VARIABLE DENSITY GROUNDWATER FLOW:

From current challenges to future possibilities

Professor Craig T. Simmons

Hydrogeology Journal (2005) 13: 116–119

OUTLINE

• Variable density flow: Why, what, how and when?

• Groundwater applications

• Current challenges

• Illustrative examples of research

• Future possibilities

Variable density flow:Why, what, how and when?

When is variable density flow important in groundwater flow processes?

Density variation: changing concentration, temperature or pressure of the fluid

Adams and Bachu [2002]

Variable density groundwater system Relevant papers Sea water intrusion, fresh-saline water interfaces in coastal aquifers

Yechieli et al., (2001) Kooi et al., (2000) Post and Kooi (2003) Underwood et al., (1992) Voss and Souza (1987) Huyakorn et al., (1987) Pinder and Cooper (1970) Werner and Gallagher (2006)

Subterrenean groundwater discharge Langevin (2003) Kaleris et al., (2002)

Infiltration of leachates from waste disposal sites, dense contaminant plumes

Liu and Dane (1996) Zhang and Schwartz (1995) Oostrom et al., (1992a,b) Koch and Zhang (1992) Schincariol and Schwartz (1990) Pashcke and Hoopes (1984) Le Blanc (1984) Frind (1982)

DNAPL flow and transport Li and Schwartz (2004) Lemke et al., (2004) Oostrom et al., (2003)

Density driven transport in the vadose zone Ying and Zheng (1999) Ouyang and Zheng (1999)

Flow through salt formations in high level disposal sites, heat and solute movement near salt domes

Jackson and Watson (2001) Williams and Ranganathan (1994) Hassanizadeh and Leijnse (1988)

Heat and fluid flow in geothermal systems Oldenburg and Pruess (1999) Gvirtzman et al., (1997)

Sedimentary basin mass and heat transport processes, diagenesis processes

Garven et al., (2003) Sharp et al., (2001) Raffensperger and Vlassopoulous (1999) Wood and Hewett (1984)

Palaeohydrogeology of sedimentary basins Senger (1993) Gupta and Bair (1997)

Processes beneath playas, sabkhas and playa lakes Yechieli and Wood (2002) Sanford and Wood (2001) Simmons et al., (1999) Wooding et al., (1997a,b) Duffy and Al-Hassan (1988)

Operation of saline (and irrigation) water disposal basins Simmons et al., (2002) Density affects in applied tracer tests Barth et al., (2001)

Zhang et al., (1998) Istok and Humphrey (1995) Le Blanc et al., (1991)

Typical groundwater hydraulic gradient ≈ 1m in a 1000m.

Equivalent density “driving force” is a density difference of 1 kg/m3

relative to a reference density of freshwater 1000 kg/m3.

Solution concentration of 2g/L (about 5% of seawater!). Quite dilute in comparison to many plumes one may encounter in groundwater systems.

Freeze and Cherry (1979, p. 435) note that the total dissolved solid concentrations for leachate from sanitary landfills typically range from 5,000 to 40,000 mgL-1 (i.e., approximately 0.35 - 2.8% density difference between the leachate and the groundwater).

DENSITY IS IMPORTANT

light fluid

light fluiddense fluid

dense fluid

STABLE UNSTABLE

DENSITYCONFIGURATIONS

Increasing complexity

STABLE DENSITY CONTRASTS

‘‘Henry circulation’’ [Henry, 1964]

UNSTABLE DENSITY CONTRASTS

[Freeze and Cherry, 1979]

What is free convection?

bunsen burner

beaker ofwater

freeconvectioncells

STABLE: 3000 mg/L CaCl2 @ 150 mins

UNSTABLE: 300,000 mg/L CaCl2 @ 50 mins

Why is unstable flow important?

• Total quantity of solute involved in transport process is far greater than that of diffusion

• Time scales for mixing are significantly reduced• Spatial scales for mixing are typically larger, enabling

solutes to spread over greater distances

“Papers on convection in porous mediacontinue to be published at a rate of over 100 per year….”

Nield and Bejan [2006]

1842 - 1919Lord Rayleigh (John W. Strutt)

Rayleigh, Lord (J. W. Strutt), 1916. Onconvection currents in a horizontal layer of fluidwhen the higher temperature is on the underside, Philos. Mag., Ser. 6, 32, 529-546.

24πμ

ρα>

Δ=

DCHgkRa oC

Rayleigh Number

Combarnous and Bories (1974)

Rac2 ≈ 240 – 300

Rac1 = 4π2

Groundwater Applications

Current Challenges

SOME CHALLENGESTRADITIONAL FLUID MECHANICS

Steady-state assumptions

Homogeneous layers

Length scale = layer thickness

Simple chemistry / fluids

Molecular diffusion

Rayleigh number - predictsa priori, known Rac

Laboratory scales, simple BC’s

Limited numerical simulation

SOME CHALLENGESTRADITIONAL FLUID MECHANICS

Steady-state assumptions

Homogeneous layers

Length scale = layer thickness

Simple chemistry / fluids

Molecular diffusion

Rayleigh number - predictsa priori, known Rac

Laboratory scales, simple BC’s

Limited numerical simulation

GROUNDWATER HYDROLOGY

Transient - effect of storage, growth, decay

Heterogeneity and its impact

Length scales ambiguous

Complex geochemistry / fluid-matrix

Dispersion greater than diffusion

Cannot determine some parametersfor Ra a priori, unknown Rac

Field scales - direct measurement?

Difficulties in simulation - high Ra

FIELD EVIDENCE?“Numerical experiments demonstrate the existence of a convection cell….” Duffy and Al-Hassan (1988)

“The observation that tritium exists throughout the profile isconsistent with vertical circulation resulting from the density instability” (Wood et al., 2002)

“The salt deficit may be accounted for by the slow downward convection of dense saline water beneath salt lake beds……” (Teller et al., 1982)

“Abundant data indicate high fluid and solute fluxes in shaly sediments and account for the observed level of sediment diagenesis”(Sharp et al., 1988)

“Assuming a critical Rayleigh number of 4π , the region is predicted a-priori to be unstable”(Simmons et al., 2002)

“Contamination from a waste dump at Noordwijk, Netherlands, resulted in a plume with downward velocity 45 times higher than the vertical velocity due to natural recharge” (Kooper, 1983)

2

MODELLING

Analytical solutions are a challenge, heavy dependence on numerical models.

Modelling of free convection, particularly at higher Rayleigh numbers, is problematic and challenging (e.g., Elder Problem Ra=400, Salt Lake Problem Ra=4870 - oscillatory regime).

Problems:

(i) Significant variability in results of different numerical codes(ii) Sensitivity to numerical schemes(iii) Numerical perturbations / dispersion control number, extent and behaviour of

fingers; (iv) Grid dependent results; Grid convergence cannot be achieved(v) Bifurcations (multiple solution states) and oscillatory (no steady state) solutions

MODELLING

Only modest hydraulic conductivity and salinity difference required to achieve Ra in excess required for oscillatory behaviour (Rac2 ≈ 240 – 300)! Bifurcations can occur at even lower Ra!

Rayleigh numbers in many practical problems of interest are much higher (e.g., sabkhat in Abu Dhabi Ra > 104).

We need an urgent reality check: (i) What can we reasonably expect from our models? (ii) Microscopic indicators (e.g., precise finger patterns; single solution) vs

macroscopic indicators (e.g., COM)? (iii) We must untangle what is physically real (bifurcations, oscillatory behaviour) and

what are numerical artifacts(iv) THE NATURE OF THE BEAST: MULTIPLE SOLUTIONS ARE A PART OF LIFE!

ResearchProjects

1. Numerical modelling2. Heterogeneity3. Field scale detection

Numerical Modelsand

Benchmarking

ELDER (1967) “SHORT-HEATER” PROBLEM

[Voss and Souza, 1987]

(Ra = 400)

(a)

(b)

Elder’s Experimental Results [1967](heating from below)

T=0.05 (10 yrs)

T=0.025 (5 yrs)

Diersch & Kolditz [2002]

Ra=400

L=9:525,825 nodes for halfdomain solved

Diersch & Kolditz [2002]

Number of quadrilateral elements in half domain = 2 x 4l

Depending on the numerical model and grid-resolution, the stable steady states S1, S2, and S3 are observed.

Are these different solutions real or numerical artefacts?

Eliminate all sources of numerical discrepancy, in order to get uncontaminated insights.

The multiple steady states

Governing equations:

where ψ is the streamfunction and c is the concentration.

These equations are solved with a pseudospectral method which employs sine-and cosine-series in the horizontal (x) direction and Chebyshev polynomials in the vertical (z) direction. The use of a pseudospectral method avoids all truncation error associated with differentiation.

A pseudospectral code for simulating buoyancy driven flow

Bifurcation diagram

Bifurcation diagram - observations

• At Ra = 400 there are three stable steady state solutions;• The higher states S2 (Ra=76) and S3 (Ra=172) come into

existence via a fold-bifurcation (Johannsen, 2003);• Below Ra = 76, there is only one stable steady state.

Hence, the ambiguities are physical rather than numerical; at Ra = 400, three stable steady states co-exist.

If Ra is lowered to 60, there will be only one stable steady state.We call this the Low Rayleigh Number Elder Problem.

Low Rayleigh Number Elder Problem (Ra = 60)

Heterogeneity

Traditional

Permeability field investigation

Schincariol et al., (1997)Simmons et al., (2001)Prasad and Simmons (2003)

Simmons et al., (2001)

Simmons et al., (1999, 2001, 2008)Sharp et al., (2000)

Shikaze et al., (2000) Graf and Therrien (2007)

HydraulicConductivity(Kav = 0.1 m/day)

ConcentrationFields

(ΔC = 50,000 mg/L)

All these systems have the same Rayleigh number!

Effect of fractures near sourceGraf and Therrien [2007]

Same mean and Ra

BUT

Increasingstandard deviationof permeability field

kills fingers!

[Prasad and Simmons, 2003]

low permeability zone

high permeability zone

laterally extensive low permeabilityzones provide barriers to vertical flow

finger propagatesin high permeability zone

low permeability zonereduces lateral mixing

Importance of geometrical structure of heterogeneity on instability

Dynamics Summary

Some key points…...

Traditional Ra dimensionless numbers (averages) cannot predictthe onset of instability (or growth and decay ) in highlyheterogeneous porous media

The TWO EDGE sword (and the main results to date!)Heterogeneity is the triggering mechanism (onset) for instability BUTHeterogeneity can either promote finger growth or kill them!

“Magnitude” of heterogeneity is the critical control

Geometrical structure is also important (discrete fractures vs trending etc…)- Laterally extensive low permeability lenses promote decay- Vertically continuous “conduits” enhance growth

Field Scale Detection of

Free Convection

7.5 m

Concentration (mg/L)

0 50,000

[Simmons et al., JCH, 2001]

0.35 m

0 313,000

Concentration (mg/L)

[Simmons et al., TIPM, 2002]

100 m

0.5m

Hundreds of papers on theory, modelling &

laboratory experiments on finger instabilities associated with free

convection ….

BUT

A COMPLETE LACK OF CONCLUSIVE FIELD

BASED EVIDENCE AND DATA !

Document the existence of free convective fingering in natural field settings using hydrogeophysical methods that exploit concentration differences associated with free convection

TDEM (ProTEM47) FDEM (EM31) Multi-electrode ERI

The quest to find fingers(March 3-6, 2008)

Why this site?Extensively characterizedHomogeneousDensity inversions existUnusual tritium distribution

(Wood et al., BGSA., 2002)

54ºE 54.2ºE

24.2ºN

24ºN

50ºE 55ºE

25ºN

20ºN

Field Site – Sabkha Aquifer SW of Abu Dhabi, UAE

Field Site – Sabkha Characteristics

Miocene carbonates

~20 km

~11.5 m

Stratigraphy / sediments:Wedge shaped profileUniform, fine sand (0.16-0.22 mm)Carbonates (60%) and quartz (35%)Porosity 38%Hydraulic conductivity 1.0 ± 0.2 m/d

Chemistry: (Wood et al., Chem.Geol., 2005)

Surface water ~400,000 mg/LSabkha water ~275,000 mg/LMiocene water ~100,000 mg/L(upward flux 4 mm/yr)

4 mm/yr

Mean annualprecipitation

Field Site – Hydrology

Mean annual precipitation ~31 mm/yr

Evaporative flux ~60 mm/yr

Average years: Halite salt crust develops at the surface

Large precipitation events Jan 15/16, 2008: Formation of hypersaline brines(> 60 mm, 7 weeks prior to data collection)

Inversion of 2D Dipole-dipole Data

vadose zone (~0.7m): highest resistivity‘saline fingers’ protrude in higher resistivity backgroundcannot use geologic variability to explain observations -

perfectly homogeneous

CONCEPTUAL UNDERSTANDING AND PREDICTION:

At present processes not easily amenable to prediction - heterogeneity controls onset, growth and/or decay of plumes- problems with application of Rayleigh number- modelling issues (oscillatory / bifurcation solutions, numerical dispersion controls)- urgent reality check on expectations from models- develop simplifying approaches- field scale applications

MEASUREMENT:

“Inference” for the existence of convection is a good starting point but we need to develop field techniques to measure it directlyin field settings and to gather much better data for our predictive tools

CONCLUDING REMARKS(or miles to go before we sleep!)

"Everything should be made

as simple as possible,

but never simpler!"

Albert Einstein

Future Possibilities

(i) dispersion(ii) geological constraints (e.g., sedimentary facies data / structural geology)(iii) improving the resolution of geophysical and remote sensing tools(iv) linking with the fields of tracer and isotope hydrogeology(v) thermohaline (double-diffusive) and multiple species transport problems(vi) complex geochemical reactions and fluid-matrix interactions(vii) multiphase flow in carbon sequestration processes (viii) in-situ desalination(ix) links with climate change (e.g., sealevel rise)(x) ecological links (e.g., phytoconvective groundwater analogs?)(xi) surface water – groundwater interaction, recharge/discharge processes(xii) fully coupled models (sw-vadose-gw) with variable density flow capability

FUTURE POSSIBILITIES?

THERE ARE LIKELY TO BE MANY OTHER PROBLEMS AND APPLICATIONS NOT YET CONCEIVED!

Robin A. Wooding6 March 1926 – 19 November 2007

(Photograph circa 1975)