric rise to solitary porosity waves by solving the two-phase flow equations of conservation of mass and momentum in 2D with porosity-dependent matrix viscosity. We systematically vary the initial size of a porosity perturbation from 1.8 to 120 times the compaction length. If the perturbation is of the order of a few compaction lengths, a single solitary wave will emerge, either with a positive or negative vertical matrix flux. If melt is not allowed to move separately to the matrix a diapir will emerge. In between these end members we observe a regime where the partially molten perturbation will split up into numerous solitary waves, whose phase velocity is so low compared to the Stokes velocity that the whole swarm of waves will ascend jointly as a diapir, just slowly elongating due to a higher amplitude main solitary wave. Only if the melt is not allowed to move separately to the matrix will no solitary waves build up, but as soon as two-phase flow is enabled solitary waves will eventually emerge. The required time to build them up increases nonlinearly with the perturbation radius in terms of compaction length and might be too long to allow for them in nature in many cases.
Highlights
• Solidification and cooling of an intruded dyke or sill within the middle or shallow crust generate stresses of order 200 MPa, which relax on time scales of a few years to million years.
• Stresses may exceed the brittle strength forming tensile fractures.
• Combined with the pressure gradient within the over-pressurized felsic melts, this explains the migration of felsic contact melt into shrinkage cracks (Sederholm-type veins).
Abstract
Rapid emplacement of a mafic dyke or sill at mid-crustal depth heats and possibly melts the felsic wall rock followed by solidification. Associated volume changes generate stresses, possibly enforcing brittle failure and melt migration. We model the evolution of melting, solidification, temperature, and stress including visco-elastic relaxation in 1D - dykes or -sills using realistic rock rheologies of the Weschnitz pluton (Odenwald). For deep emplacement (Case 1, 15.3 km) extensive contact melting of the wall rock occurs, for shallow emplacement (Case 2, 10 km) it is negligible. The stresses are zero at high melt fractions, but increase during solidification and cooling: The intrusion orthogonal stress is always zero. The intrusion parallel stress σ‖ within the intrusion is tensile (O(200 MPa)). It relaxes on a time scale between a few years (Case 1) and 0.6 m.y. (Case 2). Within the wall rock σ‖ is compressive during heating, but becomes tensile under solidification and cooling. Wall rock stresses relax on a time scale of months to 100 years. A Deborah number is defined based on viscous to thermal relaxation allowing generalization of our results. Adding lithostatic stresses, the total stresses of Case 1 remain below the brittle strength, while for Case 2 they may exceed it. Adding the lithostatic pressure to the melt pressure, the effective stresses exceed the brittle strength and intrusion orthogonal tensile fractures are predicted. Combined with the pressure gradient within the over-pressurized felsic melts generated in the wall rock, this explains the migration of felsic contact melt into shrinkage cracks of the mafic sill in the Weschnitz pluton.