Fingering, Unstable and Intermittent Fingering, Unstable and Intermittent Flow Regimes in Unsaturated SoilsFlow Regimes in Unsaturated Soils

Unstable and intermittent flows in porous media

Large body of evidence for fingering and flow instability in soils and similar porous media.

Intermittency (irregular temporal flow pattern) associated with flow instability is less studied.

Unstable and intermittent flows are not amenable for continuum representation by Richards’ equation and alike.

Although causes of such irregular flow regimes are attributed to various factors such as: heterogeneity, water repellency, layering, pressure build-up, etc., it is possible to unify these factors by diagnostic dimensionless groups leading to simple criteria.

Flow instability and fingering in porous media

Flow fingering and instability in soils and similar media has been observed for layered media (fine over coarse texture), water repellent layers, DNAPLs displacement, air pressure build-up in confined domains, etc.

Experiments by Glass et al. [1989] show remarkable persistence of fingers across several infiltration events (as long as water content remains non-uniform).

• Raats [1973] criterion for flow instability:

Criteria for flow instability in porous media

which is reduced to (h0-hL)<0

(Ks/L2(s-i) is always >0)

0)hh()(L

K

dL

duL0

is2

s

i S00

z

L

0dL

du

• Flow instability sets in whenever the pressure head at the soil surface is smaller than at the wetting front.

• Examples:• Air compression ahead of the wetting front.• Rate of rainfall smaller than Ks.• Layered soil (fine over coarse).• Certain redistribution conditions.

• Based on linear hydrodynamic stability analysis, Philip [1975] generalized the criteria for flow instability :

• Flow instability sets in whenever the pressure gradient G at the wetting front opposes the flow.

Criteria for flow instability in porous media

i S00

z

L0

L

hhG L0

• An equivalent criterion was identified by Hillel and Baker [1988] for a layered soil:

• When the hydraulic conductivity of the underlying coarse layer is larger than the average flux in the fine layer above.• Nicholl et al. [1994] provided a similar criterion for fingering in

fractured rock (G was later expanded to included capillarity):

inins qcosggk

qGK

Flow instability and fingering in a dry fracture

• An initially stable flow becomes unstable as soon as infiltration ends and redistribution with a certain air-entry value begins (see “drainage” along upper boundary).

• Sequence leading to development of flow-capturing fingers and the appearance of isolated liquid bridges.

Nicholl et al., 1994, Water Resour. Res. 30:2533-2546

• Philip’s [1975] linear hydrodynamic stability analysis that predicts also the wavelength of most unstable disturbance was tested by White et al., [1976] in a Hele-Shaw cell with reasonable success.

Tests of criteria for flow instability

• Best results were obtained for immiscible fluids in a system with sharp boundaries (–type). Experiments in porous media resulted in only qualitative agreement.

Flow intermittency in porous media (sand)

• Prazak et al. studied intermittency (temporally irregular flow pattern) in sand columns subject to various initial conditions (wet & dry medium).

• The resulting intermittent flux was associated with relatively large influence of gravitational forces with Bo in the range of 0.03 to 0.01.

Prazak et al. 1992 (Water Resour. Res. 28:1849-1855)

Rivulet flow in structured porous media (aggregates)

• Dexter 1993 studied flow in a non-absorbing aggregate bed (made of stones of different sizes).

• Uniform flow at the surface using a spraying nozzle.

• Outflow was collected from 74 locations.

Dexter 1993 (Water Resour. Res. 29:1859-1862)

• The number of rivulets per cross-sectional area decreased with increasing depth suggesting “random walk” with rivulet “capturing”.

• Similar to droplet avalanche, the concentrated mass flow (main rivulet) leaves behind a “cleared” path.

• Flow stability in porous media is a result of interplay between gravitational, capillary, and viscous forces.

Flow instability in porous media

cos

singrBo

2

Capillary Viscous

Gra

vity

cos

vCa

Je=Bo/Ca

• Most flow instability phenomena in soils and similar porous media are induced by gravitational forces overcoming restraining capillary forces (no instability observed for upward or horizontal flow).

• Held and Illangasekare (1995) and later Berkowitz and Ewing (1998) proposed extension of Lenormand et al. (1988) stability diagram to include gravitational forces (Bo).

• The interplay between the various forces is illustrated in this empirical phase diagram.

• Higher Bond numbers act to stabilize flow regimes.

Unstable flow regimes in porous media

cos

singrBo

2

cos

vCa

M

Typical values of dimensionless groups for flow regimes in soils (Friedman, 2000)

• Prazak et al. [1992], Su et al. [1999] and other experimental evidence clearly show that instability in soils occurs around Bo>0.05 when gravity becomes 5% (or less!) of capillarity.

• This could provide a potentially useful criterion for limits of applicability of the Richards equation based on liquid and media properties (i.e.,for pores with radius larger than 0.6 mm).

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 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 in Unsaturated Fractures

Pore scale physical considerations

r2

r1• The conditions for liquid invasion from r1 to r2 with r2> r1 require the capillary head to be h1+h given by the difference between equilibrium capillary heads.

• An increase (less negative) in capillary head at the front!

21 r

1

r

1

g

2h

• At an interface between fine and coarse layers we can postulate the following local scenario for the onset of a perturbation in the wetting front:

2 hsmall

hsmall

hlarge

h2 h3h1

wetting front

3

h1>h2<h3

1

Force balance for a hemispherical perturbation

weight

h’

Capillaryforces

• Consider a local force balance for a suspended hemispherical perturbation in the wetting front (i.e., what keeps the liquid element from “falling” through the porous matrix?)

• The weight is Fg=g(2/3h3)

SummarySummarySummarySummary

Unstable flow regimes in soils are manifested in various forms from fingering, to rivulet and intermittent flow.

Onset of instability is often associated with reduction in restraining capillary forces relative to gravitational ones either due to changes in pore size, wettability, or diminishing gradients.

Numerous evidence shows that for Bo>0.05 flows become unstable, and consequently the Richards equation may no longer be applicable.