John H. Cushman

Nonlocal dispersion in media with continuously evolving scales of heterogeneity

General nonlocal diffusive and dispersive transport theories are derived from molecular hydrodynamics and associated theories of statistical mechanical correlation functions, using the memory function formalism and the projection operator method. Expansion approximations of a spatially and temporally nonlocal convective-dispersive equation are introduced to derive linearized inverse solutions for transport coefficients. The development is focused on deriving relations between the frequency-and wave-vector-dependent dispersion tensor and measurable quantities. The resulting theory is applicable to porous media of fractal character.

Fluids in micropores. I. Structure of a simple classical fluid in a slit‐pore

Equilbrium properties of a rare‐gas fluid contained between two parallel fcc(100) planes of rigidly fixed rare‐gas atoms were computed by means of the grand‐canonical ensemble Monte Carlo method. The singlet distribution function ρ(1), and the pair‐correlation function g(2) in planes parallel to the solid layers, indicate that the structure of the pore fluid depends strongly on the distance h between the solid layers. As the separation increases from less than two atomic diameters, successive layers of fluid appear. The transitions between one and two layers and three and four layers are especially abrupt and are accompanied by changes in the character of g(2) from dense fluid‐like to solid‐like. Long‐range, in‐plane order in the fluid layers diminishes with increasing h, but is still evident in the contact layer (i.e., that nearest the solid layer) at h=16.5 atomic diameters, the largest separation considered. The structure of the contact layer reflects the solid‐layer structure and differs significantly...

A primer on upscaling tools for porous media

Over the last few decades a number of powerful approaches have been developed to intelligently reduce the number of degrees of freedom in very complex heterogeneous environs, e.g. mathematical homogenization, mixture and hybrid mixture theory, spatial averaging, moment methods, central limit or Martingale methods, stochastic-convective approaches, various other Eulerian and Lagrangian perturbation schemes, projection operators, renormalization group techniques, variational approaches, space transformational methods, continuous time random walks, and etc. In this article we briefly review many of these approaches as applied to specific examples in the hydrologic sciences.

Fractional advection-dispersion equation: a classical mass balance with convolution-Fickian flux

The classical form of the partial differential equation governing advective-dispersive transport of a solute in idealized porous media is the advection-dispersion equation (ADE). This equation is built on a Fickian constitutive theory that gives the local dispersive flux as the inner product of a constant dispersion tensor and the spatial gradient of the solute concentration. Over the past decade, investigators in subsurface transport have been increasingly focused on anomalous (i.e., non-Fickian) dispersion in natural geologic formations. It has been recognized that this phenomenon involves spatial and possibly temporal nonlocality in the constitutive theory describing the dispersive flux. Most recently, several researchers have modeled anomalous dispersion using a constitutive theory that relies on fractional, as opposed to integer, derivatives of the concentration field. The resulting ADE itself is expressed in terms of fractional derivatives, and they are described as “fractional ADEs.” Here we show that the fractional ADE is obtained as a special case of the authors’ convolution-Fickian nonlocal ADE [Cushman and Ginn, 1993]. To obtain the fractional ADE from the convolution-Fickian model requires only a judicial choice of the wave vector and frequency-dependent dispersion tensor.

Shear forces in molecularly thin films

Monte Carlo and molecular dynamics methods have been used to study the shearing behavior of an atomic fluid between two plane-parallel solid surfaces having the face-centered cubic (100) structure. A distorted, face-centered cubic solid can form epitaxially between surfaces that are separated by distances of one to five atomic diameters. Under these conditions a critical stress must be overcome to initiate sliding of the surfaces over one another at fixed separation, temperature, and chemical potential. As sliding begins, a layer of solid exits the space between the surfaces and the remaining layers become fluid.

Fluids in micropores. II. Self‐diffusion in a simple classical fluid in a slit pore

Self‐diffusion coefficients D are computed for a model slit pore consisting of a rare‐gas fluid confined between two parallel face‐centered cubic (100) planes (walls) of rigidly fixed rare‐gas atoms. By means of an optimally vectorized molecular‐dynamics program for the CYBER 205, the dependence of D on the thermodynamic state (specified by the chemical potential μ, temperature T, and the pore width h) of the pore fluid has been explored. Diffusion is governed by Fick’s law, even in pores as narrow as 2 or 3 atomic diameters. The diffusion coefficient oscillates as a function of h with fixed μ and T, vanishing at critical values of h, where fluid–solid phase transitions occur. A shift of the pore walls relative to one another in directions parallel with the walls can radically alter the structure of the pore fluid and consequently the magnitude of D. Since the pore fluid forms distinct layers parallel to the walls, a local diffusion coefficient D(i)∥ associated with a given layer i can be defined. D(i)∥ i...

Nonequilibrium statistical mechanics of preasymptotic dispersion

Turbulent transport in bulk-phase fluids and transport in porous media with fractal character involve fluctuations on all space and time scales. Consequently one anticipates constitutive theories should be nonlocal in character and involve constitutive parameters with arbitrary wavevector and frequency dependence. We provide here a nonequilibrium statistical mechanical theory of transport which involves both diffusive and convective mixing (dispersion) at all scales. The theory is based on a generalization of classical approaches used in molecular hydrodynamics and on time-correlation functions defined in terms of nonequilibrium expectations. The resulting constitutive laws are nonlocal and constitutive parameters are wavevector and frequency dependent. All results reduce to their convolution-Fickian quasi-Fickian, or Fickian counterparts in the appropriate limits.

A Monte Carlo assessment of Eulerian flow and transport perturbation models

Monte Carlo studies of flow and transport in two-dimensional synthetic conductivity fields are employed to evaluate first-order flow and Eulerian transport theories. Hydraulic conductivity is assumed to obey fractional Brownian motion (fBm) statistics with infinite integral scale or to have an exponential covariance structure with finite integral scale. The flow problem is solved via a block-centered finite difference scheme, and a random walk approach is employed to solve the transport equation for a conservative tracer. The model is tested for mass conservation and convergence of computed statistics and found to yield accurate results. It is then used to address several issues in the context of flow and transport. The validity of the first-order relation between the fluctuating velocity covariance and the fluctuating log conductivity is examined. The simulations show that for exponential covariance, this approximation is justified in the mean flow direction for log conductivity variance, σf2, of the order of unity. However, as σf2 increases, the relation for the transverse velocity component deviates from the fully nonlinear Monte Carlo results. Eulerian transport models neglect triplet correlation functions that appear in the nonlocal macroscopic flux. The relative importance of the triplet correlation term for conservative chemicals is examined. This term appears to be small relative to the convolution flux term in mildly heterogeneous media. As σf2 increases or the integral scale grows, the triplet correlation becomes significant. In purely convective transport the triplet correlation term is significant if the heterogeneity is evolving. The exact nonlocal macroscale flux for the purely convective case significantly differs from that of the convective-dispersive transport. This is in agreement with recent theoretical analysis and numerical studies, and it suggests that neglecting local-scale dispersion may lead to large errors. Localization errors in the flux term are evaluated using Monte Carlo simulations. The nonlocal in time model significantly differs from the fully nonlocal model. For small variance and integral scale there is a slight difference between the fully localized flux and the fully nonlocal convolution flux. This is also in agreement with recent theories that suggest that moments through the second for the two models are nearly identical for conservative tracers. The fully localized model does not perform well in the purely convective cases.

Inverse methods for subsurface flow: A critical review of stochastic techniques

The development of stochastic methods for groundwater flow representation has undergone enormous expansion in recent years. The calibration of groundwater models, the inverse problem, has lately received comparable attention especially and almost exclusively from the stochastic perspective. In this review we trace the evolution of the methods to date with a specific view toward identifying the most important issues involved in the usefulness of the approaches. The methods are critiqued regarding practical usefulness, and future directions for requisite study are discussed.

Nonlocal Reactive Transport with Physical and Chemical Heterogeneity: Localization Errors

The origin of nonlocality in “macroscale” models for subsurface chemical transport is illustrated. It is argued that media that are either nonperiodic (e.g., media with evolving heterogeneity) or periodic viewed on a scale wherein a unit cell is discernible must display some nonlocality in the mean. A metaphysical argument suggests that owing to the scarcity of information on natural scales of heterogeneity and on scales of observation associated with an instrument window, constitutive theories for the mean concentration should at the outset of any modeling effort always be considered nonlocal. The intuitive appeal to nonlocality is reinforced with an analytical derivation of the constitutive theory for a conservative tracer without appeal to any mathematical approximations. Deng et al. (1993) present a first-order, nonlocal, Eulerian theory for transport of a conservative solute in an infinite nondeforming domain under steady flow conditions. Hu et al. (this issue) extended these results to account for nonequilibrium linear sorption with random partition coefficient Kd but deterministic constant reaction rate Kr. These theories are localized herein, and comparisons are made between the fully nonlocal (FNL), nonlocal in time (NLT), and fully localized (FL) theories. For conservative transport, there is little difference between the first-order FL and FNL models for spatial moments up to and including the third. However, for conservative transport the first-order NLT model differs significantly from the FNL model in the third spatial moments. For reactive transport, all spatial moments differ between the FNL and FL models. The second transverse-horizontal and third longitudinal-horizontal moments for the NLT model differ from the FNL model. These results suggest that localized first-order transport models for conservative tracers are reasonable if only lower-order moments are desired. However, when the chemical reacts with its environment, the localization approximation can lead to significant error in all moments, and a FNL model will in general be required for accurate simulation.