Michael J. Nicholl

Saturated flow in a single fracture: evaluation of the Reynolds Equation in measured aperture fields

Fracture transmissivity and detailed aperture fields are measured in analog fractures specially designed to evaluate the utility of the Reynolds equation. The authors employ a light transmission technique with well-defined accuracy ({approximately}1% error) to measure aperture fields at high spatial resolution ({approximately}0.015 cm). A Hele-Shaw cell is used to confirm the approach by demonstrating agreement between experimental transmissivity, simulated transmissivity on the measured aperture field, and the parallel plate law. In the two rough-walled analog fractures considered, the discrepancy between the experimental and numerical estimates of fracture transmissivity was sufficiently large ({approximately} 22--47%) to exclude numerical and experimental errors (< 2%)as a source. They conclude that the three-dimensional character of the flow field is important for fully describing fluid flow in the two rough-walled fractures considered, and that the approach of depth averaging inherent in the formulation of the Reynolds equation is inadequate. They also explore the effects of spatial resolution, aperture measurement technique, and alternative definitions for link transmissivities in the finite-difference formulation, including some that contain corrections for tortuosity perpendicular to the mean fracture plane and Stokes flow. Various formulations for link transmissivity are shown to converge at high resolution ({approximately} 1/5 the spatial correlation length) in the smoothly varying fracture. At coarser resolutions, the solution becomes increasingly sensitive to definition of link transmissivity and measurement technique. Aperture measurements that integrate over individual grid blocks were less sensitive to measurement scale and definition of link transmissivity than point sampling techniques.

A modified invasion percolation model for low‐capillary number immiscible displacements in horizontal rough‐walled fractures: Influence of local in‐plane curvature

The authors develop and evaluate a modified invasion percolation (MIP) model for quasi-static immiscible displacement in horizontal fractures. The effects of contact angle, local aperture field geometry, and local in-plane interracial curvature between phases are included in the calculation of invasion pressure for individual sites in a discretized aperture field. This pressure controls the choice of which site is invaded during the displacement process and hence the growth of phase saturation structure within the fracture. To focus on the influence of local in-plane curvature on phase invasion structure, they formulate a simplified nondimensional pressure equation containing a dimensionless curvature number (C) that weighs the relative importance of in-plane curvature and aperture-induced curvature. Through systematic variation of C, they find in-plane interracial curvature to greatly affect the phase invasion structure. As C is increased from zero, phase invasion fronts transition from highly complicated (IP results) to microscopically smooth. In addition, measurements of fracture phase saturations and entrapped cluster statistics (number, maximum size, structural complication) show differential response between wetting and nonwetting invasion with respect to C that is independent of contact angle hysteresis. Comparison to experimental data available at this time substantiates predicted behavior.

Liquid phase structure within an unsaturated fracture network beneath a surface infiltration event: Field experiment

[1] We conducted a simple field experiment to elucidate structure (i.e., geometry) of the liquid phase (water) resulting from ponded infiltration into a pervasive fracture network that dissected a nearly impermeable rock matrix. Over a 46 min period, dyed water was infiltrated from a surface pond while electrical resistance tomography (ERT) was employed to monitor the rapid invasion of the initially dry fracture network and subsequent drainage. We then excavated the rock mass to a depth of ∼5 m, mapping the fracture network and extent of dye staining over a series of horizontal pavements located directly beneath the pond. Near the infiltration surface, flow was dominated by viscous forces, and the fracture network was fully stained. With increasing depth, flow transitioned to unsaturated conditions, and the phase structure became complicated, exhibiting evidence of fragmentation, preferential flow, fingers, irregular wetting patterns, and varied behavior at fracture intersections. ERT images demonstrate that water spanned the instrumented network rapidly on ponding and also rapidly drained after ponding was terminated. Estimates suggest that our excavation captured from ∼15 to 1% or less of the rock volume interrogated by our infiltration slug, and thus the penetration depth from our short ponding event could have been quite large.

Factors controlling satiated relative permeability in a partially-saturated horizontal fracture

Recent work demonstrates that phase displacements within horizontal fractures large with respect to the spatial correlation length of the aperture field lead to a satiated condition that constrains the relative permeability to be less than one. The authors use effective media theory to develop a conceptual model for satiated relative permeability, then compare predictions to existing experimental measurements, and numerical solutions of the Reynolds equation on the measured aperture field within the flowing phase. The close agreement among all results and data show that for the experiments considered here, in-plane tortuosity induced by the entrapped phase is the dominant factor controlling satiated relative permeability. They also find that for this data set, each factor in the conceptual model displays an approximate power law dependence on the satiated saturation of the fracture.

The interaction of two fluid phases in fractured media

In fractured porous media, interactions between immiscible fluid phases within the fractures place a critical control on system behavior. A key component of the interactions is the geometry, or structure, of the respective phases. Over the past 10 years, process-based experiments have greatly increased our understanding of phase structure development within individual fractures. In the past 2 years, new calculational models that incorporate some of this understanding have further demonstrated the influence of phase structure on flow and transport within the phases, and inter-phase mass transport. These computational models can now be applied to consider the efficacy and parameterization of constitutive relations for a subset of two-phase situations. Full understanding of the morphology, connectivity, and temporal dynamics of phase structure in rough-walled fractures is yet to be developed, and is a promising area for further research.

Simulation of flow and transport in a single fracture: Macroscopic effects of underestimating local head loss

Fluid flow in a single fracture is commonly simulated using the Reynolds equation. Recent work suggests that this depth-averaged approach underestimates head loss in regions of changing aperture. Implementing an ad hoc correction in the numerical formulation of the Reynolds equation allows us to modify local head loss, and calibrate simulation results to existing experimental data. Calibrated flow fields provide an improved estimate of longitudinal dispersivity, demonstrating the importance of adequately describing local head loss.

Application of a Darcian Approach to Estimate Liquid Flux in a Deep Vadose Zone

near-surface environment. Also, many of the complicating effects found in the near-surface environment will Approaches for estimating liquid flux in the shallow (0–2 m) vadose be damped or eliminated with increasing depth. For zone are hindered by the high degree of spatial and temporal variability present near the land surface. It is hypothesized that high-frequency these reasons, flux measurement at depth would appear variations in flux will be damped with depth. This study was conducted to be an attractive alternative at such sites. However, to estimate deep liquid flux using the Darcian approach at a waste borehole instruments for direct measurement of deep disposal site in south-central Idaho that is underlain by a complex flux do not exist at this time. Environmental tracers sequence of unsaturated basalt flows intercalated with thin sedimen(e.g., Scanlon et al., 1997; Phillips, 2001) may be used tary layers. Flux is estimated by combining in situ water potential to provide information on average historical flux at measurements from sedimentary interbeds located at depths of 34 some sites, but this approach is not conducive to moniand 73 m below land surface (bls) with laboratory estimates for the toring activities. Conversely, Darcian approaches will unsaturated hydraulic conductivity. Tensiometer data at seven locahave a more widespread applicability and are amenable tions indicated nearly constant conditions for 30 mo, while nine of to monitoring. the other 10 sites showed small gradual trends. Assumption of a unit hydraulic gradient led to flux estimates ranging from 0.2 to 10 000 cm Darcian approaches are founded on an assumption of yr 1. Estimates in the 34-m interbed ranged across four orders of one-dimensional vertical flow. One then needs sufficient magnitude while flux estimates for the 73-m interbed ranged three information on either the in situ moisture content or orders of magnitude. While the tensiometer data appear to reflect in water potential to calculate flux from laboratory-derived situ conditions and are a sensitive indicator of hydrologic conditions in unsaturated hydraulic conductivity. Previous applicathe deep vadose zone, the laboratory-developed hydraulic properties tions of this approach have used tensiometers (Stephens introduce a high degree of uncertainty, potentially affecting predicand Knowlton, 1986), thermocouple psychrometers tions by orders of magnitude. There is a need to develop techniques (Andraski, 1997), or heat dissipation sensors (Montazer for assessing flux rates for the range of applicable field conditions to et al., 1986) to measure water potential gradients along improve the confidence in deep flux estimates. boreholes at depths of 2, 5, and 200 m, respectively. Tensiometric data have a distinct advantage in that it is a direct measure of water potential, whereas the other F the transport of waterborne contamitwo methods are calibration-dependent, indirect meanants through the vadose zone requires estimates sures. Available techniques for in situ measurement of for liquid flux between the land surface and the water moisture content are not only calibration dependent, table. While there is an extensive body of literature but also physically difficult to install at depths beyond regarding the estimation of flux at shallow depths, the a few meters. deeper vadose zone has received much less attention. Conventional tensiometers require a continuous waInstruments used for direct measurement of flux in the ter column that extends from the measurement point near-surface environment (0–2 m) include pan lysimeto the sensor location at or near the land surface. The ters (e.g., Jordan, 1968), tension lysimeters (Byre et al., vaporization of water in the water column limits the 1999), and vadose zone flux meters (Wagenet, 1986; depth of emplacement to about 8 m and has precluded Gee et al., 2002). There are also Darcian approaches in the use of tensiometer data for Darcian estimates of which shallow measurements of moisture content or flux below that depth. This problem was recently overwater potential ( ) are combined with laboratory develcome by development of the advanced tensiometer oped relations for the unsaturated hydraulic conductiv(Hubbell and Sisson, 1998), which has been successfully ity to estimate flux (Stephens and Knowlton, 1986). deployed to make direct measurements of water potenHowever, the utility of all such measurements is limited tial at depths up to 145 m. The advanced tensiometer has by the inherent spatial and temporal variability of flux two parts, a permanently installed porous cup assembly in the shallow vadose zone (Wagenet, 1986). with casing that extends to land surface and a removable At sites with thick vadose zones, flux estimates obelectronic pressure transducer assembly for installation tained at depth may be more representative of mass from land surface. Positioning the sensor close to the transfer to the water table than those obtained from the measurement point (porous cup) eliminates the need for a water column extending to land surface, thus reJ.M. Hubbell, J.B. Sisson (ret.), and D.L. McElroy, Idaho National moving the restriction on depth of emplacement. Engineering and Environmental Laboratory, Geosciences Research Here, we present a first attempt to estimate flux at Department, P.O. Box 1625, MS 2107, Idaho Falls ID 83415; M.J. Nicholl, Geosciences Dep., Univ. of Nevada, Las Vegas, NV 89122. depth using long-term monitoring data obtained from Received 10 July 2003. Special Section: Uncertainty in Vadose Zone the deployment of advanced tensiometers. Instruments Flow and Transport Properties. *Corresponding author (jmh@ inel.gov). Abbreviations: bls, below land surface; ESRP, Eastern Snake River Plain; INEEL, Idaho National Engineering and Environmental LaboPublished in Vadose Zone Journal 3:560–569 (2004).  Soil Science Society of America ratory; RWMC, Radioactive Waste Management Complex; SDA, Subsurface Disposal Area. 677 S. Segoe Rd., Madison, WI 53711 USA

Correction to “A modified invasion percolation model for low–capillary number immiscible displacements in horizontal rough‐walled fractures: Influence of local in‐plane curvature” by Robert J. Glass, Michael J. Nicholl, and Lane Yarrington

In the paper “A modified invasion percolation model for low–capillary number immiscible displacements in horizontal rough-walled fractures: Influence of local in-plane curvature” by Robert J. Glass, Michael J. Nicholl, and Lane Yarrington (Water Resources Research, 34(12), 3215–3234, 1998), the invaded and uninvaded sides in Figure 2 were mislabeled in the legend (white should be invaded and gray should be uninvaded) resulting in a misrepresentation of the angle g in Figure 2. The corrected Figure 2 and caption follow.

Uncertainty in Vadose Zone Flow and Transport Prediction

This special section of Vadose Zone Journal considers uncertainty in vadose zone flow and transport prediction. The recent emergence of this topic is a consequence of a major change in the motivation for predictive modeling in the vadose zone. Until the 1980s, predictive modeling of vadose zone flow

Teaching Darcy’s Law Through Hands-On Experimentation

Darcy’s Law is one of the essential concepts in hydrogeology. Before moving on to more complex problems, students must first thoroughly understand the basic principles of one-dimensional fluid flow through saturated porous media that are embodied in Darcy’s Law. We believe that the best way for students to learn these principles is through experimentation. In this paper, we introduce an experimental apparatus and laboratory exercises designed to facilitate student exploration of Darcy’s Law. Our permeameter design is simple and inexpensive to construct from readily available materials; it is also nearly indestructible and easy to use. The exercises we present are flexible and suitable for high-school or university students. Students performing the exercises will gain an understanding of the relationships among hydraulic gradient, pore size, porosity, fluid viscosity, particle size (mean and distribution), and volumetric flow rate.