I DENTIFICATION OF main flow structures for highly CHANNELED FLOW IN FRACTURED MEDIA by solving the inverse problem R. Le Goc (1)(2), J.-R. de Dreuzy (1)

IDENTIFICATION OF main flow structures for highly CHANNELED FLOW IN FRACTURED MEDIA by solving the inverse problem

R. Le Goc (1)(2) , J.-R. de Dreuzy (1) and P. Davy (1)(1) Geosciences Rennes, UMR 6118, CNRS,

Université de Rennes 1, Rennes, France,

(2) Itasca Consultants SAS, Lyon, France ([email protected])

16 April 2008 R. Le Goc, EGU 2008 2

Purpose Aquifer

characterization Simplified flow model Preferential flow paths

Hydrologic data Piezometric heads Flow measurements

Difficulties Sparse and scarce data Heterogeneous media natural media


Classical inverse problem methodology Aquifer modeled as a coarse grid

Parameters: grid permeability or geostatistical characteristics

Getting around the lack of data Less parameters

Small zone number Homogenization

More information A priori values Qualitative geological data

Problems for fractured media: Flow is channeled in a very few

structures Importance of connectivity

128x128 permeability grid. Permeability values are log correlated with a correlation length = 10.


Inverse problem for fractured media Parameterization = 1-5 channels +

background permeability channel position and transmissivity Background permeability representing all

second-order structures Getting around the lack of data

Iterative parameterization (with an increasing number of channels).

Iterative regularization based on previously identified structures.

Specificities : Identification of hydraulically

predominant fractures Method relevant to highly channeled flow

conditionsFlow in a channeled media. Flow path width is proportional to the transmissivity.


Inversion algorithm First step

data

2i ifield model

model1 iteration

N

obj ii h

d dF p

Objective Function (classical least-square formulation):

Solving direct problem using HYDROLAB, hydrogeological simulation platform

Parameter estimation in optimizing Fobj using simulated annealing


Inversion process Second step

data

0

2i ifield model

model1 iteration

2j j0 model

j1 0

param

N

obj ii h

N

j

d dF p

p p

Objective Function with regularization term

Regularization term: values from previous step as a priori values


data

0

1

2d dfield model

model1

2j j0 0

j1 0 0

2j ji-1 i-1

j1 i-1 1

+

param

iparam

N

obj dd h i

N

j

N

j i

d dF p

p p

p p

Inversion process i-th step Objective Function with

regularization term

Regularization term is build at each iteration

The refinement level is controlled by the information included in the data (accounting for under- and over-parameterization)


Stopping criteria for inversion Classical stopping criteria For each iteration we calculate (/Tsai et al.,

2003/) The residual errors (RE)

The parameters uncertainties (PU)

The structure determination between parameterization n and n-1

1

T2

sdata param

h h

N NPU J J

RE

data2i i

field model

1

N

ii h

d dRE

data2i i

model n model n-1

1

N

i data

d dSD

N


Presentation of synthetic test cases Synthetic 2D fracture

networks 5 x 5m square domains Boundary conditions :

regional gradient + perturbations

25 steady hydraulic heads regularly spaced

16 April 2008 R. Le Goc, EGU 2008 10

Results

-2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

Abscissa

Ord

inat

e

00,11250,22500,33750,45000,56250,67500,78750,9000

0 1000 2000 3000-6

-4

-2

0

2

4

para

met

er v

alue

s

iteration

connected border 1 connected border 2 coordinate on border 1 coordinate on border 2 log of channel transmissivity log offracture transmissivity

0 1000 2000 300010-510-410-310-210-1100101102103104105106107

obje

ctiv

e fu

nctio

n va

lue

iteration

16 April 2008 R. Le Goc, EGU 2008 11

Results

0 50000 100000 150000 200000 250000 3000000,2

0,3

0,4

0,5

0,6

0,70,80,9

1

Ob

ject

ive

fun

ctio

n v

alu

e

iteration

-2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

Abscissa

Ord

inat

e

00,062500,12500,25000,50000,62500,75001,0001,2501,5001,7502,000

0 50000 100000 150000 200000

-6

-4

-2

0

2

4

6

8

10

12

14

pa

ram

ete

r va

lue

s

iteration

1st channel connected border 1 connected border 2 coordinate on border 1 coordinate on border 2 transmissivity

2nd channel connected border 1 connected border 2 coordinate on border 1 coordinate on border 2 inner point x inner point y transmissivity

matrix transmissivity

Solution analysis and generation of statistical results

16 April 2008 R. Le Goc, EGU 2008 12

Conclusion Efficient methodology

Parameterization adapted to highly channeled media Models based on flow channels Iterative parameterization Iterative regularization

Results consistent with data Global minima for models requiring few parameters Local minima for more complex models statistical solutions

Channel refinement controlled by data quantity Data partially sensitive to parameters Relation between model parameterization and data

availability (planned work)

Documents

I DENTIFICATION OF main flow structures for highly CHANNELED FLOW IN FRACTURED MEDIA by solving the inverse problem R. Le Goc (1)(2), J.-R. de Dreuzy (1)