Dripping from Dripping from Unsaturated Fractures Unsaturated Fractures into Subterranean into Subterranean CavitiesCavities

Dripping from Dripping from Unsaturated Fractures Unsaturated Fractures into Subterranean into Subterranean CavitiesCavitiesDani Or and Teamrat GhezzeheiDani Or and Teamrat GhezzeheiDani Or and Teamrat GhezzeheiDani Or and Teamrat GhezzeheiPlants, 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

IntroductionIntroductionIntroductionIntroduction

Formation and detachment of drops results from motion of free liquid surfaces and involves interplay between capillary, viscous, gravitational and inertial forces.

Two extreme flow conditions of dripping have been studied extensively: rapid jetting and slow dripping.

In fractured porous media, slow dripping is induced under certain flow and humidity conditions, and within particular geometrical settings.

Dripping in subterranean cavities is of interest to karst hydrology and geochemistry, subsurface mining, and disposal of nuclear waste.

This study was motivated by the long-term effect of dripping on nuclear waste disposal canisters at Yucca Mountain Project.

Formation and detachment of drops results from motion of free liquid surfaces and involves interplay between capillary, viscous, gravitational and inertial forces.

Two extreme flow conditions of dripping have been studied extensively: rapid jetting and slow dripping.

In fractured porous media, slow dripping is induced under certain flow and humidity conditions, and within particular geometrical settings.

Dripping in subterranean cavities is of interest to karst hydrology and geochemistry, subsurface mining, and disposal of nuclear waste.

This study was motivated by the long-term effect of dripping on nuclear waste disposal canisters at Yucca Mountain Project.

Yucca Mountain Project SiteYucca Mountain Project SiteYucca Mountain Project SiteYucca Mountain Project Site

Dripping in Natural Caves & Dripping in Natural Caves & TunnelsTunnelsDripping in Natural Caves & Dripping in Natural Caves & TunnelsTunnels Dripping in natural caves and man-made tunnels is often

manifested by formation of speleothems.

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, 35:258-262)

ObjectivesObjectivesObjectivesObjectives

The objective of this study was to develop an integrated The objective of this study was to develop an integrated model for slow dripping of water at intersection of rough-model for slow dripping of water at intersection of rough-walled vertical fracture with open cavity.walled vertical fracture with open cavity.

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 ComponentsModel ComponentsModel ComponentsModel Components The model comprises the following components: 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 Flow on Rough Fracture SurfaceSurfaceFlow on Rough Fracture Flow on Rough Fracture SurfaceSurface

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. 36:1165-1177)(Or and Tuller, 2000 Water Resour. Res. 36:1165-1177)

1-D Axial Extension of a 1-D Axial Extension of a Viscous DropViscous Drop1-D Axial Extension of a 1-D Axial Extension of a Viscous DropViscous Drop

A= 0

= tAo

Drop Evolution in Lagrangian Coordinates

Xt,

At,

Longitudinal Force BalanceLongitudinal Force BalanceLongitudinal Force BalanceLongitudinal Force Balance

(Wilson, S.D.R. 1988, J. Fluid Mech. 190:561-570)

,t,t XAQVolume of a thin Volume of a thin extruded elementextruded elementVolume of a thin Volume of a thin extruded elementextruded element

Force balance Force balance for thin elementfor thin elementForce balance Force balance for thin elementfor thin element

pQgASAS

Constitutive Relationship (axial elongation flow)Constitutive Relationship (axial elongation flow)Constitutive Relationship (axial elongation flow)Constitutive Relationship (axial elongation flow)

Elongation forceElongation forceElongation forceElongation force2

p

td

AdAS

Where Where = 3· = 3·Where Where = 3· = 3·

Force at Force at element element Force at Force at element element

,t,t pQgAS

Where Where pp is perimeter and is perimeter and SS is the longitudinal stress is the longitudinal stressWhere Where pp is perimeter and is perimeter and SS is the longitudinal stress is the longitudinal stress

Boundary ConditionsBoundary ConditionsBoundary ConditionsBoundary Conditions

Emerging liquid element:Emerging liquid element:Emerging liquid element:Emerging liquid element:

ConstantAoAt tt ,

AoAoRupturing liquid element:Rupturing liquid element:Rupturing liquid element:Rupturing liquid element:

0ddt

0ddt

Experimental observations Experimental observations suggest that drop anchoring area suggest that drop anchoring area (at (at = t) is relatively constant. = t) is relatively constant.

Experimental observations Experimental observations suggest that drop anchoring area suggest that drop anchoring area (at (at = t) is relatively constant. = t) is relatively constant.

Derivation of actual anchoring Derivation of actual anchoring area will be considered separately.area will be considered separately.

Derivation of actual anchoring Derivation of actual anchoring area will be considered separately.area will be considered separately.

During breakage of the pendant During breakage of the pendant drop (as Adrop (as At,t,0), the rate of 0), the rate of

extension grows to infinityextension grows to infinity

During breakage of the pendant During breakage of the pendant drop (as Adrop (as At,t,0), the rate of 0), the rate of

extension grows to infinityextension grows to infinity

Experimental observations of drop anchoring area and

breakage

Experimental observations of drop anchoring area and

breakage

Solution to “Dripping” ODESolution to “Dripping” ODESolution to “Dripping” ODESolution to “Dripping” ODE

Combining the longitudinal force equations (considering liquid incompressibility and axial elongation) results in the following ODE:

Combining the longitudinal force equations (considering liquid incompressibility and axial elongation) results in the following ODE:

3

)t(AQg

td

)t(Ad

0QQ

1ln1

Q

1Ao61

cc2

Where,Where,Where,Where,

gAo

2

Analytical solution to the ODE, subject to the boundary conditions is available; with the element that ruptures first given by:

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. Detaching drop volume is given by:

The rupturing plane (element) is marked by c. c also denotes the time interval between two successive drops. Detaching drop volume is given by: cQV

(Wilson, 1988, J. Fluid Mech. 190:561-570)

1-D Axial Extension of a Viscous Thread 1-D Axial Extension of a Viscous Thread – Alternative Derivation– Alternative Derivation1-D Axial Extension of a Viscous Thread 1-D Axial Extension of a Viscous Thread – Alternative Derivation– Alternative Derivation

A= 0

= tAo

Drop Evolution in Lagrangian Coordinates

Xt,

At,

(Yarin et al., 1999, Phys. Fluids. 11:3201-3208)

0

z

A

t

A

03 22

z

RRg

zR

z

030

RzdAgz

Az

0

zA

t

A

030

RzdAg

t

A z

QzdAVz

0

03 RQg

t

A

3

RQg

t

A

Continuity Eq.Continuity Eq.Continuity Eq.Continuity Eq.Momentum Balance Eq.Momentum Balance Eq.Momentum Balance Eq.Momentum Balance Eq.

IntegrationIntegrationIntegrationIntegration

Isothermal EvaporationIsothermal Evaporation Isothermal EvaporationIsothermal Evaporation The drop is assumed to be hemispherical

during drop formation period

The drop is assumed to be hemispherical during drop formation period

Isothermal diffusion

r(t)

Hemispherical DropHemispherical Drop

0 2 4 6 8 10 12

FormationFormation DetachmentDetachment

Time (sec)Time (sec)

T*Rr

PvD

rd

CdD

td

rd

brr

b

b

Isothermal diffusion from Isothermal diffusion from drop surface by Fick’s Lawdrop surface by Fick’s LawIsothermal diffusion from Isothermal diffusion from drop surface by Fick’s Lawdrop surface by Fick’s Law

Resultant net fluxResultant net fluxResultant net fluxResultant net flux2

3

TR

PvD2Q

2

3

3

2)(Q

32

net

Theoretical Results: Dripping Theoretical Results: Dripping PeriodPeriodTheoretical Results: Dripping Theoretical Results: Dripping PeriodPeriod The model was evaluated for two groove angles under evaporative (E)

and non-evaporative (NE) conditions

Dryer condition decreases flux, hence, increases dripping period.

Narrow grooves sustain higher fluxes, hence, lower period

The model was evaluated for two groove angles under evaporative (E) and non-evaporative (NE) conditions

Dryer condition decreases flux, hence, increases dripping period.

Narrow grooves sustain higher fluxes, hence, lower period

Limiting minimum potential and maximum period exist for evaporative conditions.

Divergence between E and NE in narrow potential range implies sensitivity of dripping period to ventilation.

Limiting minimum potential and maximum period exist for evaporative conditions.

Divergence between E and NE in narrow potential range implies sensitivity of dripping period to ventilation.

Theoretical Results: Drop VolumeTheoretical Results: Drop VolumeTheoretical Results: Drop VolumeTheoretical Results: Drop Volume

Non-Evaporative: the drop volume is practically constant for all potentials, and groove geometry.

Evaporative: drop volume increases rapidly when the dripping period begins to diverge from non-evaporative (previous slide).

Non-Evaporative: the drop volume is practically constant for all potentials, and groove geometry.

Evaporative: drop volume increases rapidly when the dripping period begins to diverge from non-evaporative (previous slide).

Why are drops larger under high evaporation?Why are drops larger under high evaporation?Why are drops larger under high evaporation?Why are drops larger under high evaporation?

2

P

td

Ad3AS

Under dry conditions, the competition between high evaporation and low total flux results in a very low net flux feeding the drop.

Under dry conditions, the competition between high evaporation and low total flux results in a very low net flux feeding the drop.

Consequently, slower viscous extension rate dissipates less energy freeing extra force to support larger drop weight.

Consequently, slower viscous extension rate dissipates less energy freeing extra force to support larger drop weight.

Theoretical Results:Theoretical Results: Solute Solute ConcentrationConcentrationTheoretical Results:Theoretical Results: Solute Solute ConcentrationConcentration Evaporation from drops, leaves dissolved solutes behind.

Consequently, drops have higher salt concentration than the bulk liquid.

The total evaporated volume increases with increase in dripping period leading to higher solute concentration.

Evaporation from drops, leaves dissolved solutes behind. Consequently, drops have higher salt concentration than the bulk liquid.

The total evaporated volume increases with increase in dripping period leading to higher solute concentration.

VolumeDropInfluxTotal

C

C

bulk

drop

Relative solute concentration:

Relative solute concentration:

The effect of change in solute concentration on surface tension is not considered in this study.

The effect of change in solute concentration on surface tension is not considered in this study.

Decoupling Flux and EvaporationDecoupling Flux and EvaporationDecoupling Flux and EvaporationDecoupling Flux and Evaporation In ventilated tunnels or controlled laboratory experiments

evaporation and flux (film flow) can be decoupled processes.

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

Model Testing: Lab Model Testing: Lab ExperimentsExperimentsModel Testing: Lab Model Testing: Lab ExperimentsExperiments Laboratory experiments were conducted using natural rock

surface and grooved aluminum (and quartz) surfaces.

Known fixed flux was applied at high rate (no-evaporation), dripping period and drop volume were recorded

Laboratory experiments were conducted using natural rock surface and grooved aluminum (and quartz) surfaces.

Known fixed flux was applied at high rate (no-evaporation), dripping period and drop volume were recorded

ObservationsObservationsObservationsObservations Groove angle 45o; depth=5 mm; flux 1 ml/min.

Note:

Constant drop anchoring area (A0).

Nearly constant liquid-vapor interfaces (above plane).

Long formation period vs. rapid detachment.

Drop recoil volume.

Groove angle 45o; depth=5 mm; flux 1 ml/min.

Note:

Constant drop anchoring area (A0).

Nearly constant liquid-vapor interfaces (above plane).

Long formation period vs. rapid detachment.

Drop recoil volume.

0 2 4 6 8 10 12

FormationFormation DetachmentDetachment

Time (sec)Time (sec)

Natural Rock - Natural Rock - Drop Anchoring AreaDrop Anchoring AreaNatural Rock - Natural Rock - Drop Anchoring AreaDrop Anchoring Area

18.01

18.01

15.14

15.14

18.23

18.23

18.21

18.21

18.2418.24

Model Testing: Lab ResultsModel Testing: Lab ResultsModel Testing: Lab ResultsModel Testing: Lab Results The model was evaluated with a

constant drop anchoring area

aluminum: d=10mm, rock: d=12mm

Predicted dripping period agrees very well with measurement.

Drop volume is primarily determined by drop anchoring area (a function of solid-liquid properties & groove geometry).

Rough rock surfaces induced larger variability in drop volume than obtained from smooth aluminum slab with a fixed groove angle.

The model was evaluated with a constant drop anchoring area

aluminum: d=10mm, rock: d=12mm

Predicted dripping period agrees very well with measurement.

Drop volume is primarily determined by drop anchoring area (a function of solid-liquid properties & groove geometry).

Rough rock surfaces induced larger variability in drop volume than obtained from smooth aluminum slab with a fixed groove angle.

Model Testing: Model Testing: Père Noël cave - Père Noël cave - BelgiumBelgiumModel Testing: Model Testing: Père Noël cave - Père Noël cave - BelgiumBelgium

(Genty and Deflandre (1998), J. Hydrology 211:208-232)(Genty and Deflandre (1998), J. Hydrology 211:208-232)

High flowsmall drops

Dripping from stalactites was monitored for five hydrologic cycles (1991-1996).

Dripping from stalactites was monitored for five hydrologic cycles (1991-1996).

Model Testing: Model Testing: Père Noël cave - Père Noël cave - BelgiumBelgiumModel Testing: Model Testing: Père Noël cave - Père Noël cave - BelgiumBelgium

(Genty and Deflandre (1998), J. Hydrology 211:208-232)(Genty and Deflandre (1998), J. Hydrology 211:208-232)

The data reported includes: dripping rate (number of drop per 10 minutes) and corresponding average drop volume.

The data reported includes: dripping rate (number of drop per 10 minutes) and corresponding average drop volume.

Model Testing: Model Testing: Père Noël cave - Père Noël cave - BelgiumBelgiumModel Testing: Model Testing: Père Noël cave - Père Noël cave - BelgiumBelgium Model prediction are in good agreement with measurements, except at

high fluxes (onset of instability and jetting).

These results provide experimental evidence for the increase in drop volume with decreasing net flux.

Model prediction are in good agreement with measurements, except at high fluxes (onset of instability and jetting).

These results provide experimental evidence for the increase in drop volume with decreasing net flux.

(Genty and Deflandre (1998), J. Hydrology 211:208-232)(Genty and Deflandre (1998), J. Hydrology 211:208-232)

Fast drippingRe>1 - instability

SummarSummaryySummarSummaryy

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 CO2, 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), 367-379, 2000.

Liquid Profile Near the Dripping Liquid Profile Near the Dripping PlanePlaneLiquid Profile Near the Dripping Liquid Profile Near the Dripping PlanePlane

zga

zP

2

)(

Capillary ZoneCapillary Zone

Transition ZoneTransition Zone

Drop ZoneDrop Zone

z Capillary ZoneCapillary ZoneCapillary ZoneCapillary Zone

Transition ZoneTransition ZoneTransition ZoneTransition Zone

In equilibrium with ambient vapor In equilibrium with ambient vapor pressurepressureIn equilibrium with ambient vapor In equilibrium with ambient vapor pressurepressure

Drop ZoneDrop ZoneDrop ZoneDrop Zone

Idealized Drop Shape

Vertical force balance:Vertical force balance:Vertical force balance:Vertical force balance:

gVAoPtop

Horizontal force balanceHorizontal force balanceHorizontal force balanceHorizontal force balance

solidgroove FF