The main focus of my work is characterising properties of disordered materials. Here is an overview of my recent research.

Disordered Materials

Properties of Microscopically Disordered Materials

In many areas of science and engineering a key problem is to predict the effective properties of composite media from the properties and spatial distribution (or microstructure) of its constituent components. There are two broad questions in the study of random composites. Firstly, how can the complex microstructure observed in physical composites be realistically modeled. And, secondly, how can the properties (conductivity, diffusivity, permeability, elasticity etc) be predicted from measurements of composite morphology. The solution of these general problems will answer important questions in a range of fields. For example, the processes of oil recovery and groundwater contamination depend strongly on the microstructure of porous rocks. Furthermore the useful properties of a range of engineered materials are a direct consequence of their microstructure. These include; membranes, filters, silica aerogels, foamed solids in insulation and mechanical applications, and other composites such as cements and concrete. To optimize the properties of these materials, it is necessary to understand the precise role of microstructure.

Theoretical development of Microstructural Models

Past theoretical attempts to relate macroscopic physical properties to a description of microstructure of the porous material have been limited. Most methods use oversimplified representations of the pore structure for which properties can be analytically evaluated. Examples include simple micro-porous models (dilute spheres, cylindrical tubes, periodic media) and effective medium approaches. These models do not explicitly account for microstructure. We have extended microstructural models developed in the 1960's and have shown that the model accounts for features observed in a range of materials. Derivation of mechanical, conductive and single-phase flow properties of the materials have yielded excellent agreement with experimentally measured properties of disordered materials. The model can be tuned to match complex morphologies as required. The agreement now allows correlation of effective physical properties of disordered solids to microstructure. This year we have embarked on a study of the dielectric and optical properties of microstructured materials. This study will impact particularly on applications to the paper and coating industry (see section 1.3).

 
Figure 1: Examples of Model microstructures

Mark A. Knackstedt, Christoph Arns(UNSW), John Daicic(YKI), A.P. Roberts(ex-member).

Anisotropic Materials

Previously we have focused on isotropic materials. This year we have extended the work to the study of anisotropic materials. In empirical derivations the most important parameter characterising material structure is relative density or porosity. Second in importance is anisotropy; the tendency for materials to be elongated or flattened. Most man-made materials are anisotropic. Polymer foams utilise foaming agents which produce cells that are elongated in the rise direction. Natural materials like wood, bone and coral, which have evolved to meet needs in an efficient manner, are anisotropic. Cell walls in trees for example grow in response to load aligning themselves along the direction of principal stress- parallel to the trunk. Due to horizontal stratification associated with geological bedding, sedimentary rocks are almost always anisotropic. Material properties depend strongly upon orientation. The elastic modulus of wood can differ by an order of magnitude along the grain compared to across the grain. Similar differences are noted when comparing horizontal and vertical permeability of sedimentary rock. In a similar fashion to the ongoing project on isotropic materials, we evaluate the effective mechanical and transport properties of a large range of model anisotropic porous solids using bounds and by computer simulation.

 
Figure 2: Examples of Model anisotropic microstructures

Mark A. Knackstedt, Christoph Arns(UNSW), A.P. Roberts(ex-Appl. Maths)

Applications to Industrial materials

From the basic work described above we now have the potential to develop predictive tools for industry. Once the morphology of a material is defined, and given component properties (conductivity, elastic modulus, optical properties, diffusivity) of complex materials, properties can be calculated for the disordered system. Understanding the role of structure in governing the electrical, mechanical, optical properties of porous/multicomponent systems will provide the design engineer with a much-needed understanding of structure-property correlations. We have begun collaborations with two groups with strong industrial contacts with the aim to utilise the model to develop systematic, as opposed to empirical, methods for estimating properties of complex materials.

The first application is to the study of physical properties of paper and paper coatings. The physical properties of paper and paper coatings, such as strength, elasticity, friction and optical properties, are clearly of importance in production processes, quality control and in the development of improved paper products. Initially we are focusing on the optical properties of coatings and paper, important for achieving desired reflectivity and for quality control. A longer-term project has begun to study fluid penetration into paper and coatings. This ongoing project involves close links with the Forest Products Section at the Institute for Surface Chemistry (YKI) in Stockholm, Sweden.

The second application is to petrophysical log interpretation. In log interpretation one attempts to understand the inter-relationship between rock properties (e.g., porosity, permeability, oil saturation, clay content, etc.) which one cannot directly measure and their expression in different properties (e.g., resistivity, acoustic wave velocity, NMR relaxation times, dielectric permittivity) which are measurable downhole. The rock properties one is interested in are derived from combinations of the measurable quantities. Currently in interpreting well logs, different measurements invoke a different representation of the rock-pore structure. If the long-term goal is to cross-correlate different measurements, one should not use different representations of the pore space. To optimally understand logs one would model a variety of properties within a single model. This model, constrained and unique, will lead to a better understanding of the relationships between properties. Currently we are considering optimal morphologies for the description of various rock lithologies, and studying the electrical properties of the model morphologies. This project involves an ongoing collaboration with the School of Petroleum Engineering at UNSW.

M.A. Knackstedt, T.J. Senden, B.W. Ninham, J. Daicic(YKI), F. Tiberg (YKI), C. Arns(UNSW), W. V. Pinczewski(UNSW).

Modelling Fracture in Heterogeneous Materials

The fracture characteristics of a material depend on many factors: the geometry of the material; the way the external load is applied; the physical conditions under which the rupture or fracture occurs; material properties of the sample, etc. Disorder of material properties, which is exhibited on all scales in real materials, plays a decisive role in the process of rupture and fracture. There is ample evidence from both the engineering literature and everyday observation illustrating the effect of intrinsic disorder of materials on the resultant rupture and fracture at all length scales. From an inter-granular fracture surface in metal on a micron scale to the pattern of the interconnected cracks on a dried clay-rich mud on a meter scale and a fault map on a geologic scale; in all cases the fracture surface is highly disordered.

Classical textbook fracture mechanics is based around the growth of flat planar fractures. In reality, however, many fractures are not flat, but are serrated with branches. For instance, the design of hydrofractures for the stimulation of production from oil and gas wells is based on an ideal flat penny-shaped model. This is not achieved in reality, especially in a heterogeneous material such as coal, where branched irregular fractures occur. Furthermore, modelling of the flow of fluids in fractured rocks usually assumes a regular array of flat fractures. Again, in reality, this is not a good model, particularly for important questions like connectivity of the fracture network. We use random network models to study fracture in heterogeneous materials.

Development of Three Dimensional models of Fracture in Heterogneeous Media

We study network models of fracture in heterogeneous media in two and in three dimensions at a large scale. We consider a number of statistical properties of the fracture process including macroscopic conductivities, fracture surface morphologies and crack size distributions for both two and three dimensional systems. For conductivities and surface roughness the systems exhibit similar properties. However, for identical disorder the crack size distributions exhibit important differences. We compare the breakdown characteristics of the two- and three- dimensional system with identical disorder. Little correlation is noted. Most experimental measures are of two dimensional surfaces orthogonal to the fracture surface. We consider two-dimensional slice-wise information from the three-dimensional simulation. No similarities to the two-dimensional distributions are noted. The results highlight the inadequacy of two-dimensional models for fracture in real materials. We finally discuss the evolution of the crack size distribution and its possible role as a measure of disorder in material properties.

Figure 3: Fracture surfaces for system of increasing disorder. The colour indicates the time resistor is fused- red the latest and dark blue earliest.

Figure 4: Critical clusters (fracture surface plus all adjoining fractures) for breakdown criteria of differing disorder: (a) Large disorder and (b) Lesser disorder. The colour indicates the time the crack is formed; red the latest and dark blue earliest.

Experimental Studies of Disordered Media

Flow and Dispersion Studies of Polymeric Gels

While this realistic characterisation of porous material allows more accurate prediction of properties, to date no well characterised experimental system has been available to test predictions. Moreover, no system allows the independent measurement of transport and mechanical properties. Substantial progress has been made in using ternary microemulsion systems to generate model media of prescribed microstructure by the department in the 1980's. The ternary system allows one to tune experimental variables to give a specific microstructure and thus allows the independent determination of various transport properties. Polymerised ternary microemulsion systems have now been developed by Singaporean collaborators. This has allowed us to generate solid model media of prescribed microstructure. The solid polymeric materials are transparent, allowing the direct visualisation of transport properties in these media. In collaboration with the Chemical Engineering Department in Lund, Sweden, we have studied diffusion in these model porous materials. We are currently studying more complex flow characteristics of these materials.

M.A. Knackstedt, C. Mattisson (Lund), B.W. Ninham, L.M. Gan (Singapore)

Flow in Fractured Media

Fractures play a fundamental role in flow and transport in rock. The presence of fractures, natural or man-made, is crucial to the economics of oil recovery from underground reservoirs and the development of geothermal and groundwater resources. In all cases the fractures provide high permeability pathways for fluid flow in reservoirs that are otherwise of very low permeabilities and may not be able to produce at high rates. In reservoir rocks, fracture porosities which are generally low, in the range of while the pore porosity is generally larger than . Despite the significance of fractures, the field of characterization of fractured rocks is poorly developed compared to that of porous media without fractures.

Few attempts have been made to quantify the hydraulic characteristics of fractured samples. Due to the need for large computational grids, numerical methods are unable to tackle this complex dual system. Therefore studies of flow in fractured reservoirs commonly divide the reservoir into two continua- a matrix system and a fracture system. In this clearly naive formulation the matrix is assumed to be disconnected and fluid flow occurs only through the fracture network. Experimental studies of flow properties in fractured samples have been limited to similar simple geometries. Fractures are manually imbedded by cutting samples and then mating the two cut surfaces to form a single discrete fracture. Flow properties are trivial for this geometry.

We have begun an effort to experimentally study flow in fractured media exhibiting realistic fractures. We have prepared and characterised fractured porous solids with a realistic disordered fracture geometry. Results show that approximating the topology of a rough fracture by a single discrete fracture can introduce errors of an order of magnitude or more in the prediction of the permeability. Morevoer, the samples we have prepared are transparent. We are now studying dispersion and multiphase flow using holographic laser interferometry techniques and photoluminescent volumetric imaging. The implications of this work to miscible and immiscible flows are manifold.

T.J. Senden, M.A. Knackstedt and C. Mattisson (Lund)

Large-scale modeling of multi-phase flows in heterogeneous porous media

Understanding flow phenomenon at the field (or reservoir) scale is crucial to the economics of oil and gas recovery processes, geothermal energy extraction and groundwater pollution abatement in underground reservoirs. In petroleum reservoir engineering for example, significant errors in petroleum reservoir performance prediction have been caused by inaccurate characterisation of the reservoir. This can prevent the full potential of a reservoir from being achieved and lead to significant amounts of oil and gas remaining unrecovered or recovered inefficiently. A second source of error is a lack of understanding of the pore-scale mechanisms responsible for mobilization and recovery of oil. We have considered the problem of how to accurately characterise geological heterogeneity, and the implication to reservoir flow properties. Results to date hint at the drastic effect incorporation of a realistic heterogeneity has on the problem of fluid displacement in natural porous media. The result implies many results for displacement processes will have to be reworked if they are to accurately model processes in natural porous media. This work involves extensive collaborations with the School of Petroleum Engineering,UNSW, and CSIRO Division of Petroleum Resources.

Two-phase flow on Correlated Media

Percolation theory has been widely applied to the description of two-phase flow phenomena in porous rocks. We have studied percolation on grids with long-range correlations. We find that new interpretation is required for the percolation threshold. For example, the position of the percolation threshold is no longer well defined. Invasion percolation studies have also been made. A two phase model has been used to investigate relative permeability measurements for displacements in porous media with spatial correlation.

S.J. Marrink, Mark A. Knackstedt, L. Paterson(CSIRO), W.V. Pinczewski(UNSW).

Figure: Invasion percolation with trapping on uncorrelated property map and fLm property field. In (a) and (c) we show the original property maps for random and Levy fields respectively; the darker sites correspond to higher permeabilities. In (b) and (d) we compare the displacement patterns; the displacing fluid is shown in black. The trapped phase is yellow.

Development of three-phase flow simulator

A knowledge of three-phase relative permeabilities is essential to predicting the performance of a number of important oil recovery processes in which gas is injected to recover oil. Measuring and interpreting three-phase data even at the laboratory scale has been difficult, so three-phase data are usually estimated from two-phase information useing empirical models and correlations. These models are inadequate for three-phase flow and fail to account for important pore-scale physics in three-phase flow (e.g., thin film drainage). They are therefore unable to correctly describe the coupling between pore-scale flow and heterogeneity.

Network models, in which the pore space is divided into pores and throats and which may include viscous effects and film flow, offer a fundamental apprach to the study of three-phase flow. The aim of our present work is to develop a network model for three-phase flow capable of realistically predicting relative permeability, capillary pressure and residual phase saturations (relative permeability end-points) in heterogeneous porous media. Specific objectives include:

• Investigate the important interaction between multi-phase fluid flow and reservoir heterogeneity.
• Produce a pore-scale model to simulate and up-scale core flood tests used by the oil industry to design full field developments. The model will be used to determine relative permeabilities, capillary pressures and residual saturations which will be compared with published laboratory data for actual reservoir core samples.
• Use the model to test the validity of empirical models for predicting three-phase relative permeabilies and residual oil saturations, and develop improved predictive models for use in reservoir simulators, the industry standard for the design of field-scale oil recovery developments.

The key impediment to the development of a reliable simulator for the flow of three fluid phases is the large number of competing physical processes occuring over many length scales. One must therefore make numerous simplifying assumptions to render the problem tractable on today's computers. Work is in progress at numerous laboratories around the world to determines which approximations are reasonable. We have developed new algorithms for the simulation of three phase flow in porous media that are around 25 times more rapid than traditional algorithms. These enable us to directly simulate larger models with fewer approximations than before. This will aid us to determine which processes are important, and where the simplifying assumptions can be made.

A.Sheppard

Pore-core scale Heterogeneity

Laboratory measurements of relative permeability and residual oil saturation are used as inputs to reservoir scale simulators. Variations in laboratory test measurements can be large. To date the study of sub-core scale flow has assumed that the pore network exhibits little if any correlation. Recent work has indicated that heterogeneity exists down to the pore-scale. We have recently embarked on a project to examine sub-core scale heterogeneity both theoretically and experimentally.

Experimentally we will investigate several techniques to measure heterogeneity on a wide range of core material. These techniques include:

• Constant rate mercury injection
• X-ray CT scanning
• Thin section imaging.

We aim to develop statistical models that mimic the pore size distributions and the spatial correlations for input to network models. Substantial work on the characterisation of sedimentary rock at the reservoir scale from the analysis of borehole data and outcrop studies has led to a new approach to the modeling of spatial variability in geological formations. We will examine whether the reservoir-scale heterogeneity models can be extended down to the pore scale, or whether some new characterisation is appropriate. This integrated approach involving laboratory measurements and network modeling may lead to the development of a virtual core-testing laboratory. Based on preliminary experimental CT scan results, we observe a critical role played by spatial correlations in the rock properties that occur over all length scales.

T.J. Senden, A. Sheppard, S.J. Marrink, M. Knackstedt, W.V. Pinczewski(UNSW).

Generation of Fractional Brownian Motion

Statistical descriptions of heterogeneity in sedimentary rocks have shown the extent of long-range correlations. Fractional Brownian motion and fractional Levy motion has been introduced as a description of the underlying heterogeneity. Several methods have been proposed for generating approximate fractional Brownian motion. Substantial progress has already been made in conjunction with researchers at CSRIO DPR in Melbourne, toward more accurate and efficient methods for the generation of fractional Brownian and fractional Levy motion.

S.J. Marrink, A. Sheppard, J. Gunning (CSIRO), L. Paterson(CSIRO).