Stochastic discrete model of karstic networks

Advances in Water Resources 27 (2004) 751–760

www.elsevier.com/locate/advwatres

Stochastic discrete model of karstic networks

O. Jaquet a,*, P. Siegel a, G. Klubertanz a, H. Benabderrhamane b

a Colenco Power Engineering Ltd., Mellingerstr. 207, Baden 5405, Switzerlandb Agence Nationale pour la Gestion des D�echets Radioactifs (ANDRA), 92298 Chatenay Malabry, France

Received 10 January 2003; received in revised form 30 July 2003; accepted 25 March 2004

Available online 7 June 2004

Abstract

Karst aquifers are characterised by an extreme spatial heterogeneity that strongly influences their hydraulic behaviour and the

transport of pollutants. These aquifers are particularly vulnerable to contamination because of their highly permeable networks of

conduits. A stochastic model is proposed for the simulation of the geometry of karstic networks at a regional scale. The model

integrates the relevant physical processes governing the formation of karstic networks. The discrete simulation of karstic networks is

performed with a modified lattice-gas cellular automaton for a representative description of the karstic aquifer geometry. Conse-

quently, more reliable modelling results can be obtained for the management and the protection of karst aquifers. The stochastic

model was applied jointly with groundwater modelling techniques to a regional karst aquifer in France for the purpose of resolving

surface pollution issues.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Karst geometry; Coupled effects; Stochastic model; Numerical simulation; Finite elements

1. Introduction

Karst is defined as an irregular and disordered land-

form exhibiting particular hydrological characteristics

such as a coarse hydrographic network, point infiltra-tions, very large springs, etc. These characteristics may

form in highly soluble rocks, like sedimentary carbonate

rocks, with well developed secondary porosity (i.e., fis-

sures, fractures, channels and conduits).

Across the world, sedimentary karstic formations

constitute aquifers with important water reserves. It is

estimated that 25% of the global population is supplied

largely or entirely by groundwater from karst aquifers[14]. These aquifers are characterised by extreme spatial

heterogeneity due to the presence of networks of highly

permeable channels and conduits embedded in low-

permeable fractured rocks (matrix). The geometry of the

highly permeable drainage network considerably influ-

ences the hydraulic behaviour and transport of pollu-

tants in the entire karst aquifer. Compared to other

types of aquifers, their specific spatial structure renders

*Corresponding author. Tel.: +41-56-483-1576; fax: +41-56-483-

1881.

E-mail address: [email protected] (O. Jaquet).

0309-1708/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2004.03.007

them particularly vulnerable to pollution as contami-

nants can be transported over long distances with little

dilution [38].

Knowledge of the geometry of the conduit network is

essential for any numerical modelling of flow andtransport in karstic systems. Numerous authors [1,5,9–

11,16–18,32] have proposed deterministic methods for

the modelling of the formation of karst. These models

are complex and difficult in their application because

they require the definition of many parameters and

would lead to unacceptable run-times for industrial

applications when modelling the geometry of karst

aquifers at a regional scale in three dimensions.An alternative approach for modelling karst geome-

try is the use of geostatistical simulation methods [4,31].

These methods are applied in the modelling of frac-

ture systems in the context of fluid flow studies for

fractured rocks [2,3]. To our knowledge, few attempts

have been made to develop geostatistical models

for karst structures [6,21,23]. With these spatial models

it is difficult to capture the karstic complexity as therelevant physical processes are neglected. Either the

resulting geometry is too simplistic or the models fail

to reproduce the intrinsic connectivity of karstic net-

works.

mail to: [email protected]

752 O. Jaquet et al. / Advances in Water Resources 27 (2004) 751–760

In order to overcome these difficulties, a stochastic

(spatio-temporal) approach is proposed for the model-

ling of the geometry of karst aquifers. The model allows

for a simplified description of physico-chemical pro-

cesses which is then used to generate karstic networks byapplying a formulation of the Langevin equation. This

stochastic differential equation is solved using a

numerical method of the lattice-gas cellular automaton

type. This simulation method provides for the discrete

characterisation of the spatially variable behaviour of

geometric and hydraulic properties encountered in karst

aquifers. The corresponding enhancement of the input

parameter fields leads to more reliable hydrogeologicalmodelling results.

The stochastic approach was applied to a regional

karstic limestone aquifer in France. The vulnerability

of this aquifer with respect to potential surface pollu-

tion related to an industrial project was assessed using

hydrogeological modelling; the karstic complexity of

the aquifer was incorporated in an explicit (discrete)

manner within the framework of the modelling pro-cess.

The paper first describes the conceptual model chosen

for the formation of karst geometry, followed by the

development of the stochastic model principles. Then,

its numerical implementation is explained and finally an

application of the model to a real case study is pre-

sented.

2. Conceptual model

The development of a model for the geometry of

karstic networks requires a conceptualisation of theirformation. Three physico-chemical mechanisms are rel-

evant for modelling karstic conduits: advection, disper-

sion and dissolution [25]. These mechanisms can be

described using the classical solute transport equation

[34]. They are governed mainly by parameters linked to

the geometry of the medium, i.e., hydraulic conductivity

and porosity. A special feature of karstic rocks is the

interdependence of flow and medium geometry by wayof dissolution. The so-called feedback loop [27,28] is due

to the following coupled effects:

• the flow velocity depends on the hydraulic conductiv-

ity and on the hydraulic gradient;

• the hydraulic conductivity is a function of the geom-

etry of the voids;

• the direction and the size of the velocity vector influ-ence this geometry through the dissolution phenom-

ena.

From a conceptual point of view, the initial rock

body is assumed to contain connected discontinuities

(fissures, fractures, etc.) within which groundwater flow

can take place. The water circulation tends to concen-

trate preferentially along channels issuing from the

intersections of neighbouring discontinuities. Progres-

sively, conduits are created under the influence of dis-

solution phenomena. With time, a karstic network ofregional scale can form when the hydrogeological con-

ditions are favourable.

The proposed model takes into account dissolution

phenomena as the fluid is assumed to contain particles––

i.e., fictive entities representing the dissolution pro-

cesses––that hold the property of ingesting the traversed

rock. The flow velocity is assumed large enough in order

that the variation of the solute concentration remainsconstant.

Under the influence of the flow field, these particles

are preferentially dispersed along the more permeable

discontinuities of the medium: the more heavily fis-

sured zones are first attacked and conduits are formed

rapidly within these zones. The larger the conduit, the

more the following particles have the tendency to se-

lect these conduits. Thus, the size of a given conduit isa function of the number of particles that have passed

through it. Particles gradually consume the rock along

preferential pathways, and a heterogeneous network

of conduits is created along which flow is concen-

trated.

3. Langevin equation

Kolmogoroff [29] demonstrated that the employment

of deterministic and probabilistic solution approaches

for the transport equation (without coupled effects) is

equivalent from a purely formal point of view. A sto-

chastic process of the random-walk type can be applied

for solving the transport equation. Such processes cor-respond to a Lagrangian representation of the forma-

tion of karstic networks, i.e., their evolution is obtained

by describing the movement of particles as a function of

time. This description can be given by the stochastic

differential equation of Langevin [30]. Its general form

can be expressed as follows [15]:

dX ðtÞdt

¼ vðX ðtÞ; tÞ þHðX ðtÞ; tÞnðtÞ ð1Þ

where X ðtÞ is the random function of particle position

(m), vðX ðtÞ; tÞ the fluid velocity at position X ðtÞ and time

t (m s�1), HðX ðtÞ; tÞ the fluctuation term (–), and nðtÞ thewhite noise (m s�1).

When modelling the formation of karstic conduitnetworks, the fluid velocity (deterministic term) repre-

sents the flow velocity of karstic waters and the proba-

bilistic term is assumed to describe simultaneously the

spatial heterogeneities of the medium and the effects of

the dissolution. Due to its strong variability [25], the

opening of fissures and karstic conduits is selected as the

O. Jaquet et al. / Advances in Water Resources 27 (2004) 751–760 753

main parameter. The value of this dynamic parameter at

a given location is a function of the initial opening and

of the number of particles that have passed through it by

a given point in time.

The number of particles is related to the effects ofdissolution. The effect of spatial heterogeneity is mod-

elled by preferentially directing, in a probabilistic man-

ner, the particles into those conduits with larger sizes.

With the gradual increase of any conduit’s size, an

increasing number of particles tend to travel through it.

This results in further widening of the conduits through

dissolution which makes them even more attractive to

subsequent particles. These coupled effects make itpossible to model the feedback loop previously de-

scribed.

One consequence of the spatial and temporal depen-

dence of the opening sizes of fissures and karstic con-

duits is that the Langevin equation becomes non-linear,

which precludes its solution using analytical methods.

The opening parameter is a function of the random-

walk process history––a given particle has the possibilityto follow the same trajectory as the preceding one––

which means that the particles recognise previously

travelled trajectories to some degree. In probability

theory this random process with memory (i.e., a non-

Markovian process) corresponds to a reinforced random

walk [7].

The application of the Langevin equation to the

modelling of karstic networks, therefore, requires anumerical method.

4. Numerical approach

The basic idea for modelling the formation of kar-

stic geometry is to solve the Langevin equation by

applying the lattice-gas cellular automata formalism in

terms of spatial and temporal discretisation. This

numerical approach integrates the spatial and temporal

processes required to capture the geometry of karstic

networks.

Cellular automata are discrete models for describingdynamical phenomena. These models evolve in discrete

time steps on a regular lattice by nearest-neighbour

interactions according to simple rules. Lattice-gas cel-

lular automata are models of a gas on a lattice in which

particles jump from one site to another at each time

step. The conservation of mass and momentum is at-

tained by permitting particles to collide when they meet.

Lattice-gas cellular automata are applied as simplemodels for characterising complex hydrodynamics

[8,37].

The karstic model differs from the lattice gas ap-

proach used in hydrodynamics as no direct interactions

between particles occur and hence no exclusion principle

holds on the lattice. Collisions between particles are

replaced by probabilistic interactions of the particles

with the trajectories of the ones that have preceded

them. This means that at each time step and for each

lattice site, particles tend to jump probabilistically to-

wards the direction of the largest opening (fissure orconduit). The tracking of the particle trajectories con-

stitutes a network of conduits, or, in other words, a

model of the karstic geometry. The combination of

stochastic and lattice gas concepts and the integration of

the main physical mechanisms of karst formation make

the proposed simulation method a modified lattice-gas

cellular automaton.

For karst simulation, one starts with the Langevinequation using the stochastic differential form of Ito

[15]:

dX ðtÞ ¼ vðX ðtÞ; tÞdt þHðX ðtÞ; tÞdW ðtÞ ð2Þ

where W ðtÞ is the Brownian motion (m).

The discrete form of this equation is expressed as

follows:

X ðtiþ1Þ � X ðtiÞ ¼ vðX ðtiÞ; tiÞðtiþ1 � tiÞ þHðX ðtiÞ; tiÞ� ðW ðtiþ1Þ � W ðtiÞÞ ð3Þ

The model is two-dimensional with a spatial discreti-

sation given by a regular square lattice. The use of a

square lattice is adequate since it is karstic geometry

rather than hydrodynamics that is being modelled. Themodel’s velocity field is simplified and assumed to be

known. For each time step, the particle can move up to

two times the lattice spacing; the first jump is related to

the flow velocity while the second is linked to effects of

heterogeneity and dissolution. Since the lattice is regu-

lar, H equals one. Accordingly, the discrete form of the

equation applied is written as follows (note that X ðtiÞ iswritten as Xi):

X lþðnþrÞk;kþðmþsÞkiþ1 � X l;k

i ¼ vn;mðX l;ki ; tiÞDti þ W ðr;sÞ

i k ð4Þ

where l; k are the discretisation indices of the lattice

according to directions x and y, ðn;mÞ 2 fð1; 0Þ; ð0; 1Þ;ð�1; 0Þ; ð0;�1Þ or ð0; 0Þg the indices of the flow velocity

vector, W ðr;sÞi the random vector with values ðr; sÞ (–),

and k the spacing of the lattice (m).

The memory effect is solely directional; i.e., the par-ticles tend to follow the previous trajectories with a

constant displacement value. The values of the random

vector, W ðr;sÞi , correspond to one of the doublets [1 0],

[0 1], [)1 0] or [0 )1] with the probabilities p, q, r and

1� p � q� r. The directional probability is propor-

tional to the cube of the values of the neighbouring

conduit diameters. This choice is based on the Poiseuille

law where transmissivity and fracture opening are linked

through a cubic function [33]:


p ¼ ð1þ ah1Þ3P4c¼1

ð1þ ahcÞ3; q ¼ ð1þ ah2Þ3P4

c¼1

ð1þ ahcÞ3;

r ¼ ð1þ ah3Þ3P4c¼1

ð1þ ahcÞ3; 1� p � q� r ¼ ð1þ ah4Þ3P4

c¼1

ð1þ ahcÞ3

ð5Þ

where a is the persistence (–), and hc the diameter of

conduit c (m).

The persistence is defined as a multiplying factor

whose role it is to relate the initial medium heterogeneity

to the geometry of the simulated conduits. The value ofpersistence was calibrated using geometric data of the

karstic networks of the Sieben Hengste in Switzerland

[20]. With values ranging from 1 to 10 it was possible to

reproduce the total length of the Sieben network.

The flux of dissolved matter per unit volume of rock is

influenced by [25]: (a) geometry of the karstic medium,

through the ratio Sp=Vr where Sp is the contact surface

between water and rock and Vr the volume of rock, (b)flow velocity and (c) saturation concentration. As the

range of variability for the ratio Sp=Vr can reach several

orders of magnitude, it is assumed that the dissolution is

mainly influenced by the geometry of the karstic medium

(i.e., the opening of fissures and karstic conduits) and the

flow velocity. The memory effect allows the coupling of

the widening of the conduits with the number of parti-

cles. As the conduits grow wider, more particles tend torun through them, which describes the relationship be-

tween the amount of flowing water and the conduit size.

The flow velocity is assumed large enough to render the

effects of the saturation concentration negligible.

Therefore, the following linear relation considers the

enlargement of fissures and karstic conduits as a function

of the number of passages of particles:

hic ¼ I � NP ic þ hinitc ð6Þ

where I is the opening increase per passage (m), NP ic the

number of passages of particles at time i for a conduit c(–), and hinitc the initial opening (m).

The range of values for the conduit opening stems

from diameters measured in the network of the Sieben

Hengste. The smallest observed diameter was 0.05 m

and 95% of the observed diameters were less than 6 m.

The range of the number of passages was taken from

network simulations performed for the Sieben Hengste.

These ranges lead to an estimate of 6 · 10�4 m for theincrease in the diameter of an opening for every passage

of a particle.

For the simulation, steady-state conditions are con-

sidered to be reached when no additional conduits are

created; i.e., the total conduit length remains constant.

In addition, the maximum diameter is not to exceed 6 m.

The hydraulic conductivity of the conduits is obtained

using the following relation [25]:

Kc ¼ 2 log1:9

r

� � ffiffiffiffiffiffiffiffiffi2ghc

pð7Þ

where Kc is the hydraulic conductivity of conduit c(m s�1); r the relative rugosity (equal to 0.2) (–), and gthe gravity acceleration (m s�2).

5. GARST

A 2-dimensional simulation code (GARST: ‘‘GAz sur

R�eseau pour la Simulation de la Tubulure karstique’’)

was developed for the modelling of karstic conduit

networks. With the help of this spatial-temporal simu-

lation method, the tracking of the particles with time

from their entry into the network to their exit is guar-anteed. The trajectories drawn by the passages of the

particles in the network constitute a model of the karstic

geometry. The following assumptions were made for this

model:

• The karst network is 2-dimensional and horizontal.

• The infiltration zone (left side) is homogeneous with a

steady-state flow-rate.• The advection velocity of the particles is constant.

• The increase of the conduit diameter due to dissolu-

tion is linear (independent of the solute concentra-

tion).

• The right side is the exfiltration zone.

When initial conditions are homogeneous, it is as-

sumed that the initial network is composed of fracturesdistributed along all the segments of the lattice, whereby

a segment connects two nodes of the lattice. Each frac-

ture possesses the same initial opening. This fracture

becomes a conduit only after the passage of the first

particle.

The lateral boundary conditions are of a periodic

type: a particle exiting on one side is replaced by a

particle entering the opposite side of the model. Thelattice gas model comprises three parameters: (a) en-

trance flow-rate of particles, (b) particle advection-

velocity and (c) persistence. Variations of the persistence

make for a distinct network geometry for each simula-

tion (see Fig. 1).

The comparison of the network of conduits as ob-

tained from numerical simulations with the geometry

surveys of speleologists remains a difficult task. Even ifsimulated conduits are considered at no less than the

centimetre scale, the survey data is restricted to actually

explored conduits whereby accessibility requires a min-

imum diameter of about 0.3 m. A simulation method

conditioned to the results of a geometry survey could

provide a solution for this issue. During the course of

Fig. 1. Synthetic case with velocity along the horizontal direction after 1000 time steps for a model size of 10,000 nodes, with persistence values

of 1 (a) and 10 (b).


simulation, the trajectories of the particles should be

capable of reproducing exactly the geometry of the ex-

plored network as well as its hydraulic properties. Some

heuristic tests have been conducted already, but a gen-

eral approach remains to be developed.

6. Case study

As part of the realisation of the Meuse/Haute Marne

underground research laboratory, ANDRA (National

Radioactive Waste Management Agency of France) has

conducted an environmental impact assessment study

with respect to the surface facilities associated with the

laboratory. The consequences of potential aquifer pol-lution related to surface activities (mainly during con-

struction or transportation) need to be evaluated

regarding groundwater supplies to the popula-

tion downstream of the site. This safety evaluation

requires an understanding of the groundwater circula-

tion in the karst aquifers located in the vicinity of the

site.

A hydrogeological model was developed by Jaquetet al. [19] for the Barrois Karstic Limestones (BKL).

This model provides the tools to predict the aquifer re-

sponse in the case of an accidental pollution on site. In a

first step it required the definition of hydrogeological

units in terms of their geometric and hydraulic proper-

ties. This information was drawn from geological and

hydrogeological studies, including the interpretation of

borehole data, seismic surveys and speleologic investi-gations. In the given case, a multitude of information of

varying quality was available from different sources, see

e.g., [12,35].

As shown in Fig. 2, seven hydrogeological units were

considered in the region (units 1–7) as well as some

important faults. Numerous springs throughout the

model area also entered the model. The epikarst (unit 2)

is a thin, fractured and highly permeable layer on top of

the limestone formations. It outcrops at the surface if

unit 1 is absent. Units 1 and 2 concentrate the water flow

close to the surface and direct it towards the karstic

networks of the limestone units (unit 3: calcaires cari�es,unit 5: calcaires de Dommartin and unit 7: calcaires

lithographiques). These units function as aquifers in the

BKL formation. They are highly karstified and each

presumably contains a karstic network of conduits

embedded in a low-permeable rock matrix (fractured

limestones). The three units are separated by semi-per-

meable layers of Oolithe de Bure and Pierre Chaline, i.e.,

marl units (non-karstic) which may be crossed via ver-tical karstic conduits connecting the networks.

The simulation of the karstic networks in the BKL

was performed in two dimensions using the code

GARST (cf. Section 5). Regarding the discretisation in

space, the choice of the spacing of the lattice (500 m) was

based on the average density of observed karstic mani-

festations for the BKL both at the surface and in

boreholes. The available information was integrated asfollows [19]: (a) the velocity field was deformed

according to a regional hydraulic potential field so that

during the simulation, the generated karstic conduits

tended to be attracted to the locations of the springs,

and (b) geological heterogeneities were introduced as

initial conditions. The geological information of concern

for the karst geometry of the site consisted solely of: (a)

the locations of major faults and (b) the density ofkarstic manifestations observed on the surface and in

boreholes. No information about the karstic network, in

terms of conduit geometry, was available at the outset.

This geometric information was introduced into the

model by increasing the initial opening of the lattice

segments concerned, i.e., to 0.5 m for faults and to a

diameter of 0.05–0.5 m according to the density of

Table 1

Hydraulic conductivity of conduits obtained by karstic simulation

Number of

passages NPa

Diameter (m) Hydraulic conductivity

Kc (m/s)b

1–10 0.06 2

11–100 0.11 3

101–1000 0.64 7

1001–10000 6.00 21

aUpper limit of range is used for diameter calculations with Eq. (6).bObtained using Eq. (7).

Fig. 2. 3D finite element model of the Barrois Karstic Limestones with a cross-section displaying the hydrogeologic units and the karstic networks.


karstic manifestations. This means that conduit loca-

tions were enforced in a preferential manner; that is,

particles were inclined to follow major faults in the re-

gion as well as be drawn toward places with karstic

observations.The effect of including information in terms of initial

conditions is that, during simulation, the importance of

the random component (during particle displacement)

decreases faster with time as compared to homogeneous

initial conditions; more preferential pathways are cre-

ated. The pathways followed by the particles injected at

the upstream boundary of the modelled region reached,

after �20,000 time steps, steady-state conditions. Atthat time no additional conduits were created and the

total conduit length remained constant. In other words,

karstic simulation was in equilibrium with the available

initially input information. The CPU time required for

such a simulation was of the order of minutes.

Karstic network simulations were performed inde-

pendently for each of the three karstic layers, consider-

ing the information available for each unit. At the end,vertical conduits from each of the networks towards the

surface were generated at locations of karstic manifes-

tations at the surface. For the hydraulic conductivity of

the conduits, four classes were attributed with respect to

the simulated conduit diameter and, therefore, in

dependence of the number of particle passages through

each conduit (cf. Table 1). These hydraulic conductivity

values are comparable to values obtained by other au-thors [22,25].

The 3D finite element grid was generated using a

multilayer technique, based on a 2D grid, to integrate

the available geometric information. The thickness of

the layers was derived from boreholes, seismic surveys

and DEM (Digital Elevation Model) data. As some

layers are partially eroded, discontinuous or even miss-

ing over large parts of the model area, interpolation by

kriging was applied where necessary to ensure unit

continuity in three dimensions. The final grid contained

more than 270,000 finite elements (combining 1D, 2D

and 3D elements to discretise karst conduits, faults and

layers, respectively) and more than 309,000 nodes (cf.

Fig. 2).Transient groundwater flow, as caused by variable

infiltration, with a free surface was assumed to be gov-

erned by Darcy’s law. Leakage between aquifers and

rivers was also accounted for in the model. The com-

bined discrete channel and continuum approach [24] was

applied for the numerical modelling of groundwater

flow. This method was first proposed by Kiraly [26] to

simulate groundwater flow in karstic systems. Later, themethod of Kiraly was generalised by Perrochet [36] who

developed a geometrical framework for 4D space-time

finite elements with the capability of embedding ele-

ments of lower dimensions. With this method allowing

for the simultaneous incorporation of 1D, 2D and 3D

elements, groundwater flow modelling was performed

for the high-permeable conduits embedded in the low-

permeable units of the BKL.


The groundwater flow model was calibrated using

piezometric and flow-rate data from the field. The cali-

bration consisted of tuning selected parameters in order

to match the field observations with the modelling re-

sults. This procedure delivers estimates of parameterscalibrated to the data and, therefore, improves the initial

values available for the model parameters.

The initial hydraulic conductivity values of the con-

duits were taken from the simulation of the karst

geometry (cf. Table 1). The initial hydraulic conductiv-

ities for the other units in the model were derived from

the available hydrogeologic information. The calibra-

tion results are shown in terms of differences in thehydraulic potentials in Fig. 3. The relatively large dis-

crepancies between model results and observations

(differences between )3.5 and 15.6 m) with an average of

about 2.5 m are mainly related to: (a) not knowing the

exact locations of the conduits, (b) the coarse discreti-

sation of a rough karstic topography and (c) lateral

variations in the hydraulic conductivity of the layered

units which were disregarded.This calibration of the model parameters with field

measurements was the basis for estimating the hydraulic

parameters for the various hydrogeologic units (cf.

Table 2). Although some uncertainty related to the

complexity of the karstic system remains, the results can

be considered in good agreement with the observations.

Without an explicit description of the karst geometry,

the achievement of such modelling results would nothave been possible.

Fig. 3. Calibration results: map of discrepancies in hydraulic potentials p

neighbourhood of the site.

The method of particle tracking was applied for the

determination of potential exfiltration zones in case of

an accidental pollution on site. The locations of the first

zones affected downstream were identified with the

model (see Fig. 4). The tracking was performed using avelocity field from the groundwater flow modelling. The

particle displacement was calculated in a deterministic

manner as particles followed the fastest trajectories of

the velocity field. This method, honoring only the

advective part of the transport mechanism, allows solely

the calculation of trajectories and travel times. The

derivation of concentration curves for exfiltration zones

is foreseen; additional work is required using a transportmodel in order to account for dispersion effects observed

in karst aquifers [13].

The fastest particle travel-times obtained––from the

ANDRA site to the Mourot spring (located in the

North; cf. Fig. 4)––correspond to apparent velocities

between 270 and 570 m/day. These velocities fall within

the range of apparent velocities of 100–2000 m/day de-

duced from tracer tests conducted for the BKL.

7. Conclusions and perspectives

A stochastic model is proposed for the discrete sim-

ulation of the geometry of karstic networks at a regional

scale. Based on the Langevin equation, the model inte-grates the relevant physical processes controlling the

formation of karstic networks as well as the effects of

resented as ‘‘calculated potential minus measured potential’’ in the

Table 2

Hydrogeologic units of the Barrois Karstic Limestones with their hydraulic properties

Hydrogeologic units Hydraulic properties

Initial values Calibrated values

1. Limestone, clay and sand 10�3 m/s 4 · 10�3 m/s

2. Epikarst 10�3 m/s 6 · 10�3 m/s

3. Calcaires cari�es (limestone aquifer) Matrix: 10�8 m/s conduits (cf. Table 1) Matrix: 6· 10�9 m/s conduits: 6, 7, 9, 12 m/s

4. Oolithe de Bure (marl) 10�9 m/s 10�9 m/sa

5. Calcaires de Dommartin (limestone aquifer) Matrix: 10�8 m/s conduits (cf. Table 1) Matrix: 6· 10�9 m/s conduits: 6, 7, 9, 12 m/s

6. Pierre Chaline (marl) 10�9 m/s 10�9 m/sa

7. Calcaires lithographiques (limestone aquifer) Matrix: 10�8 m/s conduits (cf. Table 1) Matrix: 6· 10�9 m/s conduits: 6, 7, 9, 12 m/s

Faults 10�2 m2/s 10�2 m2/sa

a Parameter not estimated during calibration.

Fig. 4. Location of exfiltration zones for particles with starting points in the vicinity of the ANDRA site.


medium discontinuities such as fractures. Simulations of

the model are carried out with a modified lattice-gas

cellular automaton. They allow the characterisation of

the extreme spatial variability inherent to karst aquifers

in terms of geometry and hydraulic conductivity. These

discrete simulations provide geometric and parame-

ter input when modelling groundwater flow and trans-

port in karst aquifers using a specific finite elementmethod.

The stochastic approach was applied in the vulnera-

bility assessment for a karst aquifer with respect to po-

tential surface pollution related to an industrial project.

This study has shown the operational capabilities of the

proposed approach for modelling karst geometry at a

regional scale. In particular, through model calibration,

the results could be considered in good agreement with

field observations. This achievement is the consequence

of explicitly accounting for the karst geometry in the

groundwater modelling approach.

Several aspects call for future investigations: (a) the

conditioning of karstic simulations with geometric andhydraulic data, (b) the application of variable velocity

fields encountered in fractured media when simulating

karstic geometry and (c) a more physical description of

the dissolution process in order to relate the simulation

to the evolution of karstic systems.


Because the proposed stochastic approach exhibits

acceptable run-times, an expansion into the third

dimension is possible. The full potential of the approach

remains to be discovered, particularly in other domains

of the earth sciences.

Acknowledgements

We would like to thank Michel Maignan of the

Universit�e de Lausanne for his advice and support;

Pierre-Yves Jeannin and Lazlo Kiraly of the Centre

d’Hydrog�eologie de Neuchatel for helping us under-stand karstic hydrogeology; Christian Lantu�ejoul of theCentre de G�eostatistique de Fontainebleau for his

suggestions on probabilistics; Philippe de Forcrand of

the Eidgen€ossische Technische Hochschule Z€urich for

his introduction to lattice gas simulation; ANDRA for

financially supporting the realisation of this paper;

Val�erie Pot of the Centre INRA-INA for her con-

structive and detailed review and finally the lateGeorges Matheron, former Director of the Centre de

G�eostatistique, for an original comment made back in

1993.

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Stochastic discrete model of karstic networks