Stochastic modelling of geological heterogeneity and ...mmc2.geofisica.unam.mx/cursos/geoest/Articulos...Spatio-temporal process random walk Stochastic model Image of the geometry

22 Novembre 2002, Paris Colenco Power Engineering SA

GdR MOMAS, Journée thématique sur les modèles et méthodes probabilistes

Stochastic modellingof

geological heterogeneityand

propagation of uncertainty:

examples of applicationand open issues

Olivier Jaquet and Rabah Namar

Colenco Power Engineering SA


22 Novembre 2002, Paris



Plan

� Case study 1 : nuclear waste (F)- Objective- Model- Results- Open issues

� Case study 2 : nuclear waste (CH)...

� Case study 3 : hazardous waste (CH) ...

� Summary



Objectives

� Environmental impact assessment of the future surface facilities associated with the underground laboratory ANDRA

� Study of the system response in case of accidental contamination on site

Case 1: karst



Karstic aquifers: extreme heterogeneity

EpikarstConduit karstiqueCalcaires cariés

Oolithe de BureCalcaires de DommartinPierre ChâlineCalcaires lithographiques

Case 1: karst



Random walk approachKarst networks resulting from complex physical processes

exact location of conduits is unpredictable

Simulation of their probable location

Spatio-temporal process random walk

Stochastic model

Image of the geometry of karst networks

Case 1: karst



Stochastic modelling

� Complex phenomenology: coupled physical processes (transfer+dissolution)

� Stochastic approach: simplified description of the formation of karstnetworks

� Reinforced random walk : non-linear equation of Langevin

ξ(t) t) Θ(X(t), +t) v(X(t)),= dt

dX(t)

Case 1: karst



Stochastic modelling

� Simulation: 2D stochastic simulation of karstic networks

� Lattice gas method: discrete simulationon square grid

� Displacement of particles according to local rules = simulation of geometry at regional scale

),D(∆WΘ∆t)t,v(X∆X s)(t,iii

kl,i

kl,i λα⋅⋅+⋅=

Case 1: karst



Ornain

Rau

de

Nan

t

Saulx

Ornel

Cousance

ChevillonOsne

Orge

Orm

anço

n

Barboure

Perte-Orge

Perte-Ormancon

5023-s

5059-s

5060-s

5072-s

5082-s

5096-s

5102-s

5114-s

Fontaine-Etue

Jardin-Le-Moine

Les-Grdes

Sce-Mourot Sce-Faille

Sce-Longeaux

5056

5073-s

4627/fig_40_traj_site.eps/shu/29.03.01

Site ANDRA

Source

Réseau calcaires

lithographiques

Faille

Trajectoire10 km

Altitude451.00395.50340.00248.50229.00173.50118.00

785 830 835 840790 795 800 805 810 815 820 8251080

1085

1090

1095

1100

1105

1110

1120

1115

1125

1130

Mandresen-Barrois

Perte

Perte

5059-s

5060-s

5072-s

5102-s

5114-s

Orge

Orm

5073-s

BAR-LE-DUC

Case 1: karst

Histogramme des temps de parcours des particulessortant par les rivières

0

5

10

15

20

25

30

35

40

t < 2.5 2.5 < t < 10 10 < t < 100 100 < t < 1000 1000 < t < 10000 t >10000

Temps en années

Nom

bre

de p

artic

ules

t min : 1.032 anst max : 5 372 770 ans

Histogramme des temps de parcours des particulessortant par les sources situées à proximité du site

0

2

4

6

8

10

12

14

16

t < 0.1 0.1 < t < 1 1 < t < 100 100 < t < 1000 1000 < t < 10000 t >10000

Temps en années

Nom

bre

de p

artic

ules

t min : 0.0002 anst max : 16855 ans

Histogramme des temps de parcours des particulessortant par la source du Mourot (au Nord)

0

2

4

6

8

10

12

14

t < 0.5 0.5 < t < 10 10 < t < 100 100 < t < 1000 1000 < t < 5000 t >5000

Temps en années 4627/fig_41_histo_site.eps/27.03.01

Nom

bre

de p

artic

ules

t min : 0.24 anst max : 6784 ans

Stochastic geometry + numerical modellingfor groundwater flow and particle tracking

Rivers : 43 % < 10 a

Springs (site) : 32 % < 1 a

Springs (North) :27 % < 100 a



Open issues

� Conditional random walk

� Improving physics of coupled effects� Stabilisation of the simulated networks

Case 1: karst



Objective� Evaluation of host rock storage capabilities

Case 2: marl



Geostatistical modellingCase 2: marl



Monte Carlo ApproachBorehole data and related information

Geostatistical model of K

Conditional simulation of K

Transient groundwater flow model

Distribution of performance criteria(fluxes, travel times, ...)

uncertainty of modelling results

o/3dgeomo2.eps/shu/18.11.02

Case 2: marl



Results� Volumetric flow through repository

Case 2: marl



Open issues

� Classical Monte Carlo : lack of realisations

� Convergence of results (rare events)

� Alternatives - direct methods- quasi-Monte Carlo methods- ...

Case 2: marl



Objectives

� Application of Plurigaussian simulation method (oil industry) to a hazardous waste landfill

� Characterisation of lithofacies for numerical modelling of contaminant transfer

Case 3: molasse



Plurigaussian approach

1 : Y(x) < yc 2 : Y(x) > yc and Z(x) < zc 3 : Y(x) > yc and Z(x) > zc

Y(x) Z(x)

1

1 1

z

ypy

py pz

yc zc

pz

3

2

Case 3: molasse



Geostatistical modelling

9503

-100

0

Dept

h [m

]

0 100 200 300 400 500 600

Distance [m]

Cross-Section in the Simulated Volume

LithoType

Meanderbelt sand

Levee sandstone

Crevasse sand

Mud Palaeosols

Overbank sand

Lacustrine silt

figure25b_cut.eps/shu/06.12.01

Case 3: molasse



Open issues

� Operational (fast) techniques of upscaling transfer properties for large unstructured and heterogeneous grids (presence of faults)

� Inverse methods for selecting realisations of geostatistical models

Case 3: molasse



Summary of open issues

� Conditional random walk (with memory)

� Random walk for physical processes with coupled effects

� Efficient methods for propagating uncertainty of geological and transfer properties into numerical modelling results

Documents

Stochastic modelling of geological heterogeneity and ...mmc2.geofisica.unam.mx/cursos/geoest/Articulos...Spatio-temporal process random walk Stochastic model Image of the geometry