Permeability-porosity relationship from a geometrical model of shrinking and lattice Boltzmann and Monte Carlo simulations of flow in two-dimensional pore networks

For a broad range of applications, the most important transport property of porous media is permeability. Here we calculate the permeability of pore network approximations of porous media as simple diagenetic or shrinking processes reduces their pore spaces. We use a simple random bond-shrinkage mechanism by which porosity is decreased; a tube is selected at random and its radius is reduced by a fixed factor, the process is repeated until porosity is reduced either to zero or a preset value. For flow simulations at selected porosity levels, we use precise Monte Carlo calculations and the lattice Boltzmann method with a 9-speed model on two-dimensional square lattices. Calculations show a simple power-law behavior, k ∝ φm, where k is the permeability and φ the porosity. The value of m relates strongly to the shrinking process and extension, and hence to the skewness of the pore size distribution, which varies with shrinking, and weakly to pore sizes and shapes. Smooth shrinking produces pore space microstructures resembling the starting primitive material; one value of m suffices to describe k versus φ for any value of porosity. Severe shrinking however produces pore space microstructures that apparently forget their origin; the k-φ curve is only piecewise continuous, different values of m are needed to describe it in the various porosity intervals characterizing the material. The power-law thus is not universal, a well-known fact. An effective pore length or critical pore size parameter, lc, characterizes pore space microstructures at any level of porosity. For severe shrinking lc becomes singular, indicating a change in the microstructure controlling permeability, and thus flow, thus explaining k-φ power-law transitions. Continuation of the various k-φ pieces down to zero permeability reveals pseudo-percolation thresholds φ′c for the porosity of the controlling microstructures. New graphical representations of k/lc2 versus φ-φ′c for the various φ intervals display straight and parallel lines, with a slope of 1. Our results confirm that a universal relationship between k/lc2 and φ should not be discarded

Noncommutative Independence from the Braid Group B∞

We introduce `braidability' as a new symmetry for (infinite) sequences of noncommutative random variables related to representations of the braid group $B_\infty$. It provides an extension of exchangeability which is tied to the symmetric group $S_\infty$. Our key result is that braidability implies spreadability and thus conditional independence, according to the noncommutative extended de Finetti theorem (of C. K\"{o}stler). This endows the braid groups $B_n$ with a new intrinsic (quantum) probabilistic interpretation. We underline this interpretation by a braided extension of the Hewitt-Savage Zero-One Law. Furthermore we use the concept of product representations of endomorphisms (of R. Gohm) with respect to certain Galois type towers of fixed point algebras to show that braidability produces triangular towers of commuting squares and noncommutative Bernoulli shifts. As a specific case we study the left regular representation of $B_\infty$ and the irreducible subfactor with infinite Jones index in the non-hyperfinite $II_1$-factor $L(B_\infty)$ related to it. Our investigations reveal a new presentation of the braid group $B_\infty$, the `square root of free generator presentation' $F_\infty^{1/2}$. These new generators give rise to braidability while the squares of them yield a free family. Hence our results provide another facet of the strong connection between subfactors and free probability theory and we speculate about braidability as an extension of (amalgamated) freeness on the combinatorial level.Comment: minor changes, added 3.3-3.6, version to be published in Comm.Math.Phys. (47 pages

Tomographic Characterization of Aquifer Heterogeneity

The Darcy's Law proportionality constant, hydraulic conductivity, describes the relative ease or rate at which water can move through a permeable medium and its fine-scale heterogeneity determines preferential flow rates and pathways. Traditional aquifer tests, such as slug and pumping tests, predict hydraulic conductivity values without detailed information about aquifer heterogeneity. The multiple source and receiver signals of a hydraulic tomography aquifer test can estimate interwell heterogeneity, but it requires extensive time to collect and then invert large amounts of tomographic data. An innovative adaptation of an oscillatory pressure signal was used to reduce the data collection and processing time associated with a tomography test. The phase shift of the sinusoidal pressure signal is related to the hydraulic conductivity. Multiple offset gathers (MOG) ray paths were estimated with a spatially weighted straight ray approximation method and analyzed with data processing programs that extend the 3D homogenous spherical radial equation to the heterogeneous case. A numerical model was used to check the heterogeneous extension for accuracy. High quality zero-offset profile ray paths (ZOP) were used to determine hydraulic conductivity, K, at a relatively fine scale and interpreted into representative aquifer models between different tomographic well pairs. The aquifer models were used with MOG data to evaluate the anisotropy ratio and lateral heterogeneity of the aquifer. Two different oscillatory periods, 3 and 30-sec, were evaluated and compared to previous work at the site. Analysis indicates that the 3-sec period data were more sensitive to different anisotropy ratios and both periods are capable of resolving K zones of about one meter

Power spectrum characterization of the continuous gaussian ensemble

The continuous Gaussian ensemble, also known as the ν-Gaussian or ν-Hermite ensemble, is a natural extension of the classical Gaussian ensembles of real (ν=1), complex (ν=2), or quaternion (ν=4) matrices, where ν is allowed to take any positive value. From a physical point of view, this ensemble may be useful to describe transitions between different symmetries or to describe the terrace-width distributions of vicinal surfaces. Moreover, its simple form allows one to speed up and increase the efficiency of numerical simulations dealing with large matrix dimensions. We analyze the long-range spectral correlations of this ensemble by means of the δn statistic. We derive an analytical expression for the average power spectrum of this statistic, Pkδ̅ , based on approximated forms for the two-point cluster function and the spectral form factor. We find that the power spectrum of δn evolves from Pkδ̅ ∝1/k at ν=1 to Pkδ̅ ∝1/k2 at ν=0. Relevantly, the transition is not homogeneous with a 1/fα noise at all scales, but heterogeneous with coexisting 1/f and 1/f2 noises. There exists a critical frequency kc∝ν that separates both behaviors: below kc, Pkδ̅ follows a 1/f power law, while beyond kc, it transits abruptly to a 1/f2 power law. For ν>1 the 1/f noise dominates through the whole frequency range, unveiling that the 1/f correlation structure remains constant as we increase the level repulsion and reduce to zero the amplitude of the spectral fluctuations. All these results are confirmed by stringent numerical calculations involving matrices with dimensions up to 10

Weil Spaces and Weil-Lie Groups

We define Weil spaces, Weil manifolds, Weil varieties and Weil Lie groups over an arbitrary commutative base ring K (in particular, over discrete rings such as the integers), and we develop the basic theory of such spaces, leading up the definition of a Lie algebra attached to a Weil Lie group. By definition, the category of Weil spaces is the category of functors from K-Weil algebras to sets; thus our notion of Weil space is similar to, but weaker than the one of Weil topos defined by E. Dubuc (1979). In view of recent result on Weil functors for manifolds over general topological base fields or rings by A. Souvay, this generality is the suitable context to formulate and to prove general results of infinitesimal differential geometry, as started by the approach developed in Bertram, Mem. AMS 900