Title:

Melt extraction from a permeable compacting mantle

In this thesis, I present one and threedimensional numerical solutions to a twophase fluid flow problem. The context of these investigations is the evolution of a viscous permeable matrix with a small fraction of melt that is representative of partial melt in the Earth's mantle. The matrix compacts under gravity as melt moves upward. In addition to the simple compaction solution, a range of solutions representing stably propagating waves are possible. I first present a coherent mathematical development of the governing equations for the threedimensional problem. I then describe a onedimensional numerical algorithm (1D2PF) that solves the secondorder inhomogeneous P.D.E. for the velocity of the viscous matrix, V, for arbitrary melt fraction distribution, φ (the volume fraction occupied by melt). Combined with a timestepping algorithm which advances the melt fraction in time, fully timedependent 1D solutions are obtained. With an initial constant base melt fraction φ0 with a superposed localised concentration of melt, I explore the evolution and formation of solitary compaction waves. Using (1D2PF) I investigate the width, amplitude and phase velocity of stable solitary waves, and examine how these parameters depend on the initial conditions, permeability coefficient (k0) and melt and matrix viscosities (ηf and ηm). I demonstrate the existence of a threshold initial width above which secondary solitary waves form, with larger widths producing longer wave trains and smaller widths producing a smallamplitude oscillatory disturbance to the background melt fraction. Experiments with k0, ηf and ηm reveal that the width of the stable solitary wave is simply proportional to the compaction length parameter δ=√k0ηm/ ηf and its velocity varies as δ16/ 9/ηm . I also show that the width of the solitary waves varies as λS=4.6δ and the amplitude follows the relation AS≃89/δ . For initial melt fractions whose distribution is wider than the threshold width, secondary waves are produced with progressively smaller amplitude, and hence slower propagation velocity. I demonstrate that smaller values of δ result in the same volume of melt being partitioned over increasing numbers of relatively thinner solitary waves. The amplitude of the initial perturbation to the background melt fraction however is shown to have no effect on the number of solitary waves produced. A train of such waves arriving at the surface could provide an explanation of intermittentvolcanic activity above a region of partial melt. In a preliminary study of twophase flow in threedimensions I have also made significant progress toward the development of a threedimensional twophase flow simulation program. To do so, I have adapted the threedimensional viscous fluid convection program (TDCON) by Houseman (1990). The new program TD2PF depends on a potentialfunction formulation similar to that of Spiegelman (1993a), in which the divergence of the matrix velocity field, D=∇·V, and the vector potential, A, are the primary variables. I have introduced new functionality to a significantly expanded threedimensional Poisson solver (program TDPOTS) but lack of time prevented a successful conclusion to the development of a general 3D solver for the divergence field D.
