Department of Applied Mathematics, **Machine** CM

Research School of Physical Sciences and Engineering

**Co-Investigators** Mark A Knackstedt and A P Roberts

Department of Applied Mathematics,

Research School of Physical Sciences and Engineering

**Transport in Bicontinuous Materials**

**What are the basic questions addressed?**

Disordered materials abound in nature and in man-made situations. Examples include ceramic composites, geologic media, polymer blends, foams and colloidal dispersions. The determination of effective properties (e.g. conduction, hydraulic transport, mechanical) of disordered composite materials is a subject of great importance in science and engineering. The microstructure of these materials is often quite complex and may possess intricate topology. The macroscopic (observable) transport properties of disordered materials are generally sensitive to the details of the microstructure. An important aspect of theoretically understanding the macroscopic behaviour of such media is to be able to generate and characterise suitable model microstructures. We are studying the effect of microstructure on the effective properties of composite materials.

**What are the results to date and the future of the work?**

Model random composites have been described by the level cuts of the superposition of random plane waves. The initial stage of the project was associated with the calculation of rigorous bounds for the conductivity of the Gaussian random field. Comparisons of theoretically calculated bounds for the effective conductivity with simulation data for a range of different composite morphologies was also made.

Conductivity of Oil-bearing Rocks

The oil industry has made extensive use of electrical resistivity measurements
in computing the porosity and the oil-bearing capacity of reservoir rocks and
electrical resistivity logging has been used to estimate the borehole yield
from sandstone aquifers. One of the most basic and useful empirical equations
to emerge from the study of sedimentary materials is Archie's law. The
equation links the conductivity *[[sigma]]r* and the porosity
*[[phi]]* of a fluid saturated media:

* [[sigma]]r = a[[sigma]]w[[phi]] ^{m}*

^{}where *[[sigma]]w* is the conductivity of the water and a and m
are empirical parameters that vary with the lithology (structure) of the rock
formation. The equation is viewed as a relation satisfied by a family of
porous systems with a range of porosities and a common geophysical history. We
have determined the Archie's law parameters a and m for media with a wide range
of porosities and for a wide variety of structures generated by level cuts of
gaussian random fields. We have compared the parameters we determine for
different structures with the empirically derived parameters for various rock
types including sandstones, limestones, carbonates and volcanic rocks.

Mechanical and Transport Properties of Model Foamed Solids

Many materials can be foamed, a process that dramatically extends the range of accessible properties. Techniques now exist for the fabrication of foamed solids from metals, polymers and glasses. Porous foams exhibit a wide variety of complex microstructures. While the engineering potential of porous foams is considerable, our ability to design and optimise pore structures is still very much ad hoc. A major limitation is any correlation between physical properties of the foam and microstructure. We evaluate the effective mechanical and thermal conductivity properties of such model porous solids using extensive computer simulation. The results are then compared with available experimental data for the thermal conductivity of insulating foams and the elastic properties of porous solids with excellent results

**What computational techniques are used and why is a supercomputer required?
**

The algorithms used to calculate the random field and simulate the effective conductivity were developed in consultation with Dr David Singleton of the ANUSF. The CMSSL Fortran library is employed in key areas of the implementation. Furthermore the algorithm has been compared with theoretical results for simple geometries and found to perform very well.

**Publications**

A P Roberts and M A Knackstedt, *J. Mat. Sci. Lett*, submitted.

A P Roberts and M A Knackstedt, *Scripta Metallurgica*, to be submitted.

A P Roberts and M A Knackstedt, *Geophys. Res. Letters*, submitted.