Monte-Carlo simulation of light scattering in fractal porous media

Monte-Carlo simulation of light propagation in a porous medium with a mass fractal morphology was carried out. It was shown that the simulation results can be used to analyze experimental data on light scattering and to study the porous medium structure. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 75, No. 2, pp. 193–196, March–April, 2008.     

A fractional equation for anomalous diffusion in a randomly heterogeneous porous medium

A fractional partial differential equation is derived for the spreading of matter in a saturated porous medium starting from precise hypotheses concerning the medium itself, which is a collection of intertwisted tubes with randomly distributed slopes, filled with quiescent fluid. Examining the fundamental solution of the fractional equation indicates that the second moment is not proportional to time, which is the signature of anomalous diffusion. The equation derived preserves non-negativity and also the total mass of matter.     

Numerical study on permeability characteristics of fractal porous media

The fractal Brownian motion is utilized to describe pore structures in porous media. A numerical model of laminar flow in porous media is developed, and the flow characteristics are comprehensively analyzed and compared with those of homogeneous porous media. Moreover, the roles of the fractal dimension and porosity in permeability are quantitatively described. The results indicate that the pore structures of porous media significantly affect their seepage behaviors. The distributions of pressure and velocity in fractal porous media are both non-uniform; the streamline is no longer straight but tortuous. When Reynolds number Re 1, the dimensionless permeability is independent of Reynolds number, but its further increase will lead to a smaller permeability. Moreover, due to the higher connectivity and enlarged equivalent aperture of internal channel network, the augment in porosity leads to the permeability enhancement, while it is small and insensitive to porosity variation when ε 0.6. Fractal dimension also plays a significant role in the permeability of porous media. The increase in fractal dimension leads to the enhancement in pore connectivity and a decrease in channel tortuosity,which reduces the flow resistance and improves the transport capacity of porous media.     

Strange eigenstates and anomalous transport in a Koch fractal with hierarchical interaction

Stationary states of non-interacting electrons on a Koch fractal are investigated within a tight binding approach. It is observed that if a hierarchically long range hopping is allowed, a suitable correlation between the parameters defining the Hamiltonian leads to spectacular changes in the transport properties of finite, but arbitrarily large fractals. Topologically identical structures, that are found to support the same distribution of the amplitudes of eigenstates, are conducting in some cases and insulating in the others, depending on the choice of the hierarchy parameter. The values of the hierarchical parameter themselves display a self-similar, fractal character.     

Formation, growth and precipitation of fractal molecular aggregates in porous media

Analysis of small-angle scattering data, as well as novel precipitation measurements, are used to delineate the structure of the molecular aggregates that are formed when a fluid is injected into a porous medium to displace the in-place fluid. Our analysis suggests conclusively that these aggregates are fractal formed by diffusion-limited processes. The implications for the molecular weight distribution of the aggregates and modelling their flow and precipitation in a porous medium are discussed.     

Nonclassical transport in fractal media with a diffusion barrier

We investigate the impurity transport in a randomly heterogeneous fractal medium with a diffusion barrier. The barrier is due to low permeable medium surrounding the source. The transport regimes and asymptotic (large-distance) concentration distributions are found. The presence of the diffusion barrier results in the retardation of the transport regimes at short times. As regards the asymptotic concentration distribution, the barrier influence persists for long times as well.     

A random field model for anomalous diffusion in heterogeneous porous media

Heterogeneity, as it occurs in porous media, is characterized in terms of a scaling exponent, or fractal dimension. A feature of primary interest for two-phase flow is the mixing length. This paper determines the relation between the scaling exponent for the heterogeneity and the scaling exponent which governs the mixing length. The analysis assumes a linear transport equation and uses random fields first in the characterization of the heterogeneity and second in the solution of the flow problem, in order to determine the mixing exponents. The scaling behavior changes from long-length-scale dominated to short-length-scale dominated at a critical value of the scaling exponent of the rock heterogeneity. The long-length-scale-dominated diffusion is anomalous.     

Reciprocity relations for the effective electrical conductivity of randomly inhomogeneous media in the fractal regime

An exact relation for the realization-averaged effective conductivities in the fractal region is found for two-dimensional randomly inhomogeneous media. It has the form {σ _e (τ,L)~× {1/σ _e (−τ,L)~⁻¹=σ _e ² (τ=0, L≫ξ), where ξ is the correlation length (the self-averaging scale), L is the size of the system, τ=(p-p _c )/p _c , and p _c is the percolation threshold. For L≫ ξ, the system is self-averaged, and the relation transforms into the Dykhne reciprocity relation, A. M. Dykhne, Zh. éksp. Teor. Fiz. 59, 110 (1970) [Sov. Phys. JETP 32, 63 (1971)] σ _e (τ)σ _e (−τ])=σ _e ² (τ=0)= σ ₁ σ ₂. A similar relation is obtained for media with an exponentially broad distribution of local conductivities, as well as for individual realizations of some deterministic structures. Zh. éksp. Teor. Fiz. 113, 1484–1490 (April 1998)     

Modeling of flow and mass transport in granular porous media

The scope of this work is to estimate the effective mass-transfer coefficient in a two-phase system of oil and water fluid droplets, both being in a porous medium. To this end, a tracer is advected from the flowing aqueous phase to the immobile non-aqueous one. Partitioning at the fluid-fluid interface and surface diffusion are also taken into account. By using spatial/volume-averaging techniques, the appropriately simplified boundary-value problems are described and numerically solved for the flow velocity field and for the transport problem. The problem was found to be controlled by the Peclet number of the flowing phase, the dimensionless parameter Λ, containing both diffusion and partition in the two phases, as well as the geometrical properties of the porous structure. It is also verified that the usually involved unit cell-configurations underestimate the mass transport to the immobile phase.     

A transport theory approach to percolation of liquids through porous media

The percolation of a liquid through a porous material is investigated with the help of equations of the Onsager type. An expression is derived for the molecular attraction, starting from Sutherland’s potential approximation to the van der Waals interaction. Then appropriate Onsager equations incorporating this molecular attraction are written from transport theory considerations, in terms of dimensionless variables. As an application, the system of self-similar equations so derived is applied to a simplified situation.     

Stochastic langevin model for flow and transport in porous media

We present a new model for fluid flow and solute transport in porous media, which employs smoothed particle hydrodynamics to solve a Langevin equation for flow and dispersion in porous media. This allows for effective separation of the advective and diffusive mixing mechanisms, which is absent in the classical dispersion theory that lumps both types of mixing into dispersion coefficient. The classical dispersion theory overestimates both mixing-induced effective reaction rates and the effective fractal dimension of the mixing fronts associated with miscible fluid Rayleigh-Taylor instabilities. We demonstrate that the stochastic (Langevin equation) model overcomes these deficiencies.     

Ion transport in porous media studied by NMR.

Moisture and salt transport in masonry can give rise to damages. Therefore a detailed knowledge of the moisture and salt transport is essential for understanding the durability of masonry. A special NMR apparatus has been made allowing quasi-simultaneous measurements of both moisture and Na profiles in porous building materials. Using this apparatus both the absorption of a 4 M NaCl solution in a calcium silicate brick and the drying of a 3 M NaCl capillary saturated fired-clay brick have been studied. It was found that during the absorption process the Na ions clearly stay behind, which this is caused by adsorption of these ions to the pore surface. For the drying it was found that at the beginning of the drying process the ions accumulate near the surface. As the drying rate decreases, diffusion becomes dominant and the ion profile levels off again.     

Steady stokes flow with long-range correlations,fractal fourier spectrum,and anomalous transport

We consider viscous two-dimensional steady flows of incompressible fluids past doubly periodic arrays of solid obstacles. In a class of such flows, the autocorrelations for the Lagrangian observables decay in accordance with the power law, and the Fourier spectrum is neither discrete nor absolutely continuous. We demonstrate that spreading of the droplet of tracers in such flows is anomalously fast. Since the flow is equivalent to the integrable Hamiltonian system with 1 degree of freedom, this provides an example of integrable dynamics with long-range correlations, fractal power spectrum, and anomalous transport properties.     

Multi-gradient pulse investigations of fluid transport in porous media.

NMR pulsed field gradient (PFG) experiments employing the application of n gradient pulses k(1) ellipsis k(n) are discussed in a general way as an n-fold encoding of position at successive times. The experiments are then represented by a sampling of n-dimensional k-space, K(n). Various parameters of motion can be derived from the evolution of correlations within the n-dimensional position (r-)space, R(n), which is the Fourier conjugate space to K(n). A wide class of NMR experiments may be described by this formalism, where the dimension of the experiment is often reduced by imposing conditions to the free variables. This is demonstrated for the case of displacement measurements where the condition summation operatork(i) = 0 is met. The two simplest pulse sequences which allow one to correlate displacements at two different times with each other are presented. While the three-pulse version of SERPENT encodes displacements in two interleaved time intervals Delta(1) and Delta(2), the four-pulse VEXSY experiment includes a mixing time tau(m) in between both encoding intervals Delta. The behaviour of fluid transport subject to external pressure through a model porous system is demonstrated by means of numerical simulations of SERPENT and VEXSY experiments for water flowing through a packed bed of monosized spherical particles. Displacements parallel (Z) and perpendicular (X) to the main flow direction are determined and the 2-D joint probability densities and the conditional probabilities are discussed along with the correlation coefficients related to the displacements at different encoding times. It is shown that all possible correlations between Z and X(2) in VEXSY decay with time constants comparable to the average time needed for a fluid molecule to cover one bead diameter, while a negative correlation is observed between transverse (X) displacements which is explained by molecules flowing along streamlines which follow the circumference of the spherical particles. Correlations for displacements during the different times in SERPENT generally decay much slower and provide complementary information about the evolution of displacements with time.