Editor in Chief: XUE YUQUN, JACOB BEAR ... hydraulic conductivity imposed ... the study of transport in the groundwater (saturated zone), models with a .... by manual calibration on the concentrations measured at the bottom of the sand layer.
{ I
a
Communication presentee J'I,C.M.G.F.P. INTERNATIONAL CONFERENCE ON MODELLING GROUNDWATER FLOW AND POLLUTION NANJING (CHINE) - 23-26 AVRIL 1991
a
BRGM
\'(N1UPtlSE AU snVICE DE LA run
MODELLING CONTAMINANT TRANSPORT THROUGH THE UNSATURATED ZONE IN TRANSIENT STATE WITH A RANDOM WALK PARTICLES SCHEME
Dominique THIERY Note Technique
N· 16 91 4S EAU AoQt 1991
MODELING IN GROUNDWATER RESOURCES Proceedings of the International Conference on Modeling Groundwater Flow and Pollution (pp 311-316)
Editor in Chief: XUE YUQUN, JACOB BEAR NANJING UNIVERSITY PRESS, 1991, Nanjing (ISBN 7-305-01121-5)
BRGM SIi\,ge: Tour Mlrobeou, 39·43, quai Andri-Cllroen· 75739 Poril cedex 15, Fronce Ti1.: (33) 1 40.58.89.00 - Ulex : BRGM 180258 F . HI6copieur : (33) 1 40.58.89.33 R.C. 58 B 5614 Parh· SIRET: 58205614900419 Cenlle sclenlifique ellechnique :
Avenue de Coneyr, Orl6ans-la Source (lolrel) - B.P. 6009·45060 Orl6ons cedex 2, Fronce HI.: (33) 38.64.34.34· Telex: BRGM 780258 F· nl,scopleur I (33) 38.64.35.18
Modelling Contaminant Transport Through the Unsaturated Zone in Transient State with a Random Walk Particles Scheme Dominique Thiery Bureau de Recherches Geologiques et Minieres Boite Postale 6009 45060 Orleans Cedex 2 • France ABSTRACT
A new kind of contaminant transport model through the unsaturated zone has been designed. The velocities are computed in transient state with a one dimensional vertical finite difference model solving Richards equation in an operational way. The scheme has been adapted specially so that the model may use large time steps and large space gridding and respect perfectly the conservatio~ of flows. A random walk scheme in full transient state has been adapted in the model. This random walk particles scheme uses a continuous velocity field with analytical displacement during each time steps. In this scheme the dispersion is not represented by random jumps of particles-with risks of being trapped in stagnant zones-but by a random fluctuation of the velocity field. The exchanges with the porous matrix and immobile water are also simulated by a stochastic process generating absolutely no numerical dispersion. This model has been successfully applied to experimen!s of transfer of calcium, chlorine and potassium through a large plot of sand used for geoepuration. NOTATIONS Ca h z
K Q
capillary capacity pressure (negative when the soil is not saturated) vertical axis (positive upwards) hydraulic conductivity imposed external surface flux
t
time
C
concentration in the mobile phase position (vertical axis z in our model) pore velocity = V 18 m mobile water content =8-8, water content immobile water content time constant of solute exchange between mobile and immobile phase time constant for desorption
x u
8m 8
8, k
k,
m-I m m
m/s m/s s kg I m m m
I
s
ro' I m' m'l m' m'l m' s
s. • 311 •
V
S D K.
-a P
p". w R
Darcy velocity concentration in the immobile phase dispersion coefficient water content equivalent of immobile capacity dipersivity Peclet number relative to mobile phase exchange Peclet number waiting time mobile phase ratio =8.. I (8 m +K.)
mIs kg
1m
m' I s
m' 1m' m
s
INTRODUCTION A considerable development of transport models through the unsaturated zone is connected to the rising problems of groundwater pollution by pesticides and to the study of the water geoepuration processes which requITes a dear analysis of the flow mechanisms through a sandy bed. One-dimensional vertical finite differences models of flow in porous media have long existed. Recently THIERY (1988 and 1990) showed that an implicitlinearisation scheme with iterations {MERINOS model) makes it possible to obtain econoillically, results which remain very precise with large time steps and gridding between 5 and 20 em. For the solute transport, SMEDT & al. (1981) obtained good results by considering a convective-dispersive transfer in a mobile water phase with simultaneous exchange of solute between a mobile and an immobile phase. However, for the study of transport in the groundwater (saturated zone), models with a "Random Walk Method" are used more and more (Kinzelbach, 1988). These models present the avdantage of generating no numerical dispersion, even in transient state, of being decoupled from the hydrodynamic calculation and of being very flexible for the introduction of complex exchange Or interaction laws. Valocchi and Quinodoz (1989) describe a generalized random walk method for kinetically adsorbing solute transport. This paper describes a model of solute transport through the unsaturated [zone: it is a finite difference scheme with implicit linearization (MERINOS model), coupled with a random walk particles method in full transient state using analytical displacements witb a continuous velocity field. This model has been successfully applied.to field experiments of calcium slug injections and transfer through a large plot of sand used for geoepuration. HYDRODYNAMITCEQUATION Richards' equation ("local balance") is solved:
ca~~=:z[K(~~+
. 1 )]+Q/dZ
It gives at each time step the calculated flows between meshes which is Darcy velocity needed for the hydrodispersive equation. -' HYDRODJSPERSIVE EQUATION The Advection-Dispersion Equation (ADE) for a non reactive solute, exchanged between a
• 312 •
mobile (C) and immobile phases (S), writes:
/l(Bm C)
Il
(Bm
D Il~ _ Vile -Bm ~
Ilt Ilx IlxJ Il:r: k 1he exchanges between the phases are described by the relation: Il(Kd S) (C-S) Ilt k 'where the concentration S is relative to the immobile water capacity Kd. RANDOM WALK RESOLUTION The random walk method uses the movement of particles to represent the advection and dispersion phenomena. In the classical method particles are displaced with the pore velocity, u. and a gaussian random displacement with a standard deviation'; 2 D. dt is added, dt being the time step. THIERY and lUNG (1990) show that this scheme is not adapted for flows in the non saturated zone wl;kh display huge velocity gradients. In zones where the pore velocity is high, it would be necessary to u~e very small time steps for the transport in order to avoid random displacements driving particles in zone where the velocity is very small or even making them jump above such zones. To 1ake care of this problem they derive a completely different scheme: instead of simulating dispersion by a random displacement it is simulated by a random component added to the velocities of both 2 D/dt. ·ends of each mesh. The standard deviation su of the velocity random component is: su Jnside each mesh Darcy velocity varies continuously from one end to the other one. The effective porosity varies linearly from the beginning to the end of the time step. The displacement is then computed from analytical relations following a scheme dervied from POLLOCK (1988) or FILlPPI -& a!. (1989). The final position of a particle at the end of the time step is known exactly, whatever its duration, taking into account the change of parameters if changes of meshes occur during the time step. Hence a perfect mass conservation is achieved and stagnant zones are respected.
=,;
EXCHANGES WITH MATRIX OR IMMOBILE WATER Classically exchanges and decay are decoupled from advection and dispersion and readions happen at the beginning or at the end of time steps which must be small. VALOCCHI and QUINODOZ (1989) show that linear exchanges may be simulated efficiently with a continuous stochastic p~ocess by random walk which produces no numerical dispersion during the exchanges. Forinstance fixation of particles according to relation GlC/Glt=-(C-S)! k correspond to a waiting time w with exponential statistical distribution and average k (THIERY & lUNG, 1991). The particle is moving during a duration w before being fixed. The desorption C0rresponds to an exponential distribution with average Ist=k. Kd IBm. VALIDATION OF THE MODEL MERINOS model has been validated in steady state, without exchanges with 3 analytical ,solutions detailed by THIERY & lUNG (1990). A validation with' exchanges is given by THIERY & lUNG (1991) according to an analytical solution for a .constant mass injection derived from an
• 313 •
expression given by de SMEDT & a!. (1981). Figure I shows that the agreement is excellent (P= 22.5, P".=V. k/x. 8 m =I.92, R=0.50; see notations). c 1500
~1250
r-----------------t
•
~lOOO c
~ 750
~ u
?op
P.D'O-C]'''
"" °J>,.."..,......_aa>.a~"":::·:.."'=-D_.,O-.,.-·im..,..,m""o... bi.;.Ie.....,,....J ° 60 ° 0.20 >:> £/''-
250
.
1.0
fl ..
Figure I -Validation of the model
APPLICATION TO THE EXPERIMENTAL GEOEPURATION SITE OF ORLEANS. MEASUREMENT DEVICE AND TEST CARRIED OUT The Orleans site, described by ALAMY & lUNG (1989), is a 1.50 metre-thick sandy layer with an area of 16 m', This layer of previously washed Loire sand graded at 0.20 mm is placed on a 0.2 thick gravel bed. The measuring system consists of 9 vertical tensiometers and a neutron probe tube (SOLO 25). The flow at the bottom of the basin is monitored and an automatic sampletaker supplies water samples periodically for chemical analysis. On August 19th 1989 at 13 hours, a tracer test was carried out over an area of 3 m' with 1300 litres of water containing c1acium (3 gIl), potassium (5.6 gil) and chlorine (9.9 gil). Later on, two rincings were made with 2400 litres each, the first one on October 6th at 15 hours, the second one on October 6th also at 15 hours. The interpretation of water content and head variations during the first injection enable the determination of a very good conductivity-water content relations and retentions curves for 3 layers: water content at saturation 47%, residual water content 12 to 15 %, saturated hydraulic conductivity 3.4 to 5. I X 10-' mIs, exponent on conductivity relation 3.4 to 5.4.
m
FLOW SIMULATION
-;
"e,
-10
-;
~
"
.JQ
%
~
I!
·50
~
m
-10
.JQ ·50
·10
·'0
·50
~llO
~IIO
Water content .. \30
%
-130
-150
.. \50
I 13~
WATER CONTENT SIMULATION ERROR
·10
19 A\lO 1989
24h
13h
19 Aug 1989
24(1
Figure 2 -Simulated water content and simulation error in the sand layer on 19/08/1989 from 14 h to 24 h.
• 814 •
The sand layer was discretized with 15 meshes of dimension 10 cm, and a water volume of 433 :mm was applied above the surface at initial time. The time steps adjusted automatically from a minimum of 1 mn with an average value of 10 mn. Figure 2 shows that the evolution of water -content is very well simulated from 13 to 24 hours. The simulation error exceeds only locally 2% -of water content. TRANSPORT SIMULATION In the initial volume of 433 mm, 2600 particles were added, each one representing a mass of 0.5 g of calcium or of 16.5 g of chloride. The exchange cinetic has been considered instantaneous and the molecular dispersion has been neglected. The immobile water content Of has been set equal (0 residual water content identified on the retention curves, after a decrease of 0.02 m'/m'. The "alues of Of are then respectively 10% ,14 % and 13 % for the 3 layers of the model. The 2 remainjng parameters-the dispersivity a and the equivalent immobile capacity Kd have been determined by manual calibration on the concentrations measured at the bottom of the sand layer. The best "alues are given in Table I. Table 1- calibration of transport paramelers
(m)
_.
•
HALF LIFE FOR IRREVERSIBLE FIXATION OR DEGRADATION
IMMOBILE CAPACITY (factor applied on Or)
DlSPERSIYITY
Calcium
0.075
2.50
Chlorine
0.075
2.9
STANDARD DEVIATION OF ERROR (mg I J)
(h)
200 1,200
350
Figure 2 show that the concentrations of calcium and of chlorine are well simulated with the ,model for the 3 periods (injection followed by 2 rincings). It has not been determined why chlorine, 'which is usually considered as a very good tracer, was degraded during the tests perhaps by reaction 'With organic matter. Potassium could not be simulated correctly but it is considered as a poor tra-
.eer. 3000
1500·
'•"•
observ. mod,l
':"' 1000
u
•
~
• •,•0
u
"
500
\0
injection
,0 0
~
6
12h.
Calcium
l
-• 2000
rincing 2
~'
6h
: •0
u,
0
0
3
6h
Tin It>J
ob:!-eN. model
~ \500
~
. rincirq I
0.'
. Chlorin.
~2500
\0
\
rinclnq 2
1000
~
500
rincil'lQ 1
0
0 0
~
6
12h
O·
:3
16h
0
~.' 6h T1.~
(hI
:Figure 3 -Simulation of injection of calcium and chlorine followed by 2 rincings 47 days later (broken line = concentration estimated from measured conductivity)
• 315 •
SENSIBILITY ANALYSIS The co~fidence intervals of the parameters. iJave been' dbtermined from the derivative of the concentrations with respect to the parameters, The standard deviation on dispersivity (for calcium) is equal to 0,01 m, the standard deviations on immobile capacity factor are respectively 0,05 for calcium and 0.03 for chlorine (100 h for the degradati6n). This shows that the parameters. are very well identified from the data recorded from the succession of the. 3 tests. ' .CONCLUSION The model transport presented in this paper enables the efficient modelling offield experiments. of transport throngh porous media. The [random walk method nsed has the advantage of presenting no nUmerical dispersion and' of respeding perfectly the heterogeneties while being very versatile. REFERENCES A.lamy 2'1 lung 0., 1989.. -4JN9¢pu.r.q.tion
a Orleans (volume II). Protocoles experimentaux, aquisition de3 donnees•.
premiers sulvis. -Unpublished report BRGM R 30364 ENV 48 90 De Semtlt P., Wierenga p~ J. &: Van Den Beken A., 198L.-Theoretical and experimental study of solute movement through porous media with mo'bile an immobile water. -Vrige Universiteit Brussel (V. U.'B. - Hydrologie;'6) Filippi C., Lavie J., Seguin J. J., 1989. - Le Iogiclel VIKING. Calcul des trajectoires dans Un aquifere. -Rapport interneBRGM 89 SGN 675 EEE Kinzelbach W., 1988. -The random walk method in pollutant transport simulation. Groundwater flow;8,nd Qu-· aIity modeUing (ed. by E. Custodio et aI.). - 272-245. D. Reidel Dordrecht Pollock D; W., 1988. - Semi-analytical computatitm' of path lines for finite difference models. ':'Ground-water, vol. 26, n° 6 . " ,'. Thiery D., .1988. - Calculation of natural aquifer -recharge from rainfall \Yith an. unsaturated zone model solving. Richards equation. ~ Internat. Symp. AIRH on Interaction between groundwater arid surfaCe water. Ystad (Swei::len),: E. B[oomdahfEditor, pp. 45-57 . Thiery D., 1990. ~.:MBRINOS: Modelisation de l'evaPortanspiruion, du ruissel.Iement et de l'infiltraHon dans lao zone 'non saturee. - Unpublished report BRGM R 30623' E4U 48 90 Thiery D. & lung 0:, 1990. - Continuous modelisation of contaminant transport throtlgh the unsatu~ated zone withl a random walk particl.es model.' ........:. Conference on Hydrological research basins ,and the ,environment. Wagen-· .. ingen (The Netherlands) Thiery D. & lung 0., 1991. - Modelization hydrodynamique et hydrobiologique avec un schem particules d'un bassin pilote de gooepuration. - To be presented at the SHF conference, Sophia Antipolis Valocchi A. J., Quinodoz A. M.,198'9. - Application of the random walk ~ethod to simulate the transport of kine-tica11y absorbing solutes. - Groundwater contamination - proceedings of the symposium 'held during the, 3rd lASH Scientific assembly, Baltimore, MD-May 1989, IAHS Pub!. n° 185 '
a
• 316 •
