Roger Beckie

Flow of coal-bed methane to a gallery

Coal-seam methane reservoirs have a number of unique feature compared to conventional porous or fractured gas reservoirs. We propose a simplified mathematical model of methane movement in a coal seam taking into account the following features: a relatively regular cleat system, adsorptive methane storage, an extremely slow mechanism of methane release from the coal matrix into cleats and a significant change of permeability due to desorption.Parameters of the model have been combined into a few dimensionless complexes which are estimated to an order of magnitude. The simplicity of the model allows us to fully investigate the influence of each parameter on the production characteristics of the coal seam. We show that the reference time of methane release from the coal matrix into cleats – the parameter which is most poorly investigated – may have a critical influence on the overall methane production.

Measurement Scale, Network Sampling Scale, and Groundwater Model Parameters

The scales at which model parameters are measured with local field tests distributed on a sampling network are examined. Two scales are defined to characterize the problem: (1) the measurement scale associated with the resolution of a single field test, and (2) the network scale, associated with the separation between samples on a network. Using a spatial filtering approach, it is shown that a network of measurements can only resolve a larger-scale component of a parameter field. The smaller-scale component of the parameter field not seen by measurements, here called the subgrid component, can only vary on scales larger than the measurement scale and smaller than the network scale. These unobserved subgrid scales give rise to the so-called closure problem and consequent modeling errors. When a significant proportion of the parameter variability is contained in the subgrid scales, not only will the closure problem be significant, but aliasing errors will also pollute the estimate of the large-scale component of the parameter field. These concepts are illustrated by performing simple analytical and numerical calculations.

Groundwater flow dynamics and arsenic source characterization in an aquifer system of West Bengal, India

Numerical groundwater flow modeling, reverse particle tracking, and environmental tracers are used to locate the source of geogenic As affecting an aquifer in West Bengal. The aquifer is hosted by point-bar sands deposited in a meandering fluvial environment. Wells tapping the aquifer exhibit As concentrations up to 531 μg/L. High-As groundwaters are recharged in ponds marking an abandoned river channel. The source of As is traced to the underlying fine-grained channel-fill sediments. Arsenic release within these sediments is accompanied by a concomitant release of Br and DOC indicating that these species may be decay products of natural organobromines codeposited along with As. Mass transfer of As to the dissolved phase and its flushing from source sediments are described using a simplified reactive solute transport model. Based on this model, a characteristic reaction time for mass transfer is estimated at 6.7 years. Average groundwater residence times in the source are estimated to have declined from 16.6 to 6.6 years with the advent of intensive irrigation pumping. The ratio of residence and reaction times, a Damkohler number, has declined correspondingly from 2.49 to 0.99, indicating a shift from transport to reaction rate limited As mobilization. Greater insight into the As problem in SE Asia may be achieved by shifting the focus of field investigations from aquifers to potential contamination sources in aquitards.

The universal structure of the groundwater flow equations

Most groundwater flow models are not sufficiently detailed to allow an explicit representation of all dynamical scales. Instead, most models are constructed at such a coarse scale of resolution that unresolved subgrid variability often exists. It is therefore important to understand the interaction between unresolved dynamics and explicitly resolved dynamics. The concept of universality is central to this interaction and is here related to the problem of rescaling models of groundwater flow. We examine when it is possible to construct an accurate model for the explicitly resolved large scales, without an explicit description of the subgrid scale dynamics. We demonstrate how unresolved subgrid scale dynamics interact with resolved scale dynamics. We show that a universally valid resolved scale model can be constructed if the resolved dynamics are sufficiently independent of the details of the subgrid scale dynamics. In that event, a resolved scale model is composed of a universal structure and accompanying model parameters. The model parameters represent the effect of unresolved dynamics upon resolved dynamics. We examine two possible model structures for groundwater flow. We show theoretically, and numerically, that a local Darcy law is a universally valid resolved scale model if the resolved and subgrid scales of the hydraulic conductivity are separated in scale by a so-called spectral gap. If, however, the hydraulic conductivity possesses many scales of variability, then a more general nonlocal Darcy law is a more appropriate model structure. When the nonlocal Darcys law is more appropriate, numerical experiments suggest that errors using a local Darcys law with effective parameters are most significant at the smallest resolving scale of the model, and are minimal at scales between 8 and 16 times the resolving scale.

Modeling the Large‐Scale Dynamics of Saturated Groundwater Flow Using Spatial‐Filtering Theory: 1. Theoretical Development

We develop a model and a framework for understanding the explicitly resolved, large-scale dynamics of groundwater flow in fully saturated, heterogeneous porous media. The approach is based upon the large eddy simulation (LES) methodology from geophysical fluid dynamics. The idea behind the LES approach is to explicitly resolve the larger scales of flow and provide a closure model for the effect of subgrid scales upon resolved scales. The technical foundation of the method rests on a spatial filtering operation, which allows one to precisely define the scale of the model. We spatially filter Darcys law and the continuity equation to produce a model for the resolved-scale dynamics. We develop LES approximations for the resulting flow equations and a closure model for unresolved, subgrid-scale terms. We model the subgrid terms using resolved-scale quantities only and thus do not require subgrid-scale information. The accuracy of these closure models depends upon the scale at which hydraulic conductivity is “measured” by field tests. We characterize the approximations used to model subgrid-scale effects asymptotically with a small parameter, the dimensionless filter width ϵ, defined as the ratio of the filter width to the dominant length scale of the explicitly resolved variables.

Mixed finite element simulation of saturated groundwater flow using a multigrid accelerated domain decomposition technique

We present strategies to perform large-scale simulation of saturated groundwater flow using mixed finite element methods. The greatest limitation of these methods is their memory requirements. To overcome the memory limitations of the mixed methods, we use a domain decomposition strategy. We also present a multigrid acceleration for the domain decomposition. For sufficiently small problems, the conventional mixed method is more computationally efficient than the domain decomposition approach. However, the domain decomposition method allows one to simulate larger problems than is possible with the conventional mixed method.

Quantifying hyporheic exchange in a tidal river using temperature time series

[1] An investigation into groundwater-surface water interaction (GSWI) beneath a large tidally influenced river was conducted to determine the effect of tides on the development of a hyporheic zone (HZ) and to quantify mixing of river water and groundwater. Temperature measurements, coupled with independent hydraulic head measurements, were used to detect groundwater flow within the riverbed. GWSI under tidal forcing produced a 1 m deep HZ. Time-averaged riverbed temperature profiles displayed a distinct compressed convex pattern: clear evidence of net groundwater discharge. However, the instantaneous time series data indicate that riverbed temperatures were affected by tidal forcing to a depth of 1 m. Heat transport modeling revealed that instantaneous velocities within the shallow sediments of the riverbed are rather high, creating a zone of vigorous exchange during either a flooding or ebbing tide. Furthermore, the magnitude of the tidal pressure gradient was found to be significantly greater than the pressure gradient expected across 0.8 m high dunes, evidence that bed-form-driven exchange under these conditions, and this scale of observation, did not contribute to the development of the HZ. Conditions for exchange induced by shear and current bed form are favorable during ebbing tidal conditions only; flow paths are therefore limited in depth. Exchange flow paths in an estuary setting are complex; they are limited in duration and space and dominated by tidal pumping.

A comparison of methods to determine measurement support volumes

We examine methods that can be used to identify the spatial filter function associated with the measurement of quantities on two scales. We divide the methods into two general classes: deconvolution methods and sensitivity-based methods. We briefly describe the two classes of methods, provide examples of each from the literature, and introduce a new sensitivity-based method inspired by Backus-Gilbert model resolution ideas. We compare the approaches by examining the filter function that relates smaller-scale grid block parameters to parameters determined by a Theis model interpretation of drawdown data. In general, we find that sensitivity approaches are easier to implement numerically and are less subject to the problems of identifiability, stability, and uniqueness associated with deconvolution methods.

The influence of surface sorption on dispersion in parallel plate fractures

Abstract Solute transport in a parallel plate fracture is simulated using a random walk model which accounts explicitly for sorption onto the fracture walls. With the hypothesis that solute must move into the vicinity of the fracture wall in order to participate in the sorption process, three implications follow. (1) In comparison to a nonreactive solute, a sorbing solute requires a greater entrance length along the fracture before transverse homogenization is established. The increase in this entrance length is proportional to the surface retardation factor. (2) At short transport distances, surface sorption leads to a high degree of non-uniform retardation. The transport distance required to establish uniform retardation of the entire solute mass varies greatly with geometric conditions and sorption strength. (3) Surface sorption results in enhanced longitudinal spreading of the solute mass in transport regimes which favor advective transport along the fracture relative to transverse diffusion across the fracture aperture. At distances greater than that required for transverse homogenization, an effective longitudinal dispersion coefficient can be defined that describes this enhanced dispersion for a wide range of fluid velocities. The magnitude of enhanced dispersion increases with sorption strength.

Solubility controls for molybdenum in neutral rock drainage

Mineral controls on molybdate (MoO42−) solubility were studied to better understand the fate of molybdenum (Mo) in neutral rock drainage. Batch and column experiments lasting from 46–110 days predict that molybdate (MoO42−) mobility is limited by powellite (CaMoO4) and wulfenite (PbMoO4) precipitation under neutral pH conditions with aqueous Ca and Pb present. Batch experiments demonstrate that wulfenite forms almost instantaneously and effectively removes Pb from solution to concentrations below detection limits. Powellite formation is kinetically limited, but has the capacity to significantly reduce Mo concentrations in the presence of calcite. An initial inhibition of powellite formation is observed, likely due to a lack of available nucleation sites. After the nucleation phase, powellite formation from supersaturated conditions follows a second order rate expression with linear dependence on Ca2+ and MoO42−. A column experiment provides further evidence of rapid wulfenite formation and kinetically limited powellite formation. Both powellite and wulfenite have also been identified in molybdenum-bearing carbonate-rich waste rock sampled from barrel-sized field cell weathering experiments, providing direct evidence that these minerals affect Mo mobility under field conditions. In contrast to Pb and Ca, both Cu and Zn did not form distinct molybdate precipitates.