Einar L. Hinrichsen

Geometry of random sequential adsorption

By sequentially adding line segments to a line or disks to a surface at random positions without overlaps, we obtain configurations of the one- and two-dimensional random sequential adsorption (RSA) problem. We have simulated the one- and two-dimensional problem with periodic boundary condition. The one-dimensional simulations are compared with the exact analytical solutions to give an estimate of the accuracy of the simulation. In two dimensions the geometrical properties of the RSA configuration are discussed and in addition known results of the RSA process are reproduced. Various statistical distributions of the Voronoi-Dirichlet (VD) network corresponding to the RSA disk configuration are analyzed. In order to characterize pores in the RSA configuration, we introduce circular holes. There is a direct correspondence between vertices of the VD network and these holes, and also between direct/indirect geometrical neighbors and these holes. The hole size distribution is found to be a parabola. We also find general relations that connect the asymptotic behavior of the surface coverage, the correlation function, and the hole size distribution.

Effective renormalization group algorithm for transport in oil reservoirs

A real space renormalization group algorithm is used to evaluate macroscopic permeabilities. The algorithm was tested on many different distributions, with large spatial correlations and anisotropies, and the results are very close to the expected exact values.

A fast algorithm for estimating large-scale permeabilities of correlated anisotropic media

The problem of estimating large-scale permeabilities of reservoirs based on knowledge of the small-scale permeabilities is addressed. We present an accurate and fast algorithm to calculate the global permeabilities of two- or three-dimensional correlated and anisotropic block samples, thus providing a fast algorithm for obtaining grid block permeabilities for reservoir simulators from small scale data. The algorithm is tested on both two- and three-dimensional tube networks generated from real images and fractal forgeries modeling porous media. In almost all cases, the algorithm estimates the correct global permeability (calculated using exact but slow algorithms) of the network to better than 5%. The new algorithm is comparable in speed to conventional averaging techniques, such as the geometric mean, but the obtained estimates are always much better.

Self-similarity and structure of DLA and viscous fingering clusters

Diffusion-limited aggregation clusters and the structures observed in viscous fingering experiments at high capillary numbers are tree-like fractals. The different branches may be assigned a branch order in a way that exhibits scaling, and permits a self-similar characterisation in terms of the bifurcation ratio r, and the length ratio r, of branches of different orders. The fractal dimension is given by D = ln(rN)/ln(l/rL). Good agreement between experiments and simulations is found. A crossover function characterises the branch orders.

Geometrical crossover and self-similarity of DLA and viscous fingering clusters

Abstract Diffusion-limited aggregation (DLA) clusters and the structures observed in viscous fingering experiments at high capillary numbers are tree-like fractals. The different branches may be assigned a branch order in a way that exhibits scaling, and permits a self-similar characterization by the bifurcation ratio rN and the length ratio rL of branches of different orders. The fractal dimension is given by D = log (r N ) log ( 1 r L ) . Good agreement between experiments and simulations is found. A crossover function characterizes the branch orders, and we conclude that DLA is in a state of geometrical crossover: branches are linear up to their length, L, but fractally distributed on length scales much larger than L. The effective fractal dimension of DLA depends on how D is measured and over what part of the cluster.

The conductor-superconductor transition in disordered superconducting materials

Abstract We consider a network consisting of elements which are normal down to a threshold current, below which they become superconducting. The thresholds are distributed according to some broad statistical distribution. We show that the overall voltage versus current characteristics of the network have always a nonzero threshold below which the network is superconducting, even if the distribution of thresholds is continuous down to zero. At this threshold the system undergoes a second order phase transition. We report some critical exponents associated with this transition. We also recast this problem in the general framework of the collective behavior of nonlinear elements with a quenched disorder, and discuss the connection between this problem and those of rupture of brittle materials and percolation.

Critical behaviors of central-force lattices

Around the concept of central-force lattices, various mechanical critical behaviors can be investigated, that always exhibit non-locality and sometimes non-linearity: •- Linear behavior of randomly diluted, or reinforced, structures (percolation type problems).•- Non-linear behavior due to large deformation.•- Non-linear behavior due to contacts.•- Growth of a single crack…

Percolation in layered media — A conductivity approach

Long square-lattice and cubic-lattice samples consisting of many layers are simulated. Within each layer, the concentration of permeable bonds is constant whereas each layer has a different concentration chosen randomly from the interval between the percolation threshold and unit concentration. The conductivity of the random resistor network corresponding to this percolation model is calculated, both parallel and perpendicular to the layers, in both two and three dimensions. For the conductivity parallel to the layers, an effective medium calculation comes within 10% of the true conductivity. For the conductivity perpendicular to the layers, percolation theory is necessary.

Dynamics and structure of displacement fronts in two-dimensional porous media

The dynamics and structure of drainage in 2D porous media are discussed. We restrict the discussion to the extreme cases of slow injection rates, where capillary forces govern the dispacement, and fast injection rates, where viscous forces dominate the process.

Scaling of Overhang Distribution of Invasion Percolation Fronts

We analyse simulations of invasion percolation with a gradient in two dimensions. The fronts of the invaded structures are examined using the overhang size distribution introduced by Hansen et al. [1]. We observe that overhang sizes h are distributed according to a power law: nh ~ h-a where a is measured to be a = 2.3 ± 0.1. The fractal dimension of the external perimeter of invasion percolation clusters is known to be De 1.35. We argue that in general a = Da + 1, where Da is the fractal dimension of the perimeter sampled by particles coming from outside the cluster and moving in straight lines paralles to the gradient and the overhangs.