GENERAL OBJECTIVES:

  WORKPACKAGE 4.2: Inferences on the occurrence of the intra-crustal magma body.

Quoted from initial report: There are two complementary objectives of this workpackage:

 (a) development of a theoretical model for the elastic properties of a composite material consisting of melt (fluid) and crystals (solid),

 (b) development of an inversion method to constrain the percentage of melt in the magma reservoir(s) from seismic velocities obtained using the TOMOVES data.

  Mt. Vesuvius will erupt when the amount and supply of melt (low density and high viscosity material) in the magma reservoir is large. As the melt cools, crystals are formed, and the density and viscosity of the melt change. P- and S-wave velocities in the magma and the size of the magma reservoir(s) can be inferred from seismic methods. A theory to quantify the amount of melt in the magma reservoir will be developed. A map of melt content in the magma reservoir(s) will be obtained from the P- and S-wave velocities.
 

EXPECTED AND OBTAINED OBJECTIVES:

The objectives of the first and second six month phases were to set up and fully research the problem, and develop the theory and code for modeling the melt and crystal composite.

Both of these objectives have been achieved. Included here is a brief overview of the theory behind the method and an explanation of its application to the TOMOVES project.

  EFFECTIVE MEDIUM THEORY

  The principal tool in our analysis here is that of effective medium theory, to represent the overall behavior of a body as a propagating seismic wave 'sees' it, and thus to avoid having to specifically model the details of small-scale microstructure for which we have very few constraints. There have been quite a number of attempts to make quantitative estimates on how composite material properties vary with composition which broadly fall into two groups. There are those which take an essentially statistical approach and give upper and lower limits on the values, such as Hashin-Shtrikman and Voigt-Reuss bounds (Hashin and Shtrikman, 1963, and Reuss, 1929), but these are of little use in practice as the bounds are so far apart. The other general approach is to make a simplifying assumption about some aspect of the geometry or microstructure, such as a specific type of inclusion geometry which is what we do here. There are two principal existing theories on ways to implement this to encompass the whole range of phase concentrations, both of which start from the analytic solution for the elastic deformation due to the addition of a single inclusion in an infinite medium (Eshelby, 1957) e.g. addition of solid crystals to melt:

  1. SELF CONSISTENT APPROXIMATION (SCA)

  This method, pioneered by Budiansky (1965), Hill (1965) and Wu (1966), uses the solution for a single inclusion and approximates the interaction of many inclusions by replacing the background medium with that of the yet-to-be-determined effective medium. For any proportion of solid and fluid phases it gives explicit expressions for the resulting elastic moduli, which are coupled and must be solved by simultaneous iteration. This method has the advantage of being simple to compute, but one drawback lies in the interpretation of the microstructure which is that the solid inclusions are isolated below 40% fluid content, and the solid and fluid phases can only be considered to be mutually fully interconnected between 40 - 60%. For our application to magma chambers we would expect such interconnection at much lower crystal volume- fractions.

  2. DIFFERENTIAL EFFECTIVE MEDIUM (DEM) THEORY

  This theory models a two phase composite by incrementally adding inclusions of one phase to a background matrix of the other and then recomputing the new effective background material at each increment (Boucher, 1976, McLaughlin, 1977, Cleary et al., 1980, and Norris, 1985). The incremental approach allows the justification of calculations at any melt fraction irrespective of starting concentrations of two phases. This method is also implemented numerically and addresses the drawback of the SCA in that either phase can be fully interconnected at any concentration. However, the very aspect of the method that permits this brings with it the disadvantage that the starting microstructure essentially pre-determines the final background-inclusion structure.

  We attempt to take advantages of both these methods and minimise their shortcomings by using a combined effective medium method (Sheng, 1990, Hornby, 1994, Jakobsen et al., 1999), a combination of the SCA and DEM theory. Specifically we use the formulation originally written by Hornby (1994) for examining shales, and subsequently developed by Jakobsen et al. (1999) for gas hydrates. The SCA can be used to calculate the stiffness matrix for a bi-connected two phase material at a given concentration of fluid (in the 40 - 60% range) and then the DEM procedure is used to incrementally calculate the change in moving to the desired final composition which may be at any concentration. The inclusions we introduce are oblate spheroids, used because they can be characterised by a single parameter - the ratio of their semi-axes, 'alpha' (e.g. alpha = 1 for spherical pores, = 1:100 for thin films). The inclusions can be introduced in an aligned manner (where they all lie parallel), in which case the material is transversely anisotropic, or in a random, averaged fashion in which case the material is isotropic.
 

DELIVERABLES:

  The computer code implementation of the DEM theory calculations for seismic velocity and anisotropy can be made available upon request - the source code will need to be compiled and linked on the intended platform.

  Attached is a figure which shows the type of plots that can be generated; Variation of P- and S- wave velocity with melt fraction.
 

REFERENCES:

Boucher, S., 1976, Modules effectifs de materiaux composites quasi homogenes et quasi isotropes, constitues d'une matrice elastique et d'inclusions elastiques, II Cas des concentrations finies en inclusions, Rev. Metall., 22, 31 - 36.

Budiansky, B., 1965, On the elastic moduli of some heterogeneous materials, J. Mech. Phys. Solids, 13, 223 - 227.

Cleary, M. P., I.-W. Chen, and S.-M. Lee, 1980, Self-consistent techniques for heterogeneous media, Am. Soc. Civil Eng. J. Eng. Mech., 106, 861 - 887.

Eshelby, J. D., 1957, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Royal Soc. London, A241, 376 - 396.

Hashin, Z. and S. Shtrikman, 1963, A variational approach to the elastic behavior of multiphase materials, J. Mech. Phys. Solids, 11, 127 - 140.

Hill, R., 1965, A self-consistent mechanics of composite materials, J. Mech. Phys. Solids, 13, 213 - 222.

Hornby, B. E., 1994, The elastic properties of shales, Ph.D. University of Cambridge.

Jakobsen, M., J. A. Hudson, T. A. Minshull, and S. C. Singh, 1999, Elastic properties of hydrate-bearing sediments using effective medium theory, J. Geophys. Res., in press.

McLaughlin, R. A., 1977, A study of the differential scheme for composite materials, Int. J. Eng. Sci., 15, 237 - 244.

Norris, A. N., 1985, A differential scheme for the effective moduli of composites, Mechanics of Materials, 4, 1 - 16.

Reuss, A., 1929, Berechunung der fliessgrenzen von mischkristallen auf grund der plastiztatsbedingung fur einkristalle, Zeitschrift fur Angewandte Mathematik und Mechanik, 9, 49 - 58.

Sheng, P., 1990, Effective medium theory of sedimentary rocks, Physical Review B, 41, No. 7, 1236 - 1243.

Wu, T. T., 1966, The effect of inclusion shape on the elastic moduli of a two-phase material, Int. J. Solids and Structures, 2, 1- 8.

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