2 tredi-gemmes, mine joseph else, 68310 wittelsheim ......the density and viscosity-dependent...

Salt intrusion modelling in aquifers with high

permeability ratio

C. Oltean^, J.-M. Strauss & M. A. Bues^

1 Laboratoire Environnement Geotechnique & OuvragesENSG - BP40- 54501 Vandceuvre-les-Nancy, France

2 Tredi-Gemmes, Mine Joseph Else, 68310 Wittelsheim, FranceEMail: [email protected]

Abstract

In many environmentaly important cases of subsurface flow and transport, thedensity of the liquid phase depends on solute concentration. When densityvariations are large (> 5%), flow and transport are strongly coupled. Densityvariations in excess of 20% occur in salt dome and bedded-salt formation. Toanalyze such complex phenomena, a simulation code (VIDE POMM'S) hasbeen developed. It is based on the mixed hybrid finite element method(MHFEM) which permits to compute waterflow and solute transport. Thecoupling between groundwater flow and solute transport is realised byequations of state. We assume that in the mixing zone the density varieslinearly with the concentration. Moreover contrast of density and viscosity,these aquifers are constituted by three zones with a contrast of permeability(ratio 10). An other case will be simulated with a break (permeability ratioabout

1 Introduction

In many groundwater flow systems, the liquid density gradient plays animportant role in the transport of solutes. Examples include the saltwaterintrusion problem, the storage of heat in aquifers, the infiltration of salt-leaching, etc. In such flow problems the solute transport must be coupled withthe groundwater flow. For systems where the liquid density gradients arecaused by gradients in the solute mass fraction, the governing equationsconsist of two mass balances (for the liquid and for the solute), twomomentum balances (Darcy's law for the liquid and Pick's law for the solute)and state equations.

The solution of the set equations with the appropriate initial and boundaryconditions can be estimated by using some numerical models. These modelsemploy conventional finite differences or finite elements technics. Under

Transactions on Ecology and the Environment vol 14, © 1997 WIT Press, www.witpress.com, ISSN 1743-3541

310 Water Pollution

certain conditions, these classical methods can introduce spurious oscillationsand artificial diffusion. Special methods are then required in the attend toavoid this problem.

Our code is named VIDE-POMM'S (VIscosity-DEnsity-POrous-Media-Miscible-Simulations) and is based on the mixed hybrid finite elementmethod. For verifying coupled flow problems, we have followed thesuggestions of Voss & Souza^ who recommend comparison with two classicflow problems. The two classic problems are the seawater intrusion problemof Henry & and the free convection problem of Elder^. These comparisons canbe found in Oltean & Bues^ and will be quikly mentioned in section 3.1.

In this paper, we present an application of VIDE-POMM'S in more realisticcases. The configurations studied are similar to Henry's problem and representaquifers initially filled with fresh water. Moreover contrast of density, weassume that these aquifers are constituted by two zones with a contrast ofpermeability (ratio 2). An other case will be simulated with a break(permeability ratio about lO* to 10 ).

2 Formulation of flow and transport

2.1 Basic equations

The density and viscosity-dependent transport problem is described by thegeneral ground-water flow equation (Voss ), written in terms of the hydraulicpressure P (ML' T" ),

9P An AC-(1)

3t 3Cm 9tand the solute transport equation written in terms of the dissolved mass frac-tion Cm (M/M) (Herbert et al?},

(2) pe^ + V.(pl/Cm) - CmV.(pV) = V.(peD.VCm) + Qg9t

where V denotes the gradient, V. denotes the divergence and t is the time (T).

The other parameters are: solute density p (ML~ ), specific pressure storativity

Sop (ML"*T-2)-l, effective porosity £ (dimensionless), Q (ML T )and Qg

(ML' T" ) flow- and solute-source/sink terms, respectively, D (L T~ ) generalhydrodynamic dispersion tensor which is written as:

where Dm [L T~ ] represents the molecular diffusion tensor, aj^and otp (L)

denote the longitudinal and transversal dispersivities, respectively. V (LT~ ) isthe flow velocity and is computed by Darcy's law:

(4) V = -


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where k (L%) is the permeability tensor, |i (ML T'i) is the dynamic viscosity,

g (LT~2) is the gravity acceleration, z (L) is the elevation (axe Oxg) and U =

We is the actual flow velocity.The ground-water flow and transport equations are coupled by so-called

equations of state for density and dynamic viscosity. These equations arefunction of the dissolved mass fraction C^ and are described in our model bythe relationships:

(5) p = Po + -^- (Cm - Co) = Po + 7 (Cm - Co)

(6) |1 = |Lio[l. + Ti (Cm - Co) + Tz(Cm ' CoP + T](Cm " Q))3]

where po, (IQ and Co are reference density, dynamic viscosity and concentra-

tion respectively (generally Co = 0). y and TJ (i = 1,...,3) are experimentalconstants.

2.2 Numerical method

In order to simultaneously resolve both problems, most packages use either theconforming finite element method - CFEM - or the finite difference method -FDM-.

The application of one of these methods to the resolution of the hydrody-namic module requires pressure field estimation in order to determine the ve-locity field by differentiation of the pressure field. Frind & Matanga& haveshown that in certain cases (e.g.: aquifers with a low pressure gradient), thevelocity field calculated with the CFEM can give less than satisfactory results.Furthermore, the normal component of the velocity field is discontinuous fromone element to the next. Cordes & Kinzelbach^ have proposed a solution usingthe velocity field determined with the CFEM that makes it possible to obtain anew velocity field where the normal component is continuous at the interfacebetween two adjacent elements. Thus, this technique improves the velocityfield accuracy close to the imposed flux boundaries (especially in the case ofimpermeable boundaries) but does not eliminate the error due to numericaldifferentiation of the pressure.

The mixed finite element method - MFEM - (e.g.: Meissner^, Chavent &Jaffrel) or the mixed hybrid finite element method - MHFEM - (Chavent &Roberts ) provide an answer to the problems described above.

Finally, another numerical technique which has been tested by experimentscarried out on two physical models in the laboratory should be noted, e.g.:Oltean et al. ; Oltean^; Siegel^. This technique is known as the disconti-nuous element method. Even though this technique provides accurate resultswhen the advective term - hyperbolic term - is predominant, this techniqueshows certain limitations when the preponderance returns to the diffusive term- parabolic term - (Oltean & Bues ).

As a result, in order to resolve the mass transport equation, we have deve-loped and tested the technique implemented for the resolution of the momen-tum transfer, i.e.: MHFEM, taking into account the influence of density anddynamic viscosity contrasts. This method makes it possible to ensure conti-


312 Water Pollution

nuity of the normal component of the dispersive flux from one element toanother, to maintain the mass solute balance at the scale of each element andto simultaneous calculate the pressure and flow estimations.

Moreover three times more unknows that classical methods for the samemesh refinement appear and constitute the principal disadvantage of thistechnics.

3 Application in field case configurations

3.1 Previous results

The VIDE-POMM'S numerical code has been subjected to several tests in or-der to verify its performances. The first was concerned with salt water intru-sion on the entire lateral surface of a homogeneous aquifer saturated withfresh water (Henry's problem^ - an advective-dispersive problem with aconstant dispersion coefficient). The second test was concerned with localizedinjection of a pollutant into an aquifer considered to be, as in the first case,homogeneous and saturated with fresh water (Elder's problem^ - free convec-tion). The conclusions concerning the results provided by the VIDE-POMM'Snumerical code have been summarized in table 1.

Elder*

Voss & Souza*9

LeijnselO

SegollS

Oldenburg &Pruess 12

Henry's

problem ( * )

-H-

Henry's

problem (2)

++

++**

Elder's

problem

-

-

++*

-

-/++*

- Inadequation, + qualitative adequation, ++ quantitative adequation.* fine mesh, ** molecular diffusion coefficient different from imposedHenry's value.(1) Boundary conditions corresponding to Henry's problem.(2) Mixed boundary conditions.

Table 1: Adequation for VIDE-POMM'S tests.

For Henry's problem, the results obtained can be considered in agreementwith the analytical solution proposed by Henry8 and revised by Segol . Thedifferences between our results - concerning concentration distribution - andthose published by other authors (Voss & Souza ) are located in the top partof the domain studied and are due to the differences in boundary conditionsbeing taken into account. While our numerical code agrees with the boundaryconditions proposed by Henry, Voss & Souza^ preferred to focus on specificboundary conditions. It is obvious that these conditions do not agree with theboundary conditions in Henry's problem, even if the particular way deal withthis example seem more realistic from a physical point of view. By taking intoaccount in our numerical code Voss and Souza's boundary conditions^, the


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comparison between VIDE-POMM'S and SUTRA (Figure 1) shows a goodsuperimposition of the numerical results.

As far as Elder's problem is concerned - free convection - the numericalcode accounts for the density contrast very well (Figure 2 a, b, c, d and e).Even for domain discretization that can be qualified as coarse, the resultsobtained with the VIDE-POMM'S code are:(i) in agreement with Elder's experimental results on both the number of

vortices and their rotation direction,(ii) in agreement with results obtained with finer resolution grids (e.g.:

LeijnselO or Oldenburg & Pruess ),(iii) practically symmetrical, in spite of the use of a non-symmetrical mesh.

In the absence of analytical solutions for the density contrast problems andtaking into account the multiplicity and complexity of transport problems, it isprobably too early to come to a conclusion about the overall efficiency of themethod used in the design of the VIDE-POMM'S numerical model. Further-more, it seems that VIDE-POMM'S behaves much better than already existingprograms.

3.2. Configuration studied

The dimensions of aquifer are: length = 2 000 m and width = 250 m. Freshwater enters at a constant rate flow from the inland boundary or at constantpotentiel (E), mixes with intruding saltwater, and discharges to the sea throughthe coast boundary (W).

Along the coast boundary two conditions for the transport problem can beimposed (Figure 3.a):(i) constant seawater concentration along the whole coast boundary

(Dirichlet condition), i.e.:

C(0, z, t) = Co and P(0, z, t) = ps g z, V z and t,(ii) constant seawater concentration only when the seawater flow rate is

greater than fresh water flow (Mixte conditions), i.e.:C(0, z, t) = Co with z < h(t) or C(0, z, t) = C with h(t) < z < 0P(0, z, t) = ps g z.

The (i) condition which is in fact not realistic from a physical point of vue,has been shown to be erroneous with field mesurements. Salinitymeasurements of sea water close to the sea shore gave lower values thanexpected, showing that fresh water was flowing out the aquifer to the sea.Therefore this condition has not been used in the presented simulations.

The choice of the permeability distribution is:

(i) ratio-2, i.e.:k\ =4.17 10" m^for 0 < x < 75 m and k2 = 2,24 10' mfor x > 75 m,

(ii) ratio - 10, i.e.: k, = 2,85 10" nf for 0 < x < 75 m and ki = 2,24 10"

m for x > 75 m,

(iii) ratio = 10* to 10 , i.e.: kj = 2,85 10" m^ for 0 < x < 75 m, ki = 2,24

10^ m- for x > 75 m and k3 = 1,1 10"* nf for the break.

Using these values of permeability, but also the values of distance, a simplecalculation shows that the W condition, (pressure distribution), gives identical


314 Water Pollution

co"

5

oII W

0.00.0

0.8

0.2

0.0

N

0.5 1.51.0

Length (m)

(a) Domain and boundary conditions.

-VIDE-POMM'S

• Henry's Solution[Segol, 1994]

*"**«0mp

IIdIICL,

2.0

(b) Steady state results with Henry's boundary conditions.10 , , Flux

0.8

0.6

0.4w

0.2

0.0

-VIDE-POMM'S

-[Voss & Souza, 1987]

Flux

(c) Steady state results with mixte boundary conditions.

Figure 1. Concentration isopleths for Henry's problem.


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atm P = Patm ,

CL&

-150150 300

Length (m)450

(a) Domain and boundary conditions.

100 200 400 500 600

(b)

(c)

(d)

(e)

300Length (m)

Figure 2. Concentration isopleths and velocity field calculated withVIDE-POMM'S for Elder's problem ((b) and (d) - coarse discretization,

(c) and (e) - fine discretization).


316 Water Pollution

injection flow rate values whatever the distribution may be. The aquifer hasbeen divided into 3 456 triangular elements (1 728 rectangular elementsdivided by their diagonal according to the NW-SE direction) leading to 1 843nodes and 5 298 edges.

3.3 Numerical results and discussions

The concentration distribution and the velocity field are represented for thefirst 800 meters and for a simulation time of 500 years, considered as the timenecessary to reach an hydrodynamical equilibrium (steady state). In the firstcase (ratio ~ 2 - Figure 3.b), and in the vicinity of the interface between thetwo layers of different permeability, we observe that streamlines move from alayer to the other without any significant changes. This exhibits the continuityof the velocity field. In reference with an equivalent homogeneous porousmedium (equivalent permeability computed with two porous media in serial:

13 2keq = 2,28 10" m - Figure 3.c -), displacement of the pollutant is quiteidentical, even if a difference appears in the vicinity of the sea shore limitwhere the layer of fresh water outflow is reduced.

The permeability contrast of 10 leads to strong changes of the velocity fieldclose to the interface (Figure 3.d). The velocities in the k2 permeabilitydomain keeps approximately the same distribution and the same order ofmagnitude than previously. In the k% permeability domain the velocity fieldshows a real different distribution. If the order of the velocity magnitude is inaccordance with the permeability and boundary conditions used, i.e.: waterinflow in the lower part and outflow in the upper part, the orientation of thevelocity vectors is straightly related to the permeability contrast. Close to theinterface, the k% permeability domain is similar to a barrier for thedisplacement of pollutant, and causes a change of direction in the velocity fieldin the k\ domain. Therefore the layer of fresh water outflow has a thicknesslower than 20 meters, bringing modifications to the concentration isopleths.Moreover, streamlines exhibit a break slope for ordinate lower than - 150 m.The fracture (ratio 10^ to 10^ - Figure 3.e -) does not lead to main changes onthe pollutant distribution, excepted in the region close to the 0,5 isoline. On theother hand this fracture, introduced as a high permeable layer, behaves as adrainage channel and generates a new distribution of the velocity field in bothki and k2 permeability domain. In spite of this new distribution, one mayobserve that the layer of outflowing water keeps the same thickness than forthe previous case. Although the fresh water inflow are equivalent, the study ofthe fresh water/salt water interface, identified as the intersection between theisoline 0,5 and the bottom of the aquifer (Frind , Galeati et al.7), shows thatthe intrusion length may be subject to non negligible variations (in order of 14%). These results indicate that the heterogeneities of an aquifer may bringsome important perturbations when pumping is required in a coastal region.

Even if the VIDE-POMM'S code can compute with both density andviscosity contrast, only density variations were introduced for this work.

References

1. Chavent, G. & Jaffre J., 1986, Mathematical models and finite elementsfor reservoir simulation, North Holland, Amsterdam.

2. Chavent, G. & Roberts, G.E. A unified physical presentation of mixed,mixed hybrid finite elements and standard finite difference approximation


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P=Pog (depth)

)p = Psg (depth) or(a)

800 1200 1600 2000Length (m)

(b)

(c)

(d)

(e)

-250100 200 600 700 800300 400 500

Length (m)Figure 3. Concentration isopleths, velocity field and streamlines for the aquifer:(a) - domain configuration and boundary conditions, (b) - permeabilityratio = 2, (c) - equivalent permeability, (d) - permeability rat;o =10,

(e) - fracture


318 Water Pollution

for the determination of velocities in water flow problems, Adv. WaterKfjowr., 1991,14,329-348.

3. Cordes, C. & Kinzelbach, W. Continuous groundwater velocity field andpath lines in linear, bilinear, and trilinear finite elements, Water Resour.#f&, 1992,28,2903-2911.

4. Elder, J.W. Transient convection in a porous medium, J. Fluid Mech.,1967, 27, 609-623.

5. Frind, E.O. Simulation of long-term transient density-dependant transportin groundwater, Adv. Water Resour., 1982,5,73-88.

6. Frind, E.O. & Matanga, G.O. The dual formulation of flow for contami-nant transport modeling. 1. Review of theory and accuracy aspects, WaterKfjowr. #&?., 1985, 21, 159-169.

7. Galeati, G., Gambolati, G. & Neuman, S. P. Coupled and partiallycoupled Eulerian-Lagrangien model of freshwater-seawater mixing, WaterKfjowr. #&y., 1992, 28, 149-165.

8. Henry, H.R. Effects of dispersion on salt encroachment in coastalaquifers, [/.& GW. Swn/. WaffrSwpp/yfa;?., 1967, Vol. 16313-C, C71-C84.

9. Herbert, A.W., Jackson, C.P. & Lever, D.A. Coupled groundwater flowand solute transport with fluid density strongly dependent upon concentra-tion, Water Resour. Res., 1988, 24, 1781-1795.

10. Leijnse, T. Free convection for high concentration solute transport in po-rous media, in Contaminant Transport in Groundwater (ed H.E. Kobus eta/.), 341-346, Stuttgart, Germany, 1989, A.A. Balkema, Rotterdam, 1989.

11. Meissner, U. A mixed finite element model for use a potencial flow pro-blem, Int. J. Numer. Methods Eng., 1973, 6, 467-473.

12. Oldenburg, M.C. & Pruess, K. Dispersive transport dynamics in a stronglycoupled groundwater-brine flow system, Wat. Resour. Res., 1995, 31,289-302.

13. Oltean, C. Comportement du deplacement d'un front d'eau douce/eau saleeen milieu poreux sature: modelisations physique et numerique, These deDoctorat de 1'ULP, Strasbourg, France, 1995.

14. Oltean, C., Ackerer, Ph. & Bues, M.A. Solute transport in 3D laboratorymodel through an homogeneous porous medium: behavior of denseaqueous phase and simulation, in Computational Methods in WaterResources (ed Al. Peters et al), 341-346, Heidelberg, Germany, 1994,Kluwer Academic Press, Dordrecht, 1994.

15. Oltean, C. & Bues, M.A. Two 2-D simulation codes for modeling salt in-trusion in aquifers, in Computational Methods in Water Resources XI (edA.A. Aldamar et al), 717-724, Cancun, Mexico, 1996, ComputationalMechanics Publications, Southampton, 1996.

16. Segol, G. Classic groundwater simulation, proving and improving nume-rical models, PTR Prentice Hall, Englewood Cliffs, New Jersey, 1994.

17. Siegel, P. Transfert de masse en milieu poreux fortement heterogene: mo-delisation et estimation de parametres par elements finis mixtes hybrideset discontinus, These de Doctorat de 1'ULP, Strasbourg, France, 1995.

18. Voss, C.I. SUTRA: A finite element simulation model for saturated-unsa-turated, fluid density dependent groundwater flow with energy transportor chemically reactive single species solute transport, U.S. Geol. Surv.Wafer Kfjowr. /rzwjf. #6%?., 84-4369, p. 409, 1984.

19. Voss, C.I. & Souza, R.W. Variable density flow and solute transportsimulation of regional aquifers containing a narrow freshwater-saltwatertransition zone, Wat. Resour. Res., 1987, 23, 1851-1866.


2 tredi-gemmes, mine joseph else, 68310 wittelsheim ......the density and viscosity-dependent...

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