Dripping from Unsaturated Fractures into Subterranean Cavities Dani Or and Teamrat Ghezzehei Plants, Soils and Biometeorology Department & Biological and Irrigation Engineering Department Utah State University, Logan Utah Plants, Soils and Biometeorology Department & Biological and Irrigation Engineering Department Utah State University, Logan Utah

IntroductionIntroduction l Formation and detachment of drops results from motion of free liquid surfaces and involves interplay between capillary, viscous, gravitational and inertial forces. l Two extreme flow conditions of dripping have been studied extensively: rapid jetting and slow dripping. l In fractured porous media, slow dripping is induced under certain flow and humidity conditions, and within particular geometrical settings. l Dripping in subterranean cavities is of interest to karst hydrology and geochemistry, subsurface mining, and disposal of nuclear waste. l This study was motivated by the long-term effect of dripping on nuclear waste disposal canisters at Yucca Mountain Project. l Formation and detachment of drops results from motion of free liquid surfaces and involves interplay between capillary, viscous, gravitational and inertial forces. l Two extreme flow conditions of dripping have been studied extensively: rapid jetting and slow dripping. l In fractured porous media, slow dripping is induced under certain flow and humidity conditions, and within particular geometrical settings. l Dripping in subterranean cavities is of interest to karst hydrology and geochemistry, subsurface mining, and disposal of nuclear waste. l This study was motivated by the long-term effect of dripping on nuclear waste disposal canisters at Yucca Mountain Project.

Yucca Mountain Project Site

Dripping in Natural Caves & Tunnels l Dripping in natural caves and man-made tunnels is often manifested by formation of speleothems. Soda straw forest in the Cupp-Coutunn karst cave system, Southeast Turkmenistan (Courtesy:Vladimir Maltsev, Moscow). Soda straw in cement grouting tunnels. Wujiangdu Hydropower plant, Guizhou, China (Liu and He, 1998, Environ. Geology, 35: )

ObjectivesObjectives l The objective of this study was to develop an integrated model for slow dripping of water at intersection of rough- walled vertical fracture with open cavity. l The model consists of the following components: modeling water flow in unsaturated rough fracture surface, modeling drop growth and detachment as one-dimensional axisymetric viscous-extension process, evaporation of water from drop surface local force balance for determination of drop anchoring area. l The objective of this study was to develop an integrated model for slow dripping of water at intersection of rough- walled vertical fracture with open cavity. l The model consists of the following components: modeling water flow in unsaturated rough fracture surface, modeling drop growth and detachment as one-dimensional axisymetric viscous-extension process, evaporation of water from drop surface local force balance for determination of drop anchoring area.

Model Components l The model comprises the following components: (1) Flow on rough fractures surface (2) 1-D axial extension (3) Evaporation (4) Drop anchoring area

Flow on Rough Fracture Surface l Flow Regimes 1. Flow of thin films on planar fracture surfaces 2. Flow of capillary wedges in surface grooves l Flow Regimes 1. Flow of thin films on planar fracture surfaces 2. Flow of capillary wedges in surface grooves Natural Fracture Surface Idealized Representation of Fracture Surface (Or and Tuller, 2000 Water Resour. Res. 36: )

1-D Axial Extension of a Viscous Drop  A  = 0  = t Ao Drop Evolution in Lagrangian Coordinates  X t,  A t,   Longitudinal Force Balance (Wilson, S.D.R. 1988, J. Fluid Mech. 190: )  Volume of a thin extruded element Force balance for thin element Constitutive Relationship (axial elongation flow) Elongation force Where = 3·  Force at element  Where p is perimeter and S is the longitudinal stress

Boundary Conditions Emerging liquid element: Ao Rupturing liquid element: Experimental observations suggest that drop anchoring area (at  = t) is relatively constant. Experimental observations suggest that drop anchoring area (at  = t) is relatively constant. l Derivation of actual anchoring area will be considered separately. l During breakage of the pendant drop (as A t,   0), the rate of extension grows to infinity Experimental observations of drop anchoring area and breakage

Solution to “Dripping” ODE l Combining the longitudinal force equations (considering liquid incompressibility and axial elongation) results in the following ODE: Where,Where, l Analytical solution to the ODE, subject to the boundary conditions is available; with the element that ruptures first given by: The rupturing plane (element) is marked by  c.  c also denotes the time interval between two successive drops. l Detaching drop volume is given by: The rupturing plane (element) is marked by  c.  c also denotes the time interval between two successive drops. l Detaching drop volume is given by: (Wilson, 1988, J. Fluid Mech. 190: )

1-D Axial Extension of a Viscous Thread – Alternative Derivation  A  = 0  = t Ao Drop Evolution in Lagrangian Coordinates  X t,  A t,   (Yarin et al., 1999, Phys. Fluids. 11: )  Continuity Eq. Momentum Balance Eq. IntegrationIntegration

Isothermal Evaporation l The drop is assumed to be hemispherical during drop formation period Isothermal diffusion r(t) Hemispherical Drop FormationDetachment Time (sec) Isothermal diffusion from drop surface by Fick’s Law Resultant net flux

Theoretical Results: Dripping Period l The model was evaluated for two groove angles under evaporative (E) and non-evaporative (NE) conditions l Dryer condition decreases flux, hence, increases dripping period. l Narrow grooves sustain higher fluxes, hence, lower period l The model was evaluated for two groove angles under evaporative (E) and non-evaporative (NE) conditions l Dryer condition decreases flux, hence, increases dripping period. l Narrow grooves sustain higher fluxes, hence, lower period l Limiting minimum potential and maximum period exist for evaporative conditions. l Divergence between E and NE in narrow potential range implies sensitivity of dripping period to ventilation. l Limiting minimum potential and maximum period exist for evaporative conditions. l Divergence between E and NE in narrow potential range implies sensitivity of dripping period to ventilation.

Theoretical Results: Drop Volume l Non-Evaporative: the drop volume is practically constant for all potentials, and groove geometry. l Evaporative: drop volume increases rapidly when the dripping period begins to diverge from non-evaporative (previous slide). l Non-Evaporative: the drop volume is practically constant for all potentials, and groove geometry. l Evaporative: drop volume increases rapidly when the dripping period begins to diverge from non-evaporative (previous slide). Why are drops larger under high evaporation? l Under dry conditions, the competition between high evaporation and low total flux results in a very low net flux feeding the drop. l Consequently, slower viscous extension rate dissipates less energy freeing extra force to support larger drop weight.

Theoretical Results: Solute Concentration l Evaporation from drops, leaves dissolved solutes behind. Consequently, drops have higher salt concentration than the bulk liquid. l The total evaporated volume increases with increase in dripping period leading to higher solute concentration. l Evaporation from drops, leaves dissolved solutes behind. Consequently, drops have higher salt concentration than the bulk liquid. l The total evaporated volume increases with increase in dripping period leading to higher solute concentration. l Relative solute concentration: l The effect of change in solute concentration on surface tension is not considered in this study.

Decoupling Flux and Evaporation l In ventilated tunnels or controlled laboratory experiments evaporation and flux (film flow) can be decoupled processes.

Model Testing: Lab Experiments l Laboratory experiments were conducted using natural rock surface and grooved aluminum (and quartz) surfaces. l Known fixed flux was applied at high rate (no-evaporation), dripping period and drop volume were recorded l Laboratory experiments were conducted using natural rock surface and grooved aluminum (and quartz) surfaces. l Known fixed flux was applied at high rate (no-evaporation), dripping period and drop volume were recorded

ObservationsObservations  Groove angle 45 o ; depth=5 mm; flux 1 ml/min. Note:  Constant drop anchoring area (A 0 ).  Nearly constant liquid-vapor interfaces (above plane).  Long formation period vs. rapid detachment.  Drop recoil volume.  Groove angle 45 o ; depth=5 mm; flux 1 ml/min. Note:  Constant drop anchoring area (A 0 ).  Nearly constant liquid-vapor interfaces (above plane).  Long formation period vs. rapid detachment.  Drop recoil volume FormationDetachment Time (sec)

Natural Rock - Drop Anchoring Area

Model Testing: Lab Results l The model was evaluated with a constant drop anchoring area aluminum: d=10mm, rock: d=12mm l Predicted dripping period agrees very well with measurement. l Drop volume is primarily determined by drop anchoring area (a function of solid-liquid properties & groove geometry). l Rough rock surfaces induced larger variability in drop volume than obtained from smooth aluminum slab with a fixed groove angle. l The model was evaluated with a constant drop anchoring area aluminum: d=10mm, rock: d=12mm l Predicted dripping period agrees very well with measurement. l Drop volume is primarily determined by drop anchoring area (a function of solid-liquid properties & groove geometry). l Rough rock surfaces induced larger variability in drop volume than obtained from smooth aluminum slab with a fixed groove angle.

Model Testing: Père Noël cave - Belgium (Genty and Deflandre (1998), J. Hydrology 211: ) High flow small drops l Dripping from stalactites was monitored for five hydrologic cycles ( ).

Model Testing: Père Noël cave - Belgium (Genty and Deflandre (1998), J. Hydrology 211: ) l The data reported includes: dripping rate (number of drop per 10 minutes) and corresponding average drop volume.

Model Testing: Père Noël cave - Belgium l Model prediction are in good agreement with measurements, except at high fluxes (onset of instability and jetting). l These results provide experimental evidence for the increase in drop volume with decreasing net flux. l Model prediction are in good agreement with measurements, except at high fluxes (onset of instability and jetting). l These results provide experimental evidence for the increase in drop volume with decreasing net flux. (Genty and Deflandre (1998), J. Hydrology 211: ) Fast dripping Re>1 - instability

SummarySummary  A model for slow dripping at intersection of vertical rough walled rock fracture with a subterranean cavity was developed and tested.  Fracture surface flow was combined with 1D model for drop formation, extension and detachment.  Competitive effects of evaporation on dripping period, drop size, and concentration were introduced.  Drop anchoring area, approximate shape, and lateral position were derived for the groove-cavity geometry.  Results from a “real” cave were reconstructed and explained by the proposed model.  Geochemical effects on surface tension and evaporation and details of CO 2, degassing will be subjects of future work.  A model for slow dripping at intersection of vertical rough walled rock fracture with a subterranean cavity was developed and tested.  Fracture surface flow was combined with 1D model for drop formation, extension and detachment.  Competitive effects of evaporation on dripping period, drop size, and concentration were introduced.  Drop anchoring area, approximate shape, and lateral position were derived for the groove-cavity geometry.  Results from a “real” cave were reconstructed and explained by the proposed model.  Geochemical effects on surface tension and evaporation and details of CO 2, degassing will be subjects of future work. Publication: Or, D., and T.A. Ghezzehei, Dripping into subterranean cavities from unsaturated fractures under evaporative conditions, Water Resour. Res., 36(2), , 2000.

Liquid Profile Near the Dripping Plane Capillary Zone Transition Zone Drop Zone z Capillary Zone Transition Zone In equilibrium with ambient vapor pressure Drop Zone Idealized Drop Shape Vertical force balance: Horizontal force balance