Andrea Cortis

Modeling non‐Fickian transport in geological formations as a continuous time random walk

[1] Non-Fickian (or anomalous) transport of contaminants has been observed at field and laboratory scales in a wide variety of porous and fractured geological formations. Over many years a basic challenge to the hydrology community has been to develop a theoretical framework that quantitatively accounts for this widespread phenomenon. Recently, continuous time random walk (CTRW) formulations have been demonstrated to provide general and effective means to quantify non-Fickian transport. We introduce and develop the CTRW framework from its conceptual picture of transport through its mathematical development to applications relevant to laboratoryand field-scale systems. The CTRW approach contrasts with ones used extensively on the basis of the advectiondispersion equation and use of upscaling, volume averaging, and homogenization. We examine the underlying assumptions, scope, and differences of these approaches, as well as stochastic formulations, relative to CTRW. We argue why these methods have not been successful in fitting actual measurements. The CTRW has now been developed within the framework of partial differential equations and has been generalized to apply to nonstationary domains and interactions with immobile states (matrix effects). We survey models based on multirate mass transfer (mobile-immobile) and fractional derivatives and show their connection as subsets within the CTRW framework.

Pore-Scale Modeling of Viscous Flow and Induced Forces in Dense Sphere Packings

We propose a method for effectively upscaling incompressible viscous flow in large random polydispersed sphere packings: the emphasis of this method is on the determination of the forces applied on the solid particles by the fluid. Pore bodies and their connections are defined locally through a regular Delaunay triangulation of the packings. Viscous flow equations are upscaled at the pore level, and approximated with a finite volume numerical scheme. We compare numerical simulations of the proposed method to detailed finite element simulations of the Stokes equations for assemblies of 8–200 spheres. A good agreement is found both in terms of forces exerted on the solid particles and effective permeability coefficients.

Catalytic Transformation of Persistent Contaminants Using a New Composite Material Based on Nanosized Zero-Valent Iron

A new composite material based on deposition of nanosized zerovalent iron (nZVI) particles and cyanocobalamine (vitamin B12) on a diatomite matrix is presented, for catalytic transformation of organic contaminants in water. Cyanocobalamine is known to be an effective electron mediator, having strong synergistic effects with nZVI for reductive dehalogenation reactions. This composite material also improves the reducing capacity of nZVI by preventing agglomeration of iron nanoparticles, thus increasing their active surface area. The porous structure of the diatomite matrix allows high hydraulic conductivity, which favors channeling of contaminated water to the reactive surface of the composite material resulting in faster rates of remediation. The composite material rapidly degrades or transforms completely a large spectrum of water contaminants, including halogenated solvents like TCE, PCE, and cis-DCE, pesticides like alachlor, atrazine and bromacyl, and common ions like nitrate, within minutes to hours. A field experiment where contaminated groundwater containing a mixture of industrial and agricultural persistent pollutants was conducted together with a set of laboratory experiments using individual contaminant solutions to analyze chemical transformations under controlled conditions.

Laboratory experiments on dispersive transport across interfaces: The role of flow direction

We present experimental evidence of asymmetrical dispersive transport of a conservative tracer across interfaces between different porous materials. Breakthrough curves are measured for tracer pulses that migrate in a steady state flow field through a column that contains adjacent segments of coarse and fine porous media. The breakthrough curves show significant differences in behavior, with tracers migrating from fine medium to coarse medium arriving significantly faster than those from coarse medium to fine medium. As the flow rate increases, the differences between the breakthrough curves diminish. We argue that this behavior indicates the occurrence of significant, time-dependent tracer accumulation in the resident concentration profile across the heterogeneity interface. Conventional modeling using the advection-dispersion equation is demonstrated to be unable to capture this asymmetric behavior. However, tracer accumulation at the interface has been observed in particle-tracking simulations, which may be related to the asymmetry in the observed breakthrough curves.

A continuous time random walk approach to transient flow in heterogeneous porous media

[1] We propose a new physical interpretation of the diffusion process for the piezometric head h(x, t) in heterogeneous aquifers based on the continuous time random walk (CTRW) theory. For the typical heterogeneities considered in this work, we find that a CTRW based diffusion equation for h(x, t) provides better fits to the transient flow simulations than the classical diffusion equation (DE). The DE is found to be a special case of the CTRW diffusion equation. The results of this work have clear implications for the interpretation of pumping tests and what information can be extracted from them. [2] Well testing is an essential tool for groundwater aquifer assessment, the key idea being that hydraulic properties can be derived by means of transient drawdown and/or flux measurements. The commonly accepted mathematical model for groundwater flow in porous formations is based on the well known Darcy law, which relates the fluid flow to the gradient of the piezometric head, h(x, t )[ L]. Together with a mass balance and the assumptions of small compressibility, constant porosity and permeability, and small head gradients, it is possible to derive a partial differential equation of parabolic type (diffusion equation) for h @th x;t ðÞ ¼ r a x ðÞ rh x;t ðÞ ðÞ

Continuous-time random-walk model of transport in variably saturated heterogeneous porous media

We propose a unified physical framework for transport in variably saturated porous media. This approach allows fluid flow and solute migration to be treated as ensemble averages of fluid and solute particles, respectively. We consider the cases of homogeneous and heterogeneous porous materials. Within a fractal mobile-immobile continuous time random-walk framework, the heterogeneity will be characterized by algebraically decaying particle retention times. We derive the corresponding (nonlinear) continuum-limit partial differential equations and we compare their solutions to Monte Carlo simulation results. The proposed methodology is fairly general and can be used to track fluid and solutes particles trajectories for a variety of initial and boundary conditions.

Model of dispersive transport across sharp interfaces between porous materials.

Recent laboratory experiments on solute migration in composite porous columns have shown an asymmetry in the solute arrival time upon reversal of the flow direction, which is not explained by current paradigms of transport. In this work, we propose a definition for the solute flux across sharp interfaces and explore the underlying microscopic particle dynamics by applying Monte Carlo simulation. Our results are consistent with previous experimental findings and explain the observed transport asymmetry. An interpretation of the proposed physical mechanism in terms of a flux rectification is also provided. The approach is quite general and can be extended to other situations involving transport across sharp interfaces.

A physical interpretation of the deterministic fractal–multifractal method as a realization of a generalized multiplicative cascade

In this study, we attempt to offer a solid physical basis for the deterministic fractal–multifractal (FM) approach in geophysics (Puente, Phys Let A 161:441–447, 1992; J Hydrol 187:65–80, 1996). We show how the geometric construction of derived measures, as Platonic projections of fractal interpolating functions transforming multinomial multifractal measures, naturally defines a non-trivial cascade process that may be interpreted as a particular realization of a random multiplicative cascade. In such a light, we argue that the FM approach is as “physical” as any other phenomenological approach based on Richardson’s eddies splitting, which indeed lead to well-accepted models of the intermittencies of nature, as it happens, for instance, when rainfall is interpreted as a quasi-passive tracer in a turbulent flow. Although neither a fractal interpolating function nor the specific multipliers of a random multiplicative cascade can be measured physically, we show how a fractal transformation “cuts through” plausible scenarios to produce a suitable realization that reflects specific arrangements of energies (masses) as seen in nature. This explains why the FM approach properly captures the spectrum of singularities and other statistical features of given data sets. As the FM approach faithfully encodes data sets with compression ratios typically exceeding 100:1, such a property further enhances its “physical simplicity.” We also provide a connection between the FM approach and advection–diffusion processes.

Frequency-dependent viscous flow in channels with fractal rough surfaces

Frequency-dependent viscous flow in channels with fractal rough surfaces Andrea Cortis a) and James G. Berryman Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA The viscous dynamic permeability of some fractal-like channels is studied. For our particular class of geometries, the ratio of the pore surface area-to-volume tends to ∞ (but has a finite cutoff), and the universal scaling of the dynamic permeability, k(ω), needs modification. We performed accurate numerical computations of k(ω) for channels characterized by deterministic fractal wall surfaces, for a broad range of fractal dimensions. The pertinent scaling model for k(ω) introduces explicitly the fractal dimension of the wall surface for a range of frequencies across the transition between viscous and inertia dominated regimes. The new model provides excellent agreement with our numerical simulations. a) Electronic mail: acortis@lbl.gov

Bells Galore: Oscillations and circle-map dynamics from space-filling fractal functions

BELLS GALORE: OSCILLATIONS AND CIRCLE-MAP DYNAMICS FROM SPACE-FILLING FRACTAL FUNCTIONS Carlos E. Puente Department of Land, Air and Water Resources University of California, Davis, CA 95616 cepuente@ucdavis.edu Andrea Cortis Earth Sciences Division Lawrence Berkeley Laboratory, Berkeley, CA 94720 acortis@lbl.gov Bellie Sivakumar Department of Land, Air and Water Resources University of California, Davis, CA 95616 sbellie@ucdavis.edu Abstract The construction of a host of interesting patterns over one and two dimen- sions, as transformations of multifractal measures via fractal interpolating functions related to simple affine mappings, is reviewed. It is illustrated that, while space-filling fractal functions most commonly yield limiting Gaussian distribution measures (bells), there are also situations (depending on the affine mappings’ parameters) in which there is no limit. Specifically, the one- dimensional case may result in oscillations between two bells, whereas the two-dimensional case may give rise to unexpected circle map dynamics of an arbitrary number of two-dimensional circular bells. It is also shown that, de- spite the multitude of bells over two dimensions, whose means dance making regular polygons or stars inscribed on a circle, the iteration of affine maps yields exotic kaleidoscopes that decompose such an oscillatory pattern in a way that is similar to the many cases that converge to a single bell.