Research Interests of Christoph Arns

High-dimensional NMR inverse Laplace spectroscopy

Higher-dimensional spectral NMR inverse Laplace methods, which encode besides relaxation also for e.g. diffusion or internal gradient effects, promise to enable a more precise characterization of the environment in porous media, e.g. of structural quantities, saturations, wettability. The experiment requires the solution of ill-posed ill-conditioned multi-dimensional inverse Problems.

My interest is in the stable solution of such problems, the characterisation of the uncertainty of the solution, the associated resolution of those methods, and procedures for optimal selection of acquisition parameters.

Transport properties from Nuclear Magnetic Resonance

NMR responses are commonly used in reservoir characterization to estimate pore-size information, formation permeability, as well as fluid content and type. Difficulties arise in the interpretation of NMR response as an estimator of permeability due to internal gradients, diffusion coupling, surface-relaxivity heterogeneity, and a possible breakdown of correlations between pore and constriction sizes. In the context of this project we carry out a fundamental study of the relationship between the microstructure of a porous medium, its transport properties, and its NMR responses.

Within the scope of this project we develop a numerical capability for the simulation of various NMR responses of porous media. In particular, we consider relaxation responses and PGSE experiments.

We acknowledge funding through ARC grant DP0558185, including an APD fellowship for Arns.

Left: Use of topological (Sheppard & Saadatfar) and geometric partitions for the distribution of surface relaxivities.

Permeability

The calculation of permeability from tomographic images implies the solution of the Navier-Stokes equation on a regular lattice. This is achieved by a lattice Boltzmann approach. Our research is directed at fast parallel solvers and the inclusion of microporosity through an integration of the phenomenological Brinkman equation by essentially solving Darcy's equation in microporous regions.

Contaminant migration

Characterising and predicting the spread of contaminants in heterogeneous systems is a complex problem. We consider the solution of the advection-diffusion equation for complex microstructure. We are interested in the scaling behaviour of breakthrough curves and dispersion tensors for realistic microstructure, in the inclusion of microporosity, and plan to extend the scope of the project to non-ideal tracers.

Electrical conductivity

The electrical conductivity of disordered materials is given by the solution of the Laplace equation. We use a finite difference approach to solve the equations given by the already discretised tomogram. Current research focusses on the inclusion of microporosity into the solver.

Local field analysis

This project provides integral support for the network modelling of porous rock (A.P. Sheppard). Our aim is to understand the local flow properties on a coarse topological scale (network) compared to the fine geometric detail of a tomogram.

Property-porosity relationships

We derive elastic property-porosity relationships directly from microtomographic images using a finite element method. By estimating and minimizing several sources of numerical error, very accurate predictions of properties are derived in excellent agreement with experimental measurements over a wide range of the porosity. We find excellent agreement with Gassmann's equations for fluid substitution.

Partial saturation

For partially saturated systems, discretisation issues resulting in incomplete fluid pressure equilibration can be problematic. We are investigating ways to estimate/correct these effects, as well as novel algorithmic procedures to avoid these inaccuracies.

Second order analysis of curvature measures

Second-order characteristics are important in the description of various geometrical structures occurring in foams, porous media, complex fluids, and phase separation processes. The classical second order characteristics are pair correlation functions, which are well-known in the context of point fields and mass distributions. This project studies systematically these and further characteristics from a unified standpoint, based on four so-called curvature measures, volume, surface area, integral of mean curvature and Euler characteristic. We develop a statistical method which yields smoothed surrogates for pair correlation functions, namely variograms. Variograms lead to an enhanced understanding of the variability of the geometry of two-phase structures and can help in finding suitable models. They might also be correlated to the variability of physical measures.

Morphological drainage

The simulation of mercury drainage by an approach using morphological operations and percolation concepts allows the generation of realistic fluid distributions for the evalulation of physical properties at partial saturation conditions. Comparions with mercury drainage experiments (left, sample by Marios Ioannidis) show good agreement.

Statistical reconstruction and microstructure models

One way to characterise or reconstruct microstructure is by describing it as generated by a Poisson process. Integral geometry and the Kac theorem for the spectrum of the Laplace operator define the effective shape of an inclusion in a system made up of a distribution of arbitrarily shaped constituents. Reconstructing the microstructure using the effective inclusion shape leads to an excellent match to the percolation thresholds and to the mechanical and transport properties across all phase fractions. Use of the equivalent shape in effective medium formulations leads to good predictions.

Further, we consider the family of integral geometric measures during erosion and dilation operations to determine the accuracy of model reconstructions of random systems. We showed that the use of erosion/dilation operations on the original image leads to an accurate discrimination of morphology.

Apart from the above methods for characterisation and reconstruction we considered ways to reconstruct structure by way of 2-point correlation functions of various Gaussian models.

We have the ability to generate microstructure models of particles with a variety of shapes and orientations, Gaussian models, or models based on Voronoi tesselations, as well as combinations thereof. This allows us to test reconstruction algorithms or target the modelling of structure below the resolution of micro-tomography.