Liquid Bridges and Intermittent Flow Regimes in Unsaturated Fractured Porous

MediaDani Or and Teamrat GhezzeheiDani Or and Teamrat Ghezzehei Dept. of Plants, Soils, and Biometeorology · Utah State University · Logan – Dept. of Plants, Soils, and Biometeorology · Utah State University · Logan –

Utah [for the CNWRA,Geohydrology and Geochemistry Group · SWRI · San Utah [for the CNWRA,Geohydrology and Geochemistry Group · SWRI · San Antonio – Texas]Antonio – Texas]

Outline – Section 5 Unstable and intermittent Outline – Section 5 Unstable and intermittent flow flow

• Primary flow regimes in unsaturated vertical fractures.

• Onset of instability – fingering and intermittent flow.

• Stability analysis for isolated liquid bridges.

• Elongation and detachment of suspended liquid bridges.

• Experimental results – effect of path priming.

• Intermittency and a “chaotic-like” flux at a fracture

base.

• Inference of internal fracture geometry? (no avalanches)

• Fingering, rivulet, and intermittent flow regimes in soils.

• Governing forces for onset of flow instability - criteria.

• Dimensional analysis – limits on applicability of Richards’

Eq.

IntroductionIntroduction Flow processes in unsaturated fractured porous media

(FPM) are considerably different than flow in rock matrix due to enhanced gravitational forces in relatively large pore spaces.

Recent studies revealed coexistence of several flow regimes ranging from film flow on fracture surfaces (Tokunaga and Wan , 1997); Liquid bridges and liquid threads (Su et al., 1999); to intermittent rivulets (Kneafsey and Pruess, 1998).

The onset of any flow regime and transitions between various flow regimes are not fully understood.

The observed intermittent flow induced by formation and motion of liquid clusters is not amenable to representation by standard continuum theories.

Flow Processes in UnsaturatedFlow Processes in UnsaturatedNon-Horizontal FracturesNon-Horizontal Fractures

Flow Processes in UnsaturatedFlow Processes in UnsaturatedNon-Horizontal FracturesNon-Horizontal Fractures

Film Flow onRough Surfaces

Liquid Bridges and Fingers

Nicholl et al, 1994Tokunaga and Wan, 1997Or and Tuller, 1999

Moving Bridges and Liquid Threads

Su et al., 1999

Rivulets

cos

singbBo

2

cos

vCa

Bond Number - gravitational relative to capillary forces

Capillary Number – viscous relative to capillary forces

Bo~0.05; 1000 larger than in soils (Su et al., 1999)

Jeffery Number - gravitational relative to

viscous forces

Je=Bo/CaJe~1000 with typical soils Je~1 (Su et al., 1999)

Forces on Liquid in Unsaturated Fractures

Force Balance for a Static Liquid Bridge Suspended in Non-Horizontal Fracture

Starting from a circular “seed bridge” fed by a constant flux, we seek to define: Maximum bridge size. Optimal configuration.

Capillary forces=Liquid weightFtop+Fbot+Fside=Weight

Bridge shape evolves via changes in-plane and out-of-plane liquid-vapor interfacial curvatures to match force exerted by liquid weight while minimizing overall energy.

Force Balance for a Static Liquid Bridge Suspended in Non-Horizontal Fracture

Given: b - aperture, V - volume, Ct, and - spanning angle, all other quantities (D, Cb, etc.) can be geometrically defined.

WFFF bottomsidetop

DCCgr

1

C

1

b

1

C

1bt

bbt

br

bcosa

cr

bcosa

With:

12cosC2F ttop

sinsincosD4Fside

cos12cosC2F tbottom

gVW

Optimal configurations of

different liquid volumes

held in a 0.8 mm fracture

aperture with their solid-

liquid perimeter length

(representing interfacial

energy per unit volume)

as a function of bridge

spanning angle ().

Optimal Configurations of Stationary Liquid Bridges

Optimal Configurations of Stationary Liquid Bridges in Non-Horizontal Fractures Optimal Configurations of Stationary Liquid Bridges in Non-Horizontal Fractures

Observed liquid bridges in an artificial fracture made of rough glass surfaces with aperture size of 0.66 mm [Su et al., 1999].

Optimal configuration are calculated for estimated 200 mm3

liquid volume.

Width (mm)

- 10 0 10

Length

(m

m)

- 30

-20

-10

0

10

20

30

5 deg10 deg15 deg20 deg

Observed bridges are in motion at a rate of 0.5 cm/s. The calculated shape of stationary bridges will likely be less elongated under the influence of a drag force.

Su et al., 1999

What happens when interfacial forces can no longer support liquid bridge weight?

Finger Flow and Bridge Detachment in Non-Horizontal Fractures

Dripping

Liquid BridgeLiquid Bridge

Liquid Bridges and Intermittent Flow Liquid Bridges and Intermittent Flow Regimes in Unsaturated Fractures MediaRegimes in Unsaturated Fractures Media

A suspended bridge in a narrow asperity or a fault.

Geometry and curvature components of an elongating suspended liquid bridge.  

A Liquid bridge suspended from fracture A Liquid bridge suspended from fracture discontinuitydiscontinuityA Liquid bridge suspended from fracture A Liquid bridge suspended from fracture discontinuitydiscontinuity

Force components, their origin and direction in an elongating suspended liquid bridge (stresses due to viscous extension rate are not marked).

Liquid elements are labeled by a one-dimensional time-like element tracking Lagrangian coordinate (=0 the “oldest” water). 

Elongation and breakup of a suspended Elongation and breakup of a suspended liquid bridgeliquid bridgeElongation and breakup of a suspended Elongation and breakup of a suspended liquid bridgeliquid bridge

We seek to determine the largest bridge size that remains suspended, and subsequent dynamics of bridge elongation and eventual detachment (breakup).

Force components at a plane labeled in an elongating liquid bridge.

Elongation stress is the so-called Trouton result for axial extension of a Newtonian fluid thread.

Assuming a wet or “primed” surface, we neglect solid-liquid interactions (i.e., viscous drag force).

Elongation of a suspended liquid Elongation of a suspended liquid bridgebridgeElongation of a suspended liquid Elongation of a suspended liquid bridgebridge

Longitudinalstress

Elongation stress

A = 4yb is the bridge cross-sectional area3 the Trouton (compression) viscosityR>1 surface roughness indexv longitudinal extension velocity

•Suspended bridge volumes in vertical fractures as a function of aspect ratio () and three aperture sizes. (Symbols signify values of maximum bridge volume).

•The results are used as boundary conditions for the dynamic elongation and detachment phases.

Optimal Configurations and Optimal Configurations and Sizes of Elongating Liquid Sizes of Elongating Liquid BridgesBridges

Optimal Configurations and Optimal Configurations and Sizes of Elongating Liquid Sizes of Elongating Liquid BridgesBridges

• The width of the largest bridge volume and the associated liquid bridge anchoring area as a function of fracture aperture size.

•These results are used as boundary conditions for the dynamic elongation and detachment phases.

Liquid Bridge Width and Liquid Bridge Width and Anchoring Area for Maximum Anchoring Area for Maximum VolumeVolume

Liquid Bridge Width and Liquid Bridge Width and Anchoring Area for Maximum Anchoring Area for Maximum VolumeVolume

• detachment interval (sec).• detachment interval (sec).

Liquid bridge detachmentLiquid bridge detachmentinterval and detached interval and detached volumevolume

Liquid bridge detachmentLiquid bridge detachmentinterval and detached interval and detached volumevolume

Qg

rgbQgrb6 2

c

• detached volume (mm3)• detached volume (mm3)

QV cd

as functions of volumetric flux (Q) in three different fracture apertures.

as functions of volumetric flux (Q) in three different fracture apertures.

A sequence of water bridge formation, elongation, and detachment in a 0.6 mm fracture model. Note the formation of a liquid thread feeding the “detaching” bridge volume similar to observations by Su et al. [1999].

Experiments in an artificial Experiments in an artificial fracture:fracture:Liquid bridge Formation and Liquid bridge Formation and detachmentdetachment

Experiments in an artificial Experiments in an artificial fracture:fracture:Liquid bridge Formation and Liquid bridge Formation and detachmentdetachment

Measurements (symbols) and model predictions (lines) of bridge detachment intervals as a function of input flux within two aperture sizes

Measurements (symbols) and model predictions (lines) of bridge detachment intervals as a function of input flux within two aperture sizes

Experimental Experimental Results Results Experimental Experimental Results Results

Variations in Fracture Aperture & Output Variations in Fracture Aperture & Output FluxFlux Variations in Fracture Aperture & Output Variations in Fracture Aperture & Output FluxFlux

Time (sec)0 50 100 150 200 250

Flux (

mm

3/s

ec)

0

50

100

150

200

250

300

Flux (

mm

3/s

ec)

0

50

100

150

200

250

30050 mm3/ sec

10 mm3/ sec

• Local variations in fracture aperture (asperities) induce different detachment intervals and volumes.

• The resultant output flux often appears noisy and chaotic.

• Interactions between input flux and fracture internal geometry affect output flux structure.

Top viewTop view

Can the structure of the flux be used to Can the structure of the flux be used to extract information on fracture internal extract information on fracture internal geometry? geometry?

Can the structure of the flux be used to Can the structure of the flux be used to extract information on fracture internal extract information on fracture internal geometry? geometry?

Complications due to avalanches in 2-D Complications due to avalanches in 2-D spacesspacesComplications due to avalanches in 2-D Complications due to avalanches in 2-D spacesspaces

Cheng et al. 1989 (Phys. Rev. A 40:5922-5934)

The mass within the region swept out by the sliding bridge wdy contributes to the increase in bridge mass ( is the liquid mass density in a fracture or windshield).

For a bridge with mass m and radius rm1/2 the mass increases as (y)2 and the width w(y) growth as y.

The problem here is that a small discharge event can trigger a disproportionally-large avalanche that could distort inferences on fracture internal structure.

SummarySummarySummarySummary

Flow regimes in unsaturated fractures are a combination of film, finger and rivulet flows.

The gradual growth and subsequent breakup of a liquid bridge fed by film flow was modeled as a function of flux and aperture size.

Results show that flow from unsaturated fractures with complex internal geometry is intermittent with erratic and chaotic-like flux.

We explored the potential of using FFT analysis of flux at a control plane to identify features related to fracture internal structure.

The study highlights limitations of continuum approaches and suggest an alternative modeling approach based on discrete liquid elements.