B. Biswal

Three-dimensional local porosity analysis of porous media

A quantitative comparison of the pore space geometry for three natural sandstones is presented. The comparison is based on local porosity theory which provides a geometric characterization of stochastic microstructures. The characterization focusses on porosity and connectivity fluctuations. Porosity fluctuations are measured using local porosity distributions while connectivity fluctuations are measured using local percolation probabilities. We report the first measurement of local percolation probability functions for experimentally obtained three-dimensional pore space reconstructions. Our results suggest the use of local porosity distributions and percolation probabilities as a quantitative method to compare microstructures of models and experiment.

Quantitative prediction of effective material properties of heterogeneous media

Effective electrical conductivity and electrical permittivity of water-saturated natural sandstones are evaluated on the basis of local porosity theory (LPT). In contrast to earlier methods, which characterize the underlying microstructure only through the volume fraction, LPT incorporates geometric information about the stochastic microstructure in terms of local porosity distribution and local percolation probabilities. We compare the prediction of LPT and of traditional effective medium theory with the exact results. The exact results for the conductivity and permittivity are obtained by solving the microscopic mixed boundary value problem for the Maxwell equations in the quasistatic approximation. Contrary to the predictions from effective medium theory, the predictions of LPT are in better quantitative agreement with the exact results.

MICROSTRUCTURE ANALYSIS OF RECONSTRUCTED POROUS MEDIA

We compare the quantitative microstructural properties of Berea Sandstone with stochastic reconstructions of the same sandstone. The comparison is based on local porosity theory. The reconstructions employ Fourier space filtering of Gaussian random fields and match the average porosity and two-point correlation function of the experimental model. Connectivity properties of the stochastic models differ significantly from the experimental model. Reconstruction models with different levels of coarse graining also show different average local connectivity.

Exact and approximate calculations for the conductivity of sandstones

We analyze a three-dimensional pore space reconstruction of Fontainebleau sandstone and calculate from it the effective conductivity using local porosity theory. We compare this result with an exact calculation of the effective conductivity that solves directly the disordered Laplace equation. The prediction of local porosity theory is in good quantitative agreement with the exact result.

Neural Network Model for Apparent Deterministic Chaos in Spontaneously Bursting Hippocampal Slices

A neural network model that exhibits stochastic population bursting is studied by simulation. First return maps of interburst intervals exhibit recurrent unstable-periodic-orbit(UPO)-like trajectories similar to those found in experiments on hippocampal slices. Applications of various control methods and surrogate analysis for UPO detection also yield results similar to those of experiments. Our results question the interpretation of the experimental data as evidence for deterministic chaos and suggest caution in the use of UPO-based methods for detecting determinism in time-series data.

Continuum-based rock model of a reservoir dolostone with four orders of magnitude in pore sizes

A continuum-based pore-scale representation of a dolomite reservoir rock is presented, containing several orders of magnitude in pore sizes within a single rock model. The macroscale rock fabric from a low-resolution x-ray microtomogram was combined with microscale information gathered from high-resolution two-dimensional electron microscope images. The low-resolution x-ray microtomogram was segmented into six separate rock phases in terms of mineralogy, matrix appearances, and open- versus crystal-filled molds. These large-scale rock phases were decorated (modeled) with geometric objects, such as different dolomite crystal types and anhydrite, according to the high-resolution information gathered from the electron microscope images. This procedure resulted in an approximate three-dimensional representation of the diagenetically transformed rock sample with respect to dolomite crystal sizes, porosity, appearance, and volume of different matrix phases and pore/matrix/cement ratio. The resulting rock model contains a pore-size distribution ranging from moldic macropores (several hundred micrometers in diameter) down to mudstone micropores (1 m in diameter). This allows us to study the effect and contribution of different pore classes to the petrophysical properties of the rock. Higher resolution x-ray tomographs of the same rock were used as control volumes for the pore-size distribution of the model. The pore-size analysis and percolation tests performed in three dimensions at various discretization resolutions indicate pore-throat radii of 1.5 to 6 m for the largest interconnected pore network. This also highlights the challenge to determine appropriate resolutions for x-ray imaging when the exact rock microstructure is not known.

Reliability of unstable periodic orbit based control strategies in biological systems

Presence of recurrent and statistically significant unstable periodic orbits (UPOs) in time series obtained from biological systems is now routinely used as evidence for low dimensional chaos. Extracting accurate dynamical information from the detected UPO trajectories is vital for successful control strategies that either aim to stabilize the system near the fixed point or steer the system away from the periodic orbits. A hybrid UPO detection method from return maps that combines topological recurrence criterion, matrix fit algorithm, and stringent criterion for fixed point location gives accurate and statistically significant UPOs even in the presence of significant noise. Geometry of the return map, frequency of UPOs visiting the same trajectory, length of the data set, strength of the noise, and degree of nonstationarity affect the efficacy of the proposed method. Results suggest that establishing determinism from unambiguous UPO detection is often possible in short data sets with significant noise, but derived dynamical properties are rarely accurate and adequate for controlling the dynamics around these UPOs. A repeat chaos control experiment on epileptic hippocampal slices through more stringent control strategy and adaptive UPO tracking is reinterpreted in this context through simulation of similar control experiments on an analogous but stochastic computer model of epileptic brain slices. Reproduction of equivalent results suggests that far more stringent criteria are needed for linking apparent success of control in such experiments with possible determinism in the underlying dynamics.