Brian Berkowitz

Characterizing flow and transport in fractured geological media: A review

Abstract We analyze measurements, conceptual pictures, and mathematical models of flow and transport phenomena in fractured rock systems. Fractures and fracture networks are key conduits for migration of hydrothermal fluids, water and contaminants in groundwater systems, and oil and gas in petroleum reservoirs. Fractures are also the principal pathways, through otherwise impermeable or low permeability rocks, for radioactive and toxic industrial wastes which may escape from underground storage repositories. We consider issues relating to (i) geometrical characterization of fractures and fracture networks, (ii) water flow, (iii) transport of conservative and reactive solutes, and (iv) two-phase flow and transport. We examine the underlying physical factors that control flow and transport behaviors, and discuss the currently inadequate integration of conceptual pictures, models and data. We also emphasize the intrinsic uncertainty associated with measurements, which are often interpreted non-uniquely by models. Throughout the review, we point out key, unresolved problems, and formalize them as open questions for future research.

SCALING OF FRACTURE SYSTEMS IN GEOLOGICAL MEDIA

Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hy- drocarbon reservoir management, and earthquake haz- ard assessment. Relevant publications are therefore spread widely through the literature. Although it is rec- ognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law expo- nents and fractal dimensions from observations, al- though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these tech- niques and suggest guidelines for the accurate and ob- jective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after crit- ical appraisal of published studies, to show a wide vari- ation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The phys- ical causes of power law scaling and variation in expo- nents and fractal dimensions are still poorly understood.

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.

PERCOLATION THEORY AND ITS APPLICATION TO GROUNDWATER HYDROLOGY

The theory of percolation, originally proposed over 30 years ago to describe flow phenomena in porous media, has undergone enormous development in recent years, primarily in the field of physics. The principal advantage of percolation theory is that it provides universal laws which determine the geometrical and physical properties of the system. This survey discusses developments and results in percolation theory to date, and identifies aspects relevant to problems in groundwater hydrology. The methods of percolation theory are discussed, previous applications of the theory to hydrological problems are reviewed, and future directions for study are suggested.

Flow in rock fractures: The local cubic law assumption reexamined

We investigate the validity of applying the “local cubic law” (LCL) to flow in a fracture bounded by impermeable rock surfaces. A two-dimensional order-of-magnitude analysis of the Navier-Stokes equations yields three conditions for the applicability of LCL flow, as a leading-order approximation in a local fracture segment with parallel or nonparallel walls. These conditions demonstrate that the “cubic law” aperture should not be measured on a point-by-point basis but rather as an average over a certain length. Extending to the third dimension, in addition to defining apertures over segment lengths, we find that the geometry of the contact regions influences flow paths more significantly than might be expected from consideration of only the nominal area fraction of these contacts. Moreover, this latter effect is enhanced by the presence of non-LCL regions around these contacts. While contact ratios of 0.1–0.2 are usually assumed to have a negligible effect, our calculations suggest that contact ratios as low as 0.03–0.05 can be significant. Analysis of computer-generated fractures with self-affine walls demonstrates a nonlinear increase in contact area and a faster-than-cubic decrease in the overall hydraulic conductivity, with decreasing fracture aperture; these results are in accordance with existing experimental data on flow in fractures. Finally, our analysis of fractures with self-affine walls indicates that the aperture distribution is not lognormal or gamma as is frequently assumed but rather truncated-normal initially and increasingly skewed with fracture closure.

Percolation theory and network modeling applications in soil physics

The application of percolation theory to porous media is closely tied to network models. A network model is a detailed model of a porous medium, generally incorporating pore-scale descriptions of the medium and the physics of pore-scale events. Network models and percolation theory are complementary: while network models have yielded insight into behavior at the pore scale, percolation theory has shed light, at the larger scale, on the nature and effects of randomness in porous media. This review discusses some basic aspects of percolation theory and its applications, and explores work that explicitly links percolation theory to porous media using network models. We then examine assumptions behind percolation theory and discuss how network models can be adapted to capture the physics of water, air and solute movement in soils. Finally, we look at some current work relating percolation theory and network models to soils.

On Characterization of Anomalous Dispersion in Porous and Fractured Media

A key characterization of dispersion in aquifers and other porous media has been to map the effects of inhomogeneous velocity fields onto a Fickian dispersion term (D) within the context of the conventional advection-dispersion equation (ADE). Recent compilations of data have revealed, however, that the effective D coefficient is not constant but varies systematically with the length or timescale over which transport occurs. A natural strategy to encompass this “anomalous” behavior into the context of the conventional ADE is to make D time dependent. This approach, to use D(t) to handle the same anomalous dispersion phenomena, has also been common in the field of electronic transport in disordered materials. In this paper we discuss the intrinsic inadequacy of considering a time-dependent dispersivity in the conventional ADE context, and show that the D = D(t) generalization leads to quantifiably incorrect solutions. In the course of proving this result we discuss the nature of anomalous dispersion and provide physical insight into this important problem in hydrogeology via analysis of a class of kinetic approaches. Particular emphasis is placed on the effects of a distribution of solute “delay times” with a diverging mean time, which we relate to configurations of preferential pathways in heterogeneous media.

Analysis of fracture network connectivity using percolation theory

Connectivity aspects of fracture networks are analyzed in terms of percolation theory. These aspects are of fundamental importance in characterization, exploitation, and management of fractured formations. General connectivity and power law relationships are determined that characterize the density of fractures and average number of intersections per fracture necessary to ensure network connectivity, the likelihood of a fractured formation being hydraulically connected, and the probability that any specific fracture is connected to the conducting portion of the network. Monte Carlo experiments with a two-dimensional fracture network model confirm the percolation theory predictions. These relationships may prove useful in formulating theoretically tractable approximations of fracture nerworks that capture the essential system properties.

Transport of metal oxide nanoparticles in saturated porous media

The behavior of four types of untreated metal oxide nanoparticles in saturated porous media was studied. The transport of Fe(3)O(4), TiO(2), CuO, and ZnO was measured in a series of column experiments. Vertical columns were packed with uniform, spherical glass beads. The particles were introduced as a pulse suspended in aqueous solutions and breakthrough curves at the outlet were measured using UV-vis spectrometry. Different factors affecting the mobility of the nanoparticles such as ionic strength, addition of organic matter (humic acid), flow rate and pH were investigated. The experiments showed that mobility varies strongly among the nanoparticles, with TiO(2) demonstrating the highest mobility. The mobility is also strongly affected by the experimental conditions. Increasing the ionic strength enhances the deposition of the nanoparticles. On the other hand, addition of humic acid increases the nanoparticle mobility significantly. Lower flow rates again led to reduced mobility, while changes in pH had little effect. Overall, in natural systems, it is expected that the presence of humic acid in soil and aquifer materials, and the ionic strength of the resident water, will be key factors determining nanoparticle mobility.

Precipitation and dissolution of reactive solutes in fractures

The precipitation and dissolution of reactive solutes, transported under the action of fully developed laminar flow in saturated fractures, is analyzed assuming an irreversible first-order kinetic surface reaction for one component. Equations describing solute transport, precipitation and dissolution, and the evolution of fracture aperture were approximated and solved using combined analytical and numerical techniques; dimensionless transport parameters incorporated into the solutions were estimated from data available in the literature. Fractures with initially flat, linearly constricted, and sinusoidal apertures were investigated. The initial fracture geometry and the solute saturation content of the inflowing fluid have a profound effect on the reaction processes. The results show that the evolution of the solute transport and fracture geometry can be adequately described by the Damkohler and Peclet numbers. Two extreme transport regimes were identified: relatively uniform evolution of fracture apertures and nonuniform evolution of fracture apertures restricted to the inlet region of fractures. In the case of precipitation with half-life times of the order of seconds to years and with fluid residence times of the order of minutes to days, the time for a fracture to close completely is of the order of days to millions of years. This is consistent with the order of magnitude of hydrogeological timescales. In the model the process of dissolution is the inverse of precipitation, although the combined solute transport and reaction processes are irreversible. These results and the applied dimensionless analysis can be used as a basis for the development of more complex models of reactive solute transport, precipitation, and dissolution in saturated fractured media.