The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

/pdf/validity-of-cubic-law-for-fluid-flow-in-a-deformable-rock-2ynq4id3hq.pdf

VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

Characterizing flow and transport in fractured geological media: A review

Mathematica--A System for Doing Mathematics by Computer.

[1] A generalized formulation for kinetically controlled reactions has been developed and incorporated into a multicomponent reactive transport model to facilitate the investigation of a large variety of problems involving inorganic and organic chemicals in variably saturated media. The general kinetic formulation includes intra-aqueous and dissolution-precipitation reactions in addition to geochemical equilibrium expressions for hydrolysis, aqueous complexation, oxidation-reduction, ion exchange, surface complexation, and gas dissolution-exsolution reactions. The generalized approach allows consideration of fractional order terms with respect to any dissolved species in terms of species activities or in terms of total concentrations, which facilitates the incorporation of a variety of experimentally derived rate expressions. Monod and inhibition terms can be used to describe microbially mediated reactions or to limit the reaction progress of inorganic reactions. Dissolution-precipitation reactions can be described as surface-controlled or transport-controlled reactions. The formulation also facilitates the consideration of any number of parallel reaction pathways, and reactions can be treated as irreversible or reversible processes. Two groundwater contamination scenarios, both set in variably saturated media but with significantly different geochemical reaction networks, are investigated and demonstrate the advantage of the generalized approach. The first problem focuses on a hypothetical case study of the natural attenuation of organic contaminants undergoing dissolution, volatilization, and biodegradation in an unconfined aquifer overlaid by unsaturated sediments. The second problem addresses the generation of acid mine drainage in the unsaturated zone of a tailings impoundment at the Nickel Rim Mine Site near Sudbury, Ontario, and subsequent reactive transport in the saturated portion of the tailings.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2001WR000862

Multicomponent reactive transport modeling in variably saturated porous media using a generalized formulation for kinetically controlled reactions

Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media

Simulating the multi-seasonal response of a large-scale watershed with a 3D physically-based hydrologic model

Application of system dynamics for developing financially self-sustaining management policies for water and wastewater systems

Geologic carbon sequestration is the direct injection of CO 2 into deep geologic formations for permanent disposal. Although numerous trapping mechanisms exist in the subsurface, it is possible that CO 2 will leak from the primary sequestration target and seep out of the ground. The unsaturated zone has the potential to attenuate leaking CO 2 and decrease seepage and near-surface CO 2 concentrations. Attenuation processes include permeability trapping, ponding as dense CO 2 spreads out on the water table, solubility trapping by infiltrating or residual water, and dilution through mixing with ambient soil gas. Numerical simulations of CO 2 flowing upward through a thick model unsaturated zone were performed to investigate the sensitivity of various unsaturated zone properties on CO 2 seepage flux and near-surface CO 2 gas concentrations. These two quantities are considered drivers for health and environmental risk due to exposure to CO 2 . For the conceptual model considered, seepage flux and near-surface CO 2 gas concentrations are most strongly controlled by the leakage rate at the water table, followed by the source zone radius. Permeability and permeability anisotropy, as well as porosity and infiltration rate are also important, although to a lesser degree. Barometric pumping causes local maxima in seepage flux and near-surface CO 2 concentrations, but has negligible effect in a time-averaged sense. When the leakage source is turned off, the CO 2 plume attentuates through dissolution into infiltrating water. For the case of a constant leakage rate, the unsaturated zone can attenuate low leakage fluxes but should not be expected to attenuate large CO 2 leakage fluxes.

/pdf/on-leakage-and-seepage-from-geologic-carbon-sequestration-1gtrqo8z35.pdf

On leakage and seepage from geologic carbon sequestration sites: Unsaturated Zone Attenuation

The numerical model CompFlow is used to study the mechanisms controlling vacuum extraction, coupled with air sparging, as a means for remediation of heterogeneous formations contaminated with dense nonaqueous phase liquids (DNAPLs). Two dominant mechanisms are demonstrated to control this remediation technology. First, at early times, the gas phase directly contacts the DNAPL, particularly in the unsaturated zone, causing relatively rapid transfer of contaminant from the nonaqueous phase to the gas phase and subsequent removal by the vacuum extractor. Second, at later times, remediation is controlled by the transfer of contaminant from the nonaqueous phase to the aqueous phase below the water table. During this time the vacuum extractor pumps both liquid and vaporized water in the aqueous and gas phases. This causes the contaminant that is dissolved in the aqueous phase to migrate vertically upward across the permeability layers toward the vacuum extractor where it is removed. This intermediate to late time removal mechanism is shown to be controlled by contaminant dissolution, which is a slower transfer process than the direct DNAPL vaporization that occurs at early time. Our analysis indicates that as long as both air and water are actively flushed through the DNAPL zone, both early-time vaporization and intermediate- to late-time dissolution are effective mechanisms leading to the removal of the DNAPL. We show that it may be possible to design the remedial system so as to reduce its performance sensitivity to geologic heterogeneity. A lack of sensitivity of a remedial design to heterogeneity is highly desirable because a robust design implies that the degree of site characterization required for reasonable success will be less than that needed for a less robust scheme.

Mechanisms Controlling Vacuum Extraction Coupled With Air Sparging for Remediation of Heterogeneous Formations Contaminated by Dense Nonaqueous Phase Liquids

Abandoned and improperly sealed boreholes, monitoring wells, and water supply wells are common features at many contaminated sites. These features can act as conduits that transmit contaminants between aquifers separated by otherwise continuous aquitards. In this work the leaky boreholes are represented as highly conductive one-dimensional line elements superimposed onto a mesh of three-dimensional finite elements representing the porous medium. Simulation results are presented for a series of scenarios involving a simple hydrogeologic setting composed of an upper confined aquifer, a middle aquitard, and a lower confined aquifer. The simulations examine the effect of varying the borehole properties, and vertical hydraulic gradient across the aquitard, and the borehole location. The results show that a contaminant can rapidly migrate downward along a leaky borehole and create an extensive plume in the lower aquifer, even if the borehole is filled with aquifer sediments. If the borehole is an open feature across the aquitard, the entire plume, or a significant portion of it, that is migrating into the upper aquifer can be diverted into the lower one if the vertical hydraulic gradient across the aquitard can be diverted into the lower one if the vertical hydraulic gradient across the aquitard is sufficiently strong.more » The peak concentration arriving at a pumping well located in the lower aquifer and the time of arrival are functions of the proximity of the leaky borehole to the pumping well and its angular offset from the central flow line passing through the surficial source. Overpressurization of the lower aquifer due to injection can overcome downward preexisting hydraulic gradients across the aquitard such that contaminants can rapidly migrate upward along the leaky borehole and cause contamination of the otherwise protected upper aquifer. 26 refs., 7 figs., 1 tab.« less

Andre J. A. Unger

Papers

Simulating the multi-seasonal response of a large-scale watershed with a 3D physically-based hydrologic model

Application of system dynamics for developing financially self-sustaining management policies for water and wastewater systems

On leakage and seepage from geologic carbon sequestration sites: Unsaturated Zone Attenuation

Mechanisms Controlling Vacuum Extraction Coupled With Air Sparging for Remediation of Heterogeneous Formations Contaminated by Dense Nonaqueous Phase Liquids

Influence of Leaky Boreholes on Cross-Formational Groundwater Flow and Contaminant Transport