Open Access

Suranita Kanjilal

• Melt segregation inside the earth consists of two different processes: 1) Generation of partially molten rock and 2) separation of melt, produced from partially molten rock, from the solid residual matrix. This thesis focuses on the later process. The 2 phase flow dynamics combines the study of flow dynamics of melt and matrix. Several studies have given the background theoretical frameworks for the flow dynamics of melt inside the earth. [McKenzie, 1984] summarizes the studies of [Ahern and Turcotte, 1979; Frank, 1968; Sleep, 1975] and gives a complete set of governing equations for the 2-phase flow problem. [Bercovici et al., 2001] gives a general formulation considering the univariate system of equations related to matrix and melt flow which includes the interfacial surface force. The assumption of melt having negligible viscosity compare to the matrix has been abandoned. Therefore, based on these formulations, we have constructed our numerical model and thereafter a fortran code PERCOL2D to get an insight of melt percolation process through porous media. Additionally, we have used the Helmhotz decomposition, which splits a smooth and rapidly decaying vector field into an irrotational vector field and an incompressible vector field [Srámek, 2007], for matrix and fluid viscosity in order to lower the number of linearly independent variables to minimize the computational complications. The melt residing at inter-granular areas of lithosphere, forms an interconnected network even at low porosity. Therefore, being less dense than the matrix, melt moves up through porous media due to its buoyancy. Compaction of matrix, which occurs to compensate the melt separation, is considered in this thesis, where the effective bulk and shear viscosity of matrix are function of melt fraction. We have effective bulk viscosity of matrix as inversely proportional to melt fraction. Porosity dependence of effective bulk and shear viscosity leads to stronger melt focusing in highly porous region like mid ocean ridges [Katz, 2008] since the ratio of bulk and shear viscosity is smaller (< 10) than the constant viscosity case for the porous waves having non dimensional amplitude 5% or higher. Moreover, it is observed in [Richard et al., 2012] that the solitary wave formed in porosity dependent viscous matrix settings are steeper than the one formed in the constant matrix viscosity setting. Firstly some 1D numerical experiments with PERCOL2D have been carried out using fixed and periodic boundary conditions for zero source term (i.e. no melting or no freezing) and negligible surface tension. 3 series of model setups with different initial conditions have been carried out varying the width, non-dimensional amplitude and the background porosity value of the initial input of porous wave. A mathematical derivation for 1D solitary wave solution for the two phase flow through porosity dependent compacting media, is obtained in this thesis which is different than the study of [Barcilon and Lovera, 1989; Barcilon and Richter, 1986; Scott and Stevenson, 1984; Spiegelman, 1993a,b] as the effective viscosity of matrix is constant there. Although [Simpson and Spiegelman, 2011] gives the solitary wave solutions in 1D, 2D and 3D considering the porosity dependent effective viscosity of the matrix, but using the small background porosity approximation, they neglect the background porosity (i.e φ0) and therefore the effect of variation of compaction lengths, which causes variation in the shape and dynamics of the solitary wave. Therefore, the study [This thesis, Richard et al., 2012] can be used for more general purpose. Solitary waves in varying viscous medium, are steeper (cf fig.5.1) compared to the one in constant viscous medium and their speed decreases as an inverse function of the background porosity. Additionally, this analytical solution is used in our code PERCOL2D and also in FDCON for numerical benchmarking (1D) of PERCOL2D. The role of melt grain contiguity is considered in the revised viscosity formulation [Schmeling et al., 2012] based on elastic moduli theory of a fluid filled poro-elastic medium. This formulation is used in this thesis to produce a comparative dispersion relationship between speed of the wave and the non dimensional amplitude of porous wave, based on both the viscosity formulations (fig. 6.20) where one can see that the model based on [Bercovici et al., 2001] formulation, converges to the same dispersion relationship obtained from [Simpson and Spiegelman, 2011]. Whereas, the dispersion relationship using [Schmeling et al., 2012] formulations, shows time-dependent decrease of phase velocity with increasing amplitude and it is not yet clear that whether these solutions converge to steady state porosity waves before the porosity becomes 1.