VARIABLE DENSITY GROUNDWATER FLOW MODELLING:

Approaches, Resolutions and Future Challenges

Craig T. Simmons, Peter Bauer-Gottwein, Thomas Graf, Wolfgang Kinzelbach, Henk Kooi, Ling Li, Vincent Post, Henning Prommer,

Rene Therrien, Clifford Voss, James Ward & Adrian Werner

T he si tuation of Mount L ofty was found

from hence and from some other c ross

bearings, to be 34¡ 59' south and 138¡ 42'

east . No land was v isible so far to the

north as where the trees appeared above

the h or izon, which showed the coast to

be very low, and our s oundings wer e

fas t decr easing.

From noon to six o' clock we ran thir ty

miles to the northwar d, sk irt ing a sandy

shore at the distance of f ive , and thence

to e ight miles; the depth was then 5

fathom s, and we dr opped the anchor upon

a bot tom of sand, m ixed with pieces of

dead coral.

Centre forGroundwater S tudies

OUTLINE

• Introduction

• Importance of variable density flow

• Brief historical perspective

• Variable density flow physics

• Modelling variable density flows & Future Challenges

• Applications of variable density models (Part 2)

Introduction

Arid zones and variable density

• Arid and semi-arid climates are characterised as those areas where precipitation is less than potential evapotranspiration.

• Evaporation and transpiration remove freshwater, leaving residual salts behind.

• Characteristically low precipitation rates reduce the potential for salt to be diluted by rainfall.

• Arid and semi-arid regions make ideal “salt concentrator”hydrologic environments variable density flows are important.

• Indeed, salt flats, playas, sabkhas and saline lakes, for example, are therefore ubiquitous features of arid and semi-arid regions throughout the world [Yechieli and Wood, 2002].

http://www.adias-uae.com/sabkhamatti.html

View of sabkha (Photograph by Dr Mark Beech) Location of Sabkha Matti (Source: ADIAS)

Sabkha: An environment common to arid or semiarid environments that is characterised by coastal sedimentation above the level of high tide and by the absence of vegetation.

Evaporites, eolian deposits and tidal-flood deposits are common in sabkhas.

Importance of variable density flow

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

Brine density as a function of NaCl mass fraction, computed for the conditions of atmospheric pressure at temperature of 25oC. Source: Adams and Bachu [2002].

Brine density calculated at conditions characteristic of some typical sedimentary basins. Source: Adams and Bachu [2002].

Problems of interest

• astrophysics• chemical reactor engineering• energy storage and recovery• geophysics• geothermal reservoirs• material science• metallurgy• nuclear waste disposal• oceanography• future of our energy resources• environmental pollution • groundwater hydrology

More recently…..

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)

Brief historicalperspective

Early work on convection problem:

Some important milestones1. Benard (1900, 1901) - experiment on fluid layer heated from below2. Rayleigh (1916) - heat experiment and theoretical development3. Horton and Rogers (1945) & Lapwood (1948) - extension to porous media (heat)4. Elder (1967) - transient thermal convection5. Wooding (1959, 1962, 1963, 1969) - extension to miscible flow and solute transport6. Nield (1968) - thermohaline adaptation of HRL problem7. Combarnous and Bories (1974) - effect of layer inclination

BENARD CONVECTION (1900/1901)

1842 - 1919Lord Rayleigh (John 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.

λ = 2H

H

ρ

ρlow

high

Horton and Rogers (1945) / Lapwood (1948)

Convection in an infinite porous layer

Chigh Tlow

Clow Thigh

24πμ

ραμκ

ρα>

Δ+

Δ=

DCHgkTHgkRa oCoT

(a)

(b)

Elder’s Experiment (1967)(heating from below)

T=0.05

T=0.025

Nield (1968)

Combarnous and Bories (1974)

FLUID & HEAT (1900 - 1940)

FLUID & POROUS MEDIA & HEAT (1940’s - )

FLUID & POROUS MEDIA & SOLUTE (early 1950’s - )

FLUID & POROUS MEDIA & SOLUTE & HEAT (late 1960’s - )

GROUNDWATER APPLICATIONS – ONLY AFTER 1980!

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

Nield and Bejan (1998)

EVOLUTION

Anomaly: Saltwater intrusion!

Badon-Ghyben [1888] and Herzberg [1901] - steady position of saltwater-freshwater interfaces in coastal aquifers.

Analytical solutions for the sharp saltwater-freshwater interface in an infinitely thick confined aquifer emerged in the 1950’s [Glover, 1959; Henry, 1959].

Hele-Shaw cell analogs [Bear, 1972] were used in some of the earliest exploratory studies of seawater intrusion and clearly predated numerical solutions to these problems.

Some of the earliest numerical analyses emerged in studies by Pinderand Cooper [1970], Segol et al., [1976], Huyakorn and Taylor [1976] and Frind, [1982a,b].

SOME CHALLENGESTRADITIONAL FLUID MECHANICS

Steady-state assumptions

Homogeneous layers

Length scale = layer thickness

Fully saturated systems

Molecular diffusion

Rayleigh number - predictsa priori, known Rac

Laboratory scales, simple BC’s

Limited numerical simulation

SOME CHALLENGESGROUNDWATER HYDROLOGY

Transient - effect of storage, growth, decay

Heterogeneity and its impact

Length scales ambiguous

Variably-saturated systems

Dispersion greater than diffusion

Cannot determine some parametersfor Ra a priori, unknown Rac

Field scales - direct measurement?

Difficulties in simulation - high Ra

TRADITIONAL FLUID MECHANICS

Steady-state assumptions

Homogeneous layers

Length scale = layer thickness

Fully saturated systems

Molecular diffusion

Rayleigh number - predictsa priori, known Rac

Laboratory scales, simple BC’s

Limited numerical simulation

Models for 21st centuryEarliest work – laboratory experiments, and some simple analytical/numerical methods. Variable density flows not easily amenable to analytical solutions heavy dependence on numerical solutions!

Groundwater modelling codes employ improved numerical techniques and this coupled to faster computers with larger memories has allowed for the simulation and solution of more complex problems, with fully-coupled flow and solute transport in even 3D cases!

Simultaneous heat and transport in thermohaline convection problems [Diersch and Kolditz, 2002; Graf and Therrien, 2007b],

Heterogeneous systems including fractured rock hydrology [Graf and Therrien, 2005; Graf, 2005]

Chemically reactive transport [Freedman and Ibaraki 2002; Post and Prommer, 2007].

DIRECT 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 without data?

Indeed, it may be argued that one major limitation of the current field of variable density flow is not necessarily the availability of numerical models with the inherent capacity to solve these sorts of very complex problems, but is perhaps the availability of real data to occupy, test and verify the emerging new generation of computer models. Semi-quantitative analyses at best?

What is model purpose? Process understanding vs absolute prediction?

Field scale applications - often difficult to test models and hypotheses robustly. Useful for firming up conceptual models and/or eliminating others……..

Possibly better data for some problems (e.g. seawater intrusion)perhaps, but detailed observations of fingering and field convection are limited.

Variable density flow physics

Simmons [2005] considered a typical groundwater hydraulic gradient of say a 1m hydraulic head drop over a lateral distance of a kilometre i.e., a gradient of 1 in a 1000.

Showed that the equivalent “driving force” in density terms would be a density difference of 1 kg/m3 relative to a reference density of freshwater whose density is 1000 kg/m3.

This is a solution whose concentration is only about 2g/L (about 5% of seawater!) and this is quite dilute in comparison to many plumes one may encounter in groundwater studies.

DENSITY IS IMPORTANT

light fluid

light fluiddense fluid

dense fluid

STABLE UNSTABLE

DENSITYCONFIGURATIONS

Increasing complexity

STABLE DENSITY CONTRASTS

‘‘Henry circulation’’ [Henry, 1964]

Massmann et al., [2006] in Journal of Hydrology

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

Instabilities in porous media

[Simmons et al., 2002]

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

[Simmons et al., 2001]

Competing processes

Density gradient that drives free convection

Hydraulic gradients that drive forced convection (advection)

Diffusion/Dispersion that dissipate free convection

Heterogeneity that may enhance or kill free convection! (see later)

Capturing all of this physics correctly in a numerical model can be very challenging and often it is difficult to know if the model is accurate and reliable………

Rayleigh Numbers

Dispersion and DiffusionnGravitatio andBuoyancy =

D)HC-C(gk =

DHU =Ra

00

minmax

o

c

νθβ

Uc is the convective velocity H is the thickness of the porous layer D0 is the molecular diffusivity g is the acceleration due to gravity k is the intrinsic permeability β=ρ0

−1 (∂ρ/∂C) linear expansion coefficient of fluid density with changing fluid concentration Cmax and Cmin maximum and minimum values of concentration respectively θ is the aquifer porosity ν0=μ0/ρ0 is the kinematic viscosity of the fluid.

Mixed convection parameter

speedconvectiveForcedspeedconvectiveFree

Lh

M o =⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

ΔΔρρΔ

If M>>1, then free convection is dominant. If M<<1, then forced convection is dominant.

Where M~1 they are of comparable magnitude. Here, ∆ρ is the density difference, ρο is the

lower reference density [both free convection parameters], ∆h is the hydraulic head difference

measured over a length ∆L [both forced convection parameters].

Modelling variable density flowphenomena

Modelling

A detailed treatment on this subject was presented in review articles by Diersch and Kolditz [2002] and by Simmons [2005].

NB: The limited number of analytical solutions for variable density flow problems creates a heavy dependence on numerical simulators.

A large and growing number of numerical simulators now exist for the simulation of variable density flow phenomena.

Several of these codes will be demonstrated later…...

Simulator References for code development and verification

CFEST Gupta et al., [1987]

FAST-C Holzbecher [1998]

FEFLOW Diersch [1988], Diersch [2005]

FEMWATER Lin et al., [1997]

HEATFLOW Frind [1982]

HST3D Kipp [1987]

HYDROGEOSPHERE Therrien et al., [2004]

METROPOL Leijnse and Hassanizadeh [1989]

MITSU3D Ibaraki [1998]

MOCDENSE Sanford and Konikow [1985]

MODHMS HydroGeoLogic Inc., [2003]

NAMMU Herbert et al., [1988]

PHT3D/SEAWAT-2000 Post and Prommer [2007]

ROCKFLOW Krohn [1991], Kolditz et al. [1995], Ratke [1995]

SEAWAT-2000 Langevin and Guo [2006]

SUTRA Voss [1984]

SWIFT Reeves et al., [1986]

TOUGH2 Oldenburg and Pruess [1995]

VapourT Mendoza [1990]

VARDEN Kuiper [1983, 1985], Kontis & Mandle [1988]

Physics

• Flow / Transport– Groundwater flow– Advective/convective transport of dissolved

chemicals (ions, organic compounds, …)– Mixing/Dispersion

Density coupling (C→ ρ → V)!

Here, we use specific case of the widely used SUTRA [Saturated-Unsaturated TRAnsport Model] numerical model [Voss, 1984]

SUTRA is one of the earliest codes that emerged for the simulation of variable density flow phenomena, and is in wide use today.

The SUTRA code is a numerical solver of two general balance equations for variable-density single-phase saturated-unsaturated flow and single-species (solute or energy) transport based on Bear [1979].

MODEL EXAMPLE: SUTRA

Typical SUTRA project

www.usgs.gov

3D-Convection

Simulation using SUTRA is in two or three spatial dimensions.

Ground-water flow is simulated through numerical solution of a fluid mass-balance equation.

The ground-water system may be either saturated, or partly or completely unsaturated.

Fluid density may be constant, or vary as a function of solute concentration or fluid temperature.

SUTRA tracks the transport of either solute mass or energy in flowing ground water through a unified equation, which represents the transport of either solute or energy.

SUTRA PROCESSES

Solute transport is simulated through numerical solution of a solute mass-balance equation where solute concentration may affect fluid density.

The single solute species may be transported conservatively, or it may undergo equilibrium sorption (through linear, Freundlich, or Langmuir isotherms).

In addition, the solute may be produced or decay through first- or zero-order processes.

Energy transport is simulated through numerical solution of an energy-balance equation. The solid grains of the aquifer matrix and fluid are locally assumed to have equal temperature, and fluid density and viscosity may be affected by the temperature. Most aquifer material, flow, and transport parameters may vary in value throughout the simulated region.

Sources and boundary conditions of fluid, solute and energy may be specified to vary with time or may be constant.

SUTRA dispersion processes include diffusion and two types of fluid velocity-dependent dispersion. The standard dispersion model for isotropic media assumes direction-independent values of longitudinal and transverse dispersivity. A flow-direction-dependent dispersion process for anisotropic media is also provided.

GOVERNING EQUATIONSIn the case of variable density saturated flow with non-reactive solute transport of

total dissolved solids or chloride, and with no internal production of solute,

equations (5), (8) and (9), are greatly simplified. The fluid mass balance is:

( ) ( ) pp0 QptU

UtpS =⎥

⎦

⎤⎢⎣

⎡ρ−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛μρ

⋅−∂∂

⎟⎠⎞

⎜⎝⎛

∂ρ∂

ε+∂∂

ρ gk∇∇ (12)

The fluid velocity is given by:

( )gkv ρ−⋅⎟⎟⎠

⎞⎜⎜⎝

⎛εμρ

−= p∇ (13)

and the solute mass balance is:

( ) ( )[ ] ( )CCQUD-CtU *

pm −=⋅+ερ⋅⋅ερ+∂∂

ερ ∇∇∇ DIv (14)

GOVERNING EQUATIONS

Additionally, ρ is the fluid density [kg/m3], expressed as:

( )00 UUU

−∂

ρ∂+ρ=ρ (6)

NUMERICAL METHODSThe numerical technique uses a modified two-dimensional Galerkin finite-element method with bilinear quadrilateral elements. Solution of the equations in the time domain is accomplished by the implicit finite-difference method. Uses a velocity calculation within each finite element based on consistent spatial variability of pressure gradient and buoyancy term in Darcy’s law, equation (13). Without this ‘consistent velocity’ calculation, the standard method generates spurious vertical velocities everywhere there is a vertical gradient of concentration within a finite-element mesh, even with a hydrostatic pressure distribution [Voss, 1984; Voss and Souza, 1987]. The spurious velocities make it impossible to simulate a narrow transition zone between fresh water and seawater with the standard method, irrespective of how small a dispersivity is specified for the system.

NUMERICAL METHODS

The two governing equations, fluid mass balance and solute mass (orenergy) balance, are solved sequentially on each iteration or time step.Iteration is carried out by the Picard method with linear half-time-stepprojection of non-linear coefficients on the first iteration of each time step.Iteration to the solution for each time step is optional. Velocities required forsolution of the transport equation are the result of the flow equation solution(i.e. pressures) from the previous iteration (or time step for non-iterativesolution).

MODEL TESTING

Model test cases include:

1. The Hydrostatic Test 2. The Henry seawater intrusion problem 3. The Elder natural convection problem 4. The HYDROCOIN salt-dome problem 5. The Salt Lake problem 6. The Salt Pool Problem 7. Convection in infinite, finite, and inclined porous layers

“The numerical codes developed to simulate such variable-density processes e.g., SUTRA [Voss, 1984], MOCDENSE [Sanford and Konikow, 1985] and HST3D [Kipp, 1987] to name just a few, usually employ a set of coupled equations which may include nonlinear relations among parameters and an interdependence of solutions of the individual equations. In comparison to the cases involving homogeneous fluid properties, for many complex variable-density problems it is difficult to formulate appropriate analytical solutions that usually provide a basis for testing a numerical model.”

[Simmons et al., 1999]

“Evaluation of numerical modeling codes typically relies on internal consistency tests including mass balance indicators and external tests such as comparison with other numerical models (benchmarking) and against other observational evidence.”

[Simmons et al., 1999]

“Since model confirmation is at best only partial, increased trustworthiness in a numerical model can only be gained by the repeated testing of a code and making some judgement on its relative successes and failures. To do this requires a sufficient number of well-defined tests whose results are well known.”

Diersch & Kolditz [2002]

THE HYDROSTATIC TEST

Salt movement by diffusiononly

Examine density-on and offcases……

HENRY (1964): SEAWATER INTRUSION

Physical Setup:

Seawater intrusion into a confined aquifer studied in cross section under steady conditions.

Freshwater recharge inland flows over saltwater in the section and discharges at a vertical sea boundary.

The intrusion problem is nonlinear and may be solved by approaching the steady state gradually with a series of time steps. Initially there is no saltwater in the aquifer, and at time zero, saltwater begins to intrude the freshwater system by moving under the freshwater from the sea boundary. The intrusion is caused by the greater density of the saltwater.

Dimensions of the problem are selected to make for simple comparison with the steady-state dimensionless solution of Henry (1964), and with a number of other published simulation models. A total simulation time of t=100.0 [min], is selected, which is sufficient time for the problem to essentially reach steady state at the scale simulated.

100 time steps, Δt = 1 min, NN=231, NE=200, Δx=Δy=0.1m

HENRY (1964): SEAWATER INTRUSION

Earlier difficulty matching Henry analytic solution. Ambiguity over choice of molecular diffusion Dm.Segol [1994] provided more accurate semi-analytic results.

ELDER (1967) “SHORT-HEATER” PROBLEM

[Voss and Souza, 1987]

(a)

(b)

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

T=0.05 (10 yrs)

T=0.025 (5 yrs)

Symbol Quantity Value Units

ε Porosity 0.1 -

CINIT Initial concentration throughout 0.0 kgkg-1

∂ρ/∂C Coefficient of density variation 200 kgm-3

ρ0 Freshwater density 1000 kgm-3

k Intrinsic permeability 4.845 x 10-13 m2

αL Longitudinal dispersivity 0.0 m

αΤ Transverse dispersivity 0.0 m

g Acceleration due to gravity 9,81 ms-2

μ Dynamic viscosity of the fluid 1.0 x 10-3 kgm-1s-1

D0 Molecular diffusion coefficient 3.565 x 10-6 m2s-1

SUTRA simulation parameters for the Elder (1967b) problem (parameter valuesafter Voss and Souza, 1987).

SUTRAsolutions

[Voss and Souza, 1987]

0.600.20

0 100 200 300 400 500 600

150

100

50

0

0.600.20

0 100 200 300 400 500 600

150

100

50

0

0.20 0.20

0.60 0.60

0 100 200 300 400 500 600

150

100

50

0

0.200.20

0.60 0.60

0 100 200 300 400 500 600

150

100

50

0

0.20 0.200.60 0.60

0 100 200 300 400 500 600

150

100

50

0

0.20 0.200.60 0.60

0 100 200 300 400 500 600

150

100

50

0

0.600.20

0 100 200 300 400 500 600

150

100

50

0

0.600.20

0 100 200 300 400 500 600

150

100

50

0

0.20 0.20

0.60 0.60

0 100 200 300 400 500 600

150

100

50

0

0.20 0.200.60

0.60

0 100 200 300 400 500 600

150

100

50

0

0.200.20

0.60 0.60

0 100 200 300 400 500 600

150

100

50

0

0.20 0.200.60 0.60

0 100 200 300 400 500 600

150

100

50

0

(a)

(b)

(c)

(d)

(e)

(f)

neld teld

SUTRAsolution to Elder

problem(finer grids!)

TOUGH2solutions

High resolution results for the Elder free convection problem at elapsed times 2, 10and 20 years with discretizations (1) 60 (h) x 32 (v) uniform blocks (left panel) and(2) graded 84(h) x 42 (v) blocks (right panel). Figure after Oldenburg and Pruess(1995).

(a) (b) (c)

Elder problem results at elapsed times 4, 10, 15 and 20 years (top to bottom) withthree meshes: (a) coarse mesh (1170 grid points, 1100 finite elements), (b) fine mesh(4539 grid points, 4400 bilinear elements) and (c) very fine mesh (10108 grid points,9900 finite elements). Figure after Kolditz et al. (1998).

FEFLOWsolutions

Diersch & Kolditz [2002]

Number of quadrilateral elements in half domain =

Diersch & Kolditz [2002]

Ra=400

L=9:525,825 nodes for halfdomain solved

THE SALT DOME PROBLEM

[Konikow et al., 1996]

THE SALT DOME PROBLEM

Salt dome problemNAMMU results[Herbert et al., 1987]

Salt dome problem MOCDENSE results

THE SALT LAKE PROBLEM

[Wooding et al., 1997][Simmons et al., 1999]

g=0.855m/s2

[Simmons et al., 1999]

SEAWAT Simulation[animation by C. Langevin, USGS]

FEFLOW numerical tests

What we have learnt from this problem:

1. High Ra (~5000) in “Salt Lake problem” means system is verydynamic, with oscillatory/transient solutions

2. Only modest hydraulic conductivity and salinity difference required toachieve Ra in excess required for oscillatory solutions!

3. Grid convergence cannot be achieved [Mazzia et al., 2001]

4. Difficulty in knowing/ simulating the “perturbation” -small scale heterogeneities in the laboratory control onset

5. Finger growth/decay, penetration rate and “fatness” highly sensitive to numerical dispersion – long skinny fingers sinking fast, short fat fingers sinking slowly

THE SALT POOL PROBLEM

Johannsen et al., [2002]

Johannsen et al., [2002]

Johannsen et al., [2002]

Diersch & Kolditz [2002]

1. Due to the measurement error of the input parameter values, a direct simulation does not lead to an exact match of the experimental curves without tuning.

2. In the less dense experiment, streamlines of incoming flow follow the walls of the box and enter into the saline water body, leading to a displacement of the saltwater as a whole towards the outflow.

3. In the dense case, a completely different flow field develops. The streamlines of the incoming flow do not enter the saltwater zone but move along its top. The saltwater interface is slightly tilted and a counter current of very low velocity develops within the saltwater zone. Salt is transported into the freshwater flow by diffusion and lateral dispersionand then carried to the outlet.

4. The comparison with the experiment should therefore not only involve the breakthrough curve but also the width and position of the interface.

Johannsen et al., [2002] note the following in their conclusions:

5. The grid convergence of the numerical scheme depends strongly on the strength of the coupling of flow and transport and on the resolution of the dispersive effects.

6. Simulating saltpool, case 1 (low density) , a low spatial grid resolution was sufficient to capture the effects of advection and longitudinal dispersion, whereas the transversal dispersion has no significant effect.

7. On the contrary, the simulation of saltpool, case 2 (high density) requires a very high spatial resolution to resolve the transversaldispersive effects.

A new benchmark test based upon

Horton-Rogers-Lapwood (1945, 1948)and

Combarnous and Bories (1974)

Weatherill, D., Simmons, C.T., Voss, C.I., and Robinson, N.I., 2004. Testing density-dependent groundwater models: two-dimensional steady state unstable convection in infinite, finite and inclined porous layers, Advances in Water Resources, vol 27, pp 547-562.

Infinite porous layer

24π=Ra

Finite porous layer

Inclined porous layer

Symbol Quantity Value Unit

ΔC Concentration difference 0.1 kgkg-1

ε Porosity 0.3 -

∂ρ/∂C Coefficient of density variation 700 kgm-3

ρ0 Freshwater density 1000 kgm-3

k Intrinsic permeability 0.5 x 10-14 m2

αL Longitudinal dispersivity 0.0 m

αT Transverse dispersivity 0.0 m

g Acceleration due to gravity 9.81 ms-2

μ0 Dynamic viscosity of water 1.0 x 10-3 kgm-1s-1

D0 Molecular diffusion coefficient 1.0 x 10-9 m2s-1

Parameters used in the SUTRA simulation of the Lapwood problem.

(a)

(b)

(c)

0 50 100 150 200

4

3

2

1

0

0 50 100 150 200

4

3

2

1

0

0 50 100 150 200

4

3

2

1

0

Development of convection cellswith time in SUTRA

Ra=20

Ra=50

Ra=100

Ra=200

Ra=400

Onset of convection

Simulation stability plot

time

Oscillatory ConvectionRa > 240 - 300

( )( )( )222

22222222222

2

2

22

,min

ji

jikjkikji

+

+++⎟⎟⎠

⎞⎜⎜⎝

⎛++

=BA

BARa

BAc

π

This new test case problem suite is useful because:

1. Studied in traditional fluid mechanics literature for nearly a century

2. It has well-defined, exact analytical solutions for stability conditionsRa(crit) not offered by other current benchmarks

3. The critical threshold for oscillatory convection is well known (Ra=240-300)

4. The analytical solutions are 3D and thus offer a new benchmarkfor 3D code testing that is otherwise currently unavailable

Future challenges for modelling

Simmons, C.T., 2005. Variable density groundwater flow: From current challenges to future possibilities, Hydrogeology Journal, 2005 Special Edition on “Future of Hydrogeology”, vol 13(1), pp 116-119.

Variable density flow phenomena may be triggered, grow and decay over a very large mix of different spatial and temporal scales. The particular challenge lies in the fact that information on very small spatial scales and short time scales is needed to feed into long time and large spatial scale processes and simulations.

Measuring field scale parameters across this large range of spatial and temporal scales as input for modelling approaches is a major challenge.

Additionally, many current groundwater systems are highly transient and steady assumptions may be an oversimplification, especially where solute transport is concerned.

Furthermore, the source of dense plumes (e.g., leachate from a landfill) are often represented by simplified constant head, flux or concentration boundary conditions and yet the style of loading is expected to be important [Zhang and Schwartz, 1995].

Large mix of spatial and temporal scales

Laboratory

Field

Heterogeneity in hydraulic properties can perturb flow over many length scales (from slight differences in pore geometry to larger heterogeneities at the regional scale), triggering instabilities in density stratified systems.

This is particularly true for physically unstable situations but is also important in other situations. For example, the mixing process along a saltwater-freshwater interface can be significantly modified by nonuniform velocities.

Current research is suggesting that heterogeneity may serve as a physical perturbation in fingering processes and is therefore a critical feature controlling the onset, growth and/or decay in variable density flow processes.

Heterogeneity and dispersion

For example, low-permeability lenses can effectively dampen instability growth or even completely stabilize a weakly unstable plume. Dense plume migration, particularly at high density differences and in highly heterogeneous distributions, is therefore not easily amenable to prediction [Schincariol et al., 1997; Simmons et al., 2001; Nield and Simmons, 2006].

Whilst some studies have examined dense plume migration in fractured rock using numerical models [Shikaze et al., 1998; Graf and Therrien, 2005], there is still a need to explore how more complex and realistic fracture geometries affect variable density flow processes and to better understand the links with macroscopic dispersion [e.g., Welty et al., 2003; Kretz et al., 2003; Schotting et al., 1999].

Heterogeneity and dispersion

HydraulicConductivity(Kav = 0.1 m/day)

ConcentrationFields

(ΔC = 50,000 mg/L)

All these systems have the same Rayleigh number!

0.600.600.20

0.20

0 100 200 300 400 500 600

50

00

50

0

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0.20 0.20

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0

0.600.60

0.200.20

0 100 200 300 400 500 600

50

00

50

0

0.600.20

0.20

0 100 200 300 400 500 600

50

00

50

0

0.600.20

0.20

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00

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0.200.20

0.60 0.60

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rz8

Same mean and Ra

BUT

Increasingstandard deviation

in permeabilitykills fingers!

[Prasad and Simmons, 2002]

Effect of fractures near sourceGraf and Therrien [2007]

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 studies have examined heat and solute transport simultaneously within thermohaline convection regimes which may be important in understanding many processes including mineralization and ore formation in sedimentary basins, geothermal extraction processes and flow near salt domes. The simultaneous interaction of heat and solute creates challenges of its own and this complexity is exemplified where multiple species with differing diffusivities interact.

Multiple species studies (beyond the two species heat and salt) are fairly limited in groundwater literature. However, more recent studies [e.g., Zhang and Schwartz, 1995] suggest that the chemical composition and reactive character of a plume can greatly influence plume dynamics.

Fluids, solutes and fluid-matrix interaction

Other studies have begun to examine how chemical reactions can be coupled with density-dependent mass transport.

For example, Freedman and Ibaraki [2002] incorporated equilibrium reactions for the aqueous species, kinetic reactions between the solid and liquid phases, and full coupling of porosity and permeability changes that result from precipitation and dissolution reactions in porous media and showed that complex concentration distributions result in the variable density flow system.

Fluids, solutes and fluid-matrix interaction

Variable viscosity nature of fluids is typically ignored in variable density groundwater flow problems and suggest that plume migration pathways and rates are inaccurately predicted in the absence of the variable-viscosity relationship e.g., Ophori [1998]

Porous medium saturation. Thorenz et al., [2002] and Boufadel et al., [1997] demonstrate that significant lateral flows and coupled density-driven flow may take place in the partially saturated region above the water table and at the interface between the saturated and partially saturated zones.

The coupling of variable density flow phenomena with more complex chemical and fluid property characterisation is an area that is still in its infancy and warrants further exploration. We need new test cases that actually test those newly added features of the code.

Fluids, solutes and fluid-matrix interaction

[Simmons et al, 2002]

Complexity associated with unstable problems – oscillatory solutions at modest Ra (240-300) – not high!

Grid convergence issues [e.g., salt lake problem]

Numerical controls on generation and growth/decay processes not cleare.g., What are the physical perturbations, and c.f. numerical ones?

Simulating high density contrast and low dispersion cases is computationally demanding

Simulation of large scale 3D phenomena require “expensive” numerical meshes

Key Q: What level of simplification is permissible in our analyses?

Need access to more field data for model testing, ground truthing …….

Numerical modelling specifically…

CONCEPTUAL UNDERSTANDING AND PREDICTION:

At present processes not easily amenable to prediction - sensitivity of process to (often unknown) heterogeneity- heterogeneity scales controlling onset, growth and/or decay of plumes- problems with application of Rayleigh number- modelling issues (oscillatory solutions, sensitivity to numerical perturbation/dispersion)- fluid-solute-matrix interactions require further exploration- develop simplifying approaches?- study processes with “uncertainty” in mind & include sensitivity analyses

MEASUREMENT:

“Inference” for the existence of (unstable) variable-density flow is a good starting point but we need to develop field techniques to measure it directly in field settings and to gather better data for predictive tools

SOME REMARKS

"Everything should be made

as simple as possible,

but never simpler!"

Albert Einstein

(i) the need to better understand the relationship between variable density flow phenomena and dispersion,

(ii) better geological constraints on variable density flow analyses using lineaments, sedimentary facies data and structural properties of fractured rock aquifers,

(iii) improving the resolution of geophysical, geoprospecting and remote sensing tools for non-invasive characterisation of dense plume phenomena and heterogeneity,

(iv) linking the fields of tracer and isotope hydrogeology and variable density flow phenomena,

(v) double-diffusive and multiple species transport problems in variable density flow phenomena,

(vi) links between climate change phenomena and the response of variable density flow phenomena, such as sea level and coastline positionchange, and the impact of transgression and/or regression cycles on aquifer salinisation will be critical,

FUTURE PREDICTION?

(vii) the gas-liquid phase chemistry of carbon sequestration processes and its efficiency, safety and long term viability as a storage option will necessarily involve an understanding of variable density transport dynamics with multiple phases,

(viii) links between variable density flow phenomena and a number of hydroecological applications (e.g., how does variable density flow effect baseflow accessions and other surface-groundwater interactions, or salt budge profiles under vegetation, or the spatiotemporal distribution of stygofauna and biota in subsurface groundwater ecosystems?),

(ix) Some of the above issues point to the need for more detailed and accurate coupling of surface water, vadose zone and groundwater models in a variable density system – and which will drive an inherent increase in the complexity of the modelling approach.

AND, THERE ARE LIKELY TO BE OTHER EMERGING AREAS OF INQUIRY NOT YET CONCEIVED …………..