modeling fractured rock mass properties with dfn concepts, … · 2020-04-20 · modeling fractured...

$: Modeling fractured rock mass properties with DFN concepts, … · 2020-04-20 · Modeling fractured rock mass properties with DFN concepts, theories and issues Contributions Jean-Raynald$
Modeling fractured rock mass

properties with DFN

concepts, theories and issues

Contributions

Jean-Raynald de Dreuzy, Olivier Bour, Géosciences Rennes, CNRS, France

Julien Maillot, Etienne Lavoine, Justine Molron, Diane Doolaeghe (PhD)

Raymond Munier, Jan-Olof Selroos, Diego Mas Ivars, Martin Stigsson, SKB, Sweden

Philippe Davy

Geosciences Rennes, Univ Rennes, CNRS, France

Caroline Darcel, Romain Le Goc

Itasca Consultants s.a.s., Ecully, France

14TH CARGESE SUMMER SCHOOL: JUNE 25TH – JULY 7TH 2018 FLOW AND TRANSPORT IN POROUS AND FRACTURED MEDIA

Impact of fractured rock on ….

2

𝑠𝜕ℎ

𝜕𝑡=

𝜕

𝜕𝑥𝐾𝜕ℎ

𝜕𝑥

3 fractures

per meter!!


Application example

Safety assessment for deep nuclear waste disposal

Site : area several 𝑘𝑚2, depth ~ 500 m

Canister Layout: 𝑘𝑚 scale


Application example

Assessing the properties of the (third) geological envelope

Observations: the largest database on fracture in the world


Application exampleAssessing the properties of the (third) geological envelope

Observations: little data compared to the geological complexity



Observations Models

Fracture network

model

Connected cluster

Flowing fractures

Predictions



Predictions

Permeability

Fracture density

Observations Models

Fracture network

model

7

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

10-3

10-2

10-1

100

Perm

eab

ilit

y (

Ke

q)

MM

PM

p32



PredictionsElastic modulus

Observations Models

Fracture network

model

0 1 2

10-1

100

Elastic

modulus

fracture density parameter


Application example

Safety assessment for deep nuclear waste disposal

Predict contaminant travel time through fractures in the rock mass

Fracture model Flow model

data


DFN

DFN as a basic concept for fracture media modeling

DFN

• Geological mapping

• Geophysics

• Hydro logging

• Deformation

• Seismicity• Statistical distribution models

• Stereology

• Statistical domains / intrinsic variability

• Deterministic conditioning

• Tests

• Stochastic models conditioned by data statistics

• Process/genetic model, whose statistics are emerging properties

• Discrete vs continuum, and upscaling

• Flow paths and connectivity

• Flow intensity (equivalent permeability)

• Transfer (geothermal, contaminant)

• Rock deformation (elastic)

• Rock strength

• Induced seismicity

Data

Conceptual

models

Medium

model

Prediction for

applications

Prior

knowledge

ECM


DFN methodology Medium description as a population of

discrete (and simple) fractures

▪ ‘Close to’ target systems – ‘easy’ integration

of data

▪ Made of both statistics and deterministic

objects

But basically a statistical model

▪ Extrapolate information between data

▪ Model intrinsic and extrinsic variability

Tool for predictions

▪ Bracket possible geological behaviors, or help

for understanding data/behavior

▪ Tool to find the critical parameters or length

scales that are required to improve prediction

DFN

DFN

Data

Conceptual

models

Medium

model

Prediction for

applications

Prior

knowledge

ECM


DFN methodology

data integration


Fractured rock

geology

Fracture object

‘idealized’

Fracture population

DFN

From fractured rocks to DFN

Roughly Planar discontinuity

resulting from rock failure.

Cracks, Joints, Faults, Shear

zones, Bedding planes

Controlled by in situ field

conditions and physical

processes of fracturing

Geologist vs Physicist vs

Mining-engineer description

emphasize different aspects

Lateral dimensions >> thickness

2D planar object

Geometrically defined by position, size, orientation, shape

Process-based defined by plow, transport, and mechanical properties

Density distribution

𝑛 𝐿, 𝑙, 𝜃, 𝜙, …

number of fractures per

unit volume 𝑉

• with a given size 𝑙,

• orientation 𝜃, 𝜙

• …

East

North

13

Data

Conceptual

models


Fractured rock

geology

Fracture object

‘idealized’

Fracture population

DFN


A thin volume

characterized by

Its surface per unit

volume (𝑝32)

Some properties

(aperture, stiffness of

fracture walls or filling

material)

An idealized object

With simple or complex geometry

Whose definition depends on size (e.g. small-scale fracture, large-scale fault zones)

Ideally, consistent with hydraulic and mechanical continuity

Density distribution

𝑛 𝐿, 𝑙, 𝜃, … d𝑙 d𝜃 =

Density term 𝑑 𝑥, 𝑦 ∗

Size distr. 𝑝𝑙 𝑙 d𝑙 ∗

Orientation 𝑝𝜃 𝜃 d𝜃 ∗

Others 𝑝…. … d…

East

North

14

Data

Conceptual

models


Fractured rock

geology

Fracture object

‘idealized’

Fracture population

DFN

DFNE 2018’s Fracture Size Seminar


East

North

15

• A balance between the system complexity and the facility

• to integrate data,

• and to provide a stochastic representations in these fracture

population models.

• The notion of fracture size is fully related to the transformation

fractured surface → fracture object

• Challenge (or wishes). We rely on the predictions of a schematic model

constrained by real data 😧4TH CARGESE SUMMER SCHOOL: JUNE 25TH – JULY 7TH 2018 FLOW AND TRANSPORT IN POROUS AND FRACTURED MEDIA

Data integration

Data from different support and scales

16

Outcrop mapping Number of 2D fracture traces 𝑡 per orientation range: 𝑛2d(𝑡, 𝜃) Range of scales 1-10 m (outcrops), 100m-10km (aerial photos)

Tunnel mapping 2D traces on the tunnel wall 𝑛2d(𝑡, 𝜃) 1D fracture intensity 𝑛1d(𝜃) for fractures that fully intersect the tunnel

(scale > 5m)

Borehole fracture intensity Number of fractures that fully intersect the core, per unit borehole length,

per orientation range: 𝑛1d(θ) Investigation scale = borehole length (~1km)

Fracture scale= borehole diameter (~80mm) to xxx

Geophysical data 3D information with fracture size and orientation

But: not complete and low resolution (how many fractures are revealed by

geophysics)?

Challenge: Reducing uncertainty of DFN models by conditioning to

geophysical data

Data


DFN model for data integration

Stereology

1. A 3D model

𝑛3d 𝑙, 𝜃 = 𝑑3d 𝜃 ∗ 𝑙−𝑎3d

2. Stereology rulesObservation parameters 𝑃, fracture parameter 𝐹

𝑛(𝑃) = න𝑎𝑙𝑙

𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑃, 𝐹 𝑛3𝑑 𝐹 𝑑𝐹

Φ: angle between the fracture and the observation structure

OutcropObservation parameter: nb of fracture trace length 𝑡 on a surface 𝑆

𝑛2d 𝑡, 𝜃 =𝜋

2

Γ𝑎3d2

Γ𝑎3d¨ + 1

2

𝑑3d 𝜃 ∗ sinΦ ∗ 𝑡−𝑎3d+1

17

Data

Conceptual

models

Medium

model

Prediction for

applications

Prior

knowledge

Piggott, A. (1997), Fractal relations for the diameter and

trace length of disc-shaped fractures, J. Geophys. Res.,

102(B8), 18121-128126.

Data

Conceptual

models


Data

Conceptual

models


Stereology

1. A 3D model

𝑛3d 𝑙, 𝜃, 𝜑 = 𝑑3d 𝜃 ∗ 𝑙−𝑎3d

2. Stereology rulesObservation parameters 𝑃, fracture parameter 𝐹

𝑛(𝑃) = න𝑎𝑙𝑙

𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑃, 𝐹 𝑛3𝑑 𝐹 𝑑𝐹

Φ: angle between the fracture and the observation structure

BoreholeObservation parameter: nb of fractures𝐵 borehole diameter, 𝐿 borehole length

If 𝑙min < 𝐵 and 𝑎3d > 3

𝑛1d ℎ, 𝜃, 𝜑 =𝜋

2𝑑3d 𝜃

cosΦ𝑎3d−2 ∗ 1 − 𝜖 𝑎3d,Φ

𝑎3d − 3 𝑎3d − 2 𝑎3d − 1𝐵3−𝑎3d

18

Data

Conceptual

models

Medium

model

Prediction for

applications

Prior

knowledge

Piggott, A. (1997), Fractal relations for the diameter and

trace length of disc-shaped fractures, J. Geophys. Res.,

102(B8), 18121-128126.

x

fra

cture length, l

radius, r

Davy, P., C. Darcel, O. Bour, R. Munier, and J. R. d. Dreuzy (2006), A

note on the angular correction applied to fracture intensity profiles along

drill core, J. Geophys. Res., 111(B11), 10.1029/2005jb004121, 1-7.4TH CARGESE SUMMER SCHOOL: JUNE 25TH – JULY 7TH 2018 FLOW AND TRANSPORT IN POROUS AND FRACTURED MEDIA

0 30 60 90

10-3

10-2

10-1

Outcrops

3D

de

nsity t

erm

d3

D

dip

a3D=3.5


Checking consistency

Darcel, C., P. Davy, O. Bour, and J. De Dreuzy (2006), Discrete

fracture network for the Forsmark site, SKB Reports, R-06-79, 94 pp,

Svensk Kärnbränslehantering AB, Stockhölm.

Distribution model

𝑛3𝑑 𝑙, 𝜃 = 𝑑3d 𝜃 𝑙−𝑎3𝑑

Data

Conceptual

models

19

outcrops


0 30 60 90

10-3

10-2

10-1

Outcrops

Borehole

Rock unit

0 - 100 m

200 - 500 m

600 - 850 m

Shear zone

3D

de

nsity t

erm

d3

D

dip

a3D=3.5


Checking consistency

3D consistency between borehole information (10 cm) and outcrop mapping (50cm -5m)

Difference between the ‘background’ fracture pattern and shear zones

Increase of the fracture density by a factor 10 for low-angle dipping fracture sets

No change for high-angle dipping fracture sets

Distribution model

𝑛 𝑙, 𝜃 = 𝑑3d 𝜃 𝑙−𝑎3𝑑

outcrops

Data

Conceptual

models

20

cores



a 3D DFN density pattern from 1D cores (Forsmark, Sweden)

▪ Calculated from cored-borehole fracture density data

▪ Geological domain length investigation: 1-km * 10 boreholes

▪ Fracture size investigation: 10-cm (borehole diameter to….)

Orientations of fractures

21

Darcel, C., R. Le Goc, and P. Davy (2013), Development of the statistical

fracture domain methodology – application to the Forsmark site. SKB Rep. R-

13-54, 94 pp, Svensk Kärnbränslehantering AB, Stockhölm..

Data

Conceptual

models



a 3D DFN density pattern from 1D cores (Forsmark, Sweden)

▪ Calculated from cored-borehole fracture density data

▪ Geological domain length investigation: 1-km * 10 boreholes

▪ Fracture size investigation: 10-cm (borehole diameter to….)

Density of fractures

22

Darcel, C., R. Le Goc, and P. Davy (2013), Development of the statistical

fracture domain methodology – application to the Forsmark site. SKB Rep. R-

13-54, 94 pp, Svensk Kärnbränslehantering AB, Stockhölm..

Data

Conceptual

models

1000

800

600

400

200

0 0.1 1 10

average density

exp(-depth/60m)

exp(-depth/500m)

Classes

1

2

3

4

5

6

7

density (horizontal fractures)

de

pth

(m

)

1000

800

600

400

200

0 0.1 1 10

mean horizontal

mean vertical

density (vertical fractures)

de

pth

Classes

1

2

3

4

5

6

7


DFN methodology

scaling issues


Issue 1. Scaling laws

Filling the scale gap between measures

24

cores

Lineament maps

10-3

10-2

10-1

100

101

102

103

104

105

scale (m)

outcrop

Data

Conceptual

models



Filling the scale gap between measures

Very few data compared to the natural

complexity and modeling objectives

→ The scaling law is a key relationship, which

should be based on strong arguments

25

Data

Conceptual

models



Scaling and density parameters

A first-order local measure of the quantity (number)

of fractures

𝑛𝐷(𝑙): number of fractures per unit D-dimension

volume, per unit fracture size 𝑙

𝑛𝐷 𝑙 =Number 𝑙, 𝑙 + 𝑑𝑙

𝑉𝐷 ∗ 𝑑𝑙

An upscaling parameter

Dimensionless

Measure the ratio of fractures for different

length scales

−𝑎𝐷 𝑙1, 𝑙2 = log𝑛𝐷 𝑙1𝑛𝐷 𝑙2

/ log𝑙1𝑙2

−𝑎𝐷 𝑙 =𝑙

𝑛

𝑑𝑛𝐷𝑑𝑙

=𝑑 log 𝑛𝐷 𝑙

𝑑 log 𝑙

26

Data

Conceptual

models

-5

-4

-3

-2

-1

0

100

10-3

10-2

10-1

100

101

de

ns

ity

dis

trib

uti

on

, n

(l)

fracture trace length, l (m)

log

ari

thm

ic s

lop

e




A first-order local measure of the quantity (number)

of fractures

𝑛𝐷(𝑙): number of fractures per unit D-dimension

volume, per unit fracture size 𝑙

𝑛𝐷 𝑙 =Number 𝑙, 𝑙 + 𝑑𝑙

𝑉𝐷 ∗ 𝑑𝑙

A density parameter:

𝑑𝐷 𝑙 =𝑛𝐷 𝑙

𝑙−𝑎𝐷 𝑙

(eq. 𝑛𝐷 𝑙 = 𝑑𝐷 𝑙 𝑙−𝑎𝐷)

27

Data

Conceptual

models

-5

-4

-3

-2

-1

0

100

10-3

10-2

10-1

100

101

de

ns

ity

dis

trib

uti

on

, n

(l)


log

ari

thm

ic s

lop

e




THE power-law model

The only function without characteristic scales

𝑛𝐷 𝑙, 𝜃 = 𝑑𝐷(𝜃) ∗ 𝑙−𝑎𝐷

with 𝑑𝐷 and 𝑎𝐷 independent of 𝑙 over a (fairly

large) range of fracture sizes

28

Data

Conceptual

models

-5

-4

-3

-2

-1

0

100

10-3

10-2

10-1

100

101

de

ns

ity

dis

trib

uti

on

, n

(l)


log

ari

thm

ic s

lop

e


100 101

10-2

10-1

100

101

n2

d(l

)

outcrops 0.5m-10m

fracture trace length (m)

THE LAXEMAR FRACTURE SYSTEM (SWEDEN)

Darcel, C., et al. (2009), R-09-38 - Statistical methodology for

discrete fracture model – including fracture size, orientation

uncertainty together with intensity uncertainty and variability,

SKB.

Issue 1: Scaling law

Which scaling ?

fracture traces

29

Data

Conceptual

models



Which scaling ?

fracture traces

2.0 2.2 2.4 2.6 2.8 3.0 3.20

1

2

3

4

5

density term

, d

2d

exponent, a2d

Forskmark

Laxemar

Simpevarp

THE LAXEMAR FRACTURE SYSTEM (SWEDEN)

Davy, P., R. Le Goc, C. Darcel, O. Bour, J.-R. de Dreuzy, and

R. Munier (2010), A likely universal model of fracture scaling

and its consequence for crustal hydromechanics, J. Geophys.

Res., 115(B10), 1-13, doi: 10.1029/2009jb007043.

30

Data

Conceptual

models





Res., 115(B10), 1-13, doi: 10.1029/2009jb007043.


Which scaling ?

31

10-1 101 103 10510-16

10-13

10-10

10-7

10-4

10-1

102

n2

d(l

)

outcrops 0.5m-10m

outcrop model 1 a~2.2

outcrop model 2 a~3


Data

Conceptual

models


10-1 101 103 10510-16

10-13

10-10

10-7

10-4

10-1

102

n2

d(l

)

outcrops 0.5m-10m

outcrop model 1 a~2.2

outcrop model 2 a~3





Res., 115(B10), 1-13, doi: 10.1029/2009jb007043.


Which scaling ?

32

Data

Conceptual

models



the critical scales (i.e., the dual of scaling)

33

Data

Conceptual

models

33

The contribution of fractures to a physical process 𝚷

𝐶 𝑙 = න𝑙𝑚𝑖𝑛

𝑙𝑀𝐴𝑋

𝚷 𝒍 ∗ 𝒏𝟑𝒅 𝒍 d𝑙

𝐶 𝑙 = න𝑙𝑚𝑖𝑛

𝑙𝑀𝐴𝑋

𝚷 𝒍 ∗ 𝒏𝟑𝒅 𝒍 𝒍 d(ln 𝑙)

Physical process

Contribution of DFN to

the physical process Π




34

Data

Conceptual

models

34

Scale, 𝑙


𝐶 𝑙 = 𝑙𝑚𝑖𝑛

𝑙𝑀𝐴𝑋𝚷 𝒍 ∗ 𝒏𝟑𝒅 𝒍 𝒍 d(ln 𝑙)

𝚷 𝒍 𝒏𝟑𝒅 𝒍 ⋅ 𝒍




35

Data

Conceptual

models

35




The Queen’s regime

• Prediction depends on the capacity to

detect the position and properties of the

largest fractures

→ geophysics




36

Data

Conceptual

models

36

Scale, 𝑙








37

Data

Conceptual

models

37




The democratic regime

• The smaller fractures control the physical

process

→ what is the smaller relevant size…







38

Data

Conceptual

models

38

Scale, 𝑙








39

Data

Conceptual

models

39

Intermediary bodies

• The process is controlled by structures of

intermediate sizes

→ How to make relevant measurements…




1. a 3D model deduced from 2d data

𝑛3𝑑 𝑙 ~𝑛2𝑑 𝑙 ∗ 𝑙−1

2. the contribution of fractures to a physical process

𝐸 𝑙 = නΠ 𝑙 𝑛3𝑑 𝑙 𝑙 d(ln 𝑙)

40

Data

Conceptual

models

Physical process

40

100 101 102 103 104 105

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

103

lineaments

outcrops

n2

d(l

)





Ex. Surface-controled processes

Π 𝑙 ~𝑙2

𝑝32 Mechanical properties of fractured rocks controlled by surface friction

Permeability of dense networks

41

Data

Conceptual

models

41

100 101 102 103 104

10-4

10-3

10-2

10-1

100

101

n3

d(l

) * l 2

+1

fracture size, l (m)

lc

outcrops lineaments

𝚷 𝒍 ∗ 𝒏𝟑𝒅 𝒍 𝒍


100 101 102 103 104

10-1

100

101

102

n3

d(l

) * l 3

+1


lc

outcrops lineaments



Ex. Percolation-controlled process

Π 𝑙 ~𝑙3

Network connectivity

Permeability of networks close to the percolation threshold

Mechanical properties of frictionless fractures

42

Data

Conceptual

models

42





Ex. contribution of a fracture to deformation

𝜖~𝑆

𝑉

𝜏

𝑘𝑠+𝐸𝑜/𝑙∗=

𝑆

𝑉

𝜏

𝑘𝑠

1

1+𝑙𝑠/𝑙

Π 𝑙 ~𝑙2

1+𝑙𝑠/𝑙

43

Data

Conceptual

models

43

100 101 102 103 104

10-4

10-3

10-2

10-1

100

101

102




DFN methodology

physical rationale for scaling



A physical rationale from genetic model

Fracture networks = population dynamics

• Nucleation

• Growth

• Arrest

Data

Conceptual

models

Medium

model

Prediction for

applications

Prior

knowledge

Davy, P., R. Le Goc, and C. Darcel (2013), A model of fracture

nucleation, growth and arrest, and consequences for fracture

density and scaling, Journal of Geophysical Research: Solid

Earth, 118(4), 1393-1407, doi: 10.1002/jgrb.50120.




1. Nucleation

Nucleation rate ሶ𝑛𝑁 𝑡

Pdf of nuclei size 𝑝𝑁(𝑙)

ሶ𝑛𝑁 𝑙 = ሶ𝑛𝑁(𝑡) ∗ 𝑝𝑁 𝑙

2. Growth

~ stress intensity factor 𝐾𝑚~𝑙𝑚/2

The Charles’ law: 𝑑𝑙

𝑑𝑡= 𝐶 𝑙𝑎

Davy, P., R. Le Goc, and C. Darcel (2013), A model of fracture

nucleation, growth and arrest, and consequences for fracture

density and scaling, Journal of Geophysical Research: Solid

Earth, 118(4), 1393-1407, doi: 10.1002/jgrb.50120.

46

⇒ stationary distribution

𝑛𝐷 𝑙 =ሶ𝑛𝑁𝐶. 𝑙−𝑎 1 − 𝑃𝑁 𝑙




3. Arrest / Stop

Large amount of T-intersection




3. Arrest / Stop

The Mosaic network




3. Arrest / Stop

The Mosaic network

49

Size distribution

• Distance from one object to another ~

object size

• Densité of objects: 𝜌 = 𝑁/𝑉

• Average distance 𝑑~𝜌−1/𝐷

𝐷=space dimension

𝑛(𝑙)

𝑙 = 𝜌1/𝐷4TH CARGESE SUMMER SCHOOL: JUNE 25TH – JULY 7TH 2018 FLOW AND TRANSPORT IN POROUS AND FRACTURED MEDIA



3. Arrest / Stop

The hierarchical network

50

Size distribution• Fracture energy depends on fracture size

• An intersection should likely stop the smallest

fracture

• Size ~ average distance of larger fractures

• 𝑙~𝑁 𝑙′>𝑙

𝑉

1/𝐷

= Cumulative 𝑙 1/𝐷

𝑛(𝑙) ⇒ self-similar distribution

𝑛𝐷 𝑙 = 𝐷𝛾𝐷 𝑙− 𝐷+1




3. Arrest / Stop

The hierarchical network

51

⇒ self-similar distribution

𝑛𝐷 𝑙 = 𝐷𝛾𝐷 𝑙− 𝐷+1

20 cm

10 km


The UFM model

a collision model for fracture growth model

0.0 0.5 1.0 1.5 2.0

0

10000

20000

30000

40000

50000

60000

70000

Co

llis

ion

s r

ate

t*

ln_0.15

ln_0.3

ln_0.5

Model parameters

System dimension 𝐷=3

Nuclei size 𝑙𝑁 = 0.3

Nucleation rate ሶ𝑛𝑁

Growth law: d𝑙

𝑑𝑡= 𝐶𝐺𝑙

𝑎, 𝑎 = 3

Dimensionless time 𝑡𝑐 time for a

nuclei to have an infinite size

𝑡𝑐 =𝑙𝑁1−𝑎

𝐶𝐺∙

1

𝑎 − 1

Prior

knowledge


The UFM model

a collision model for fracture growth model

100

101

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

n(l)

Fracture length

Dimensionless time:

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1.2

Prior

knowledge

53

Model parameters

System dimension 𝐷=3

Nuclei size 𝑙𝑁 = 0.3

Nucleation rate ሶ𝑛𝑁

Growth law: d𝑙

𝑑𝑡= 𝐶𝐺𝑙

𝑎 ,𝑎 = 3

Dimensionless time 𝑡𝑐 time for a

nuclei to have an infinite size

𝑡𝑐 =𝑙𝑁1−𝑎

𝐶𝐺∙

1

𝑎 − 1




Low concentration

High concentration

and predicts criticallength scale

10-2

10-1

100

101

102

103

104

10-18

10-15

10-12

10-9

10-6

10-3

100

103

106

UFM

arrest

regime

growth

regime

den

sit

y d

istr

ibu

tio

n f

un

cti

on

fracture length

nu

cle

ati

on

regim

e

lc

that explains dataA dual physics




10-2

10-1

100

101

102

103

104

10-18

10-15

10-12

10-9

10-6

10-3

100

103

106

UFM

arrest

regime

growth

regime

den

sit

y d

istr

ibu

tio

n f

un

cti

on

fracture length

nu

cle

ati

on

re

gim

e

lc

𝑛 𝑙 =ሶ𝑛𝑜𝐶. 𝑙−𝑎 𝑛 𝑙 = 𝑑UFM 𝑙− 𝐷+1

transition Length

𝒍𝒄= 𝑑UFM𝐶𝐺

ሶ𝑛𝑁

1

𝐷+1−𝑎

A (likely) universal Fracture model (UFM) with a two-power-law scaling

distribution



A need for a better knowledge of geological processes

Etienne Lavoine’s PhD (2016-2020). Development of fracture

network models from (simplified) mechanical rules

Nucleation probability ~ 𝜎𝑚, with 𝑚 a selectivity parameter

Increase of fracture correlation, decrease of fractal dimensions56

𝑚 = 3


DFN methodology

predictions in applications


DFN predictions

the construction of the models and their critique

A DFN model is a conceptual framework, where model statistical properties are

▪ either bootstrapped from data (Poisson-process based models),

▪ or providing emerging properties (genetic models)

A DFN model contains tuning parameters

▪ Structure (statistical distributions, spatial correlations)

▪ Heuristic laws relating fracture and geological/physical properties: Transmissivity = f(size, orientation, stress),Stiffness = f(size, stress)

However, a DFN model is limited in reproducing the geological complexity

▪ How to prove that the geological system is part of the solution space.

▪ Data indicators able to discriminate between models ?

58

Data

Conceptual

models

Medium

model

Prediction for

applications

Prior

knowledge


DFN predictions

Connectivity

Percolation threshold pc (~2.5): percolation value at which the DFN is

connected towards system boundaries by a critical cluster of fractures

𝑃 =𝜋2

8∙ න𝑛 𝑙 ∙ 𝑙3 ∙ 𝑑𝑙

The percolation parameter P controls

statistically the DFN connectivity and the

size of the largest connected cluster [Bour

and Davy, 1998; Dreuzy et al., 2000]:

𝑝 = 𝑝𝑐 𝑝 ≫ 𝑝𝑐𝑝 ≪ 𝑝𝑐

No flowMany flow paths, flow

dominated by density effects

critical flow

With "red" links

Model with𝑎 = 4𝑙𝑚𝑖𝑛 = 1𝐿 = 20𝑝 ∈ [1.5; 2.5; 5]

Connectivity and percolation parameter


DFN prediction indicators

which measures to calibrate models

• The equivalent permeability

𝐾𝑒𝑞 =𝑄𝑇𝛻ℎ

✓ A flow intensity indicator, rather than an intrinsic property of the system

✓ Vary with scale 𝑳

✓ Directly defined with permeameter conditions, indirectly from pumping

tests


DFN prediction indicator

which measures to calibrate models

• The flow structure (channeling indicator)

𝑝32 𝑄 =1

V∙(σ𝑓∙ 𝑆𝑓 ∙ 𝑄𝑓 )

2

(σ𝑓∙ 𝑆𝑓 ∙ 𝑄𝑓2)

✓ Comparable with, and smaller than 𝑝32✓ Inverse of the average distance between

flow paths

✓ Measurable in boreholes or tunnels

✓ Vary with scale 𝑳

✓ 𝑝32 𝑄 is a measure of the exchange

surface between flow and rock

for geochemistry or geothermal applications



Channeling

Channeling index: 𝑝32 𝑄 =1

V∙(σ𝑓∙𝑆𝑓∙𝑄𝑓)

2

(σ𝑓∙𝑆𝑓∙𝑄𝑓2)

(distance)-1 between main flow paths

Maillot, J., P. Davy, R. L. Goc, C. Darcel, and J. R. d. Dreuzy (2016),

Connectivity, permeability, and channeling in randomly distributed and

kinematically defined discrete fracture network models, Water Resour.

Res., 52(11), 8526-8545, doi: 10.1002/2016WR018973.

62

-800

-600

-400

-200

00 10 20 90 100

Total

sealed

open

number of fractures per meterd

ep

th (

m)

-800

-600

-400

-200

0

10-4 10-5 10-6 10-7 10-8 10-9

Transmissivity m2.s-1



Channeling

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

[p32]Q

Kinematic Model

Poisson's Model

p32

Channeling index: 𝑝32 𝑄 =1

V∙(σ𝑓∙𝑆𝑓∙𝑄𝑓)

2

(σ𝑓∙𝑆𝑓∙𝑄𝑓2)

(distance)-1 between main flow paths

Maillot, J., P. Davy, R. L. Goc, C. Darcel, and J. R. d. Dreuzy (2016),

Connectivity, permeability, and channeling in randomly distributed and

kinematically defined discrete fracture network models, Water Resour.

Res., 52(11), 8526-8545, doi: 10.1002/2016WR018973.

63

constant 𝑻𝒇same size and orientation distributions


Forsmark, Sweden DFN case study


Case study (Forsmark, Sweden)

GeoDFN

𝑑 𝑙𝑐=3m

𝑃32 𝑑 → ∞ = 4.76 m-1GeoDFN UFM

All fractures

(open + sealed)

• 𝑝32 measured at 10 cm (borehole diameter): 4.76 m-1

• Transition between power laws 𝑙𝑐: 3 m

• Volume : (100 m)3

• Orientation distribution: bootstrapped from Forsmark



the issue of clogging

HydroDFN

In the application, 75% of the total fracture surface measured at the core diameter scale is sealed

25% is open or partly open

→ The HydroDFN is a subset of the GeoDFN

Fracture sealed with hematite stained

adularia (Sandström et al, 2008)



scaling transmissive DFN

𝑑 𝑙𝑐 𝑙𝑐,𝑜𝑝𝑒𝑛=10m

DFN backbone, fractures coloured by total

flow. Fracture with flow smaller than 1% of

max. flow are transparent.

HydroDFN 𝑙𝑐-open UFM

25% of GeoDFN

67

Genetic model, small fractures clogged




𝑑 𝑙𝑐=3.3m




HydroDFN 𝛼-open UFM

25% of GeoDFN

68

Genetic model, all fractures equally clogged




𝑑 𝑙𝑐




HydroDFN rmin-krmin

25% of GeoDFN

69

Poisson-process model, single power-law


100 101 102

0

5

10

15

20

25

p

system size (m)

Geo-DFN

lc-open

a-open

rmin-krmin

pc


Connectivity𝑙𝑐-open

𝛼-open

rmin-krmin

Percolation parameter


-800

-600

-400

-200

00 10 20 90 100

Total

sealed

open

number of fractures per meter

dep

th (

m)

-800

-600

-400

-200

0

10-4 10-5 10-6 10-7 10-8 10-9

Transmissivity m2.s-1

71


Scaling fracture transmissivity

Fracture transmissivity

Varies in orders of magnitude

Likely depends on fracture size

Likely depends on normal stress



Permeability scaling

72

Permeability

geometric average

100 101 102 103

10-2

10-1

100

data

scale




73

𝑙𝑐-open

𝛼-open

rmin-krmin

Structure

models

Transmissivity

models

• Constant fracture

transmissivity 𝑇𝑓• 𝑇𝑓~𝑙

• 𝑇𝑓~ e−𝜎𝑛𝜎𝑐

• 𝑇𝑓~𝑙𝛼 e

−𝜎𝑛𝜎𝑐

Permeability

geometric average

100 101 102 103

10-2

10-1

100

data

Genetic (small)

Genetic (all)

Poisson-process

scale100 101 102 103

10-2

10-1

100

data

scale


Theory


Below 𝑝𝑐, the connected backbone is a

~plane and permeability decreases as

scale−1

Above 𝑝𝑐, the connected backbone is a

complex structure with fractures in series

and parallel.

Above 𝑝𝑐, the average permeability is

highly dependent on the scaling of

transmissivity with fracture size

74

de Dreuzy, J.-R., P. Davy, and O. Bour (2001), Hydraulic properties of two-dimensional

random fracture networks following a power law length distribution 2. Permeability of

networks based on lognormal distribution of apertures, Water Resour. Res., 37(8),

10.1029/2001WR900010, 2079 - 2096.

de Dreuzy, J.-R., P. Davy, and O. Bour (2002), Hydraulic properties of two-dimensional

random fracture networks following power law distributions of length and aperture, Water

Resour. Res., 38(12), 10.1029/2001WR001009, 12-11-12-19.4TH CARGESE SUMMER SCHOOL: JUNE 25TH – JULY 7TH 2018 FLOW AND TRANSPORT IN POROUS AND FRACTURED MEDIA


Channeling and scaling

75

𝑙𝑐-open

𝛼-open

rmin-krmin

Structure

models

Transmissivity

models

• Constant fracture

transmissivity 𝑇𝑓• 𝑇𝑓~𝑙

• 𝑇𝑓~ e−𝜎𝑛𝜎𝑐

• 𝑇𝑓~𝑙𝛼 e

−𝜎𝑛𝜎𝑐

100 101 102 103

10-4

10-3

10-2

10-1

100

P1

0q

scale

lc-open, T=cte

lc-open, T=lf

lc-open, T=f()

lc-open, T=f(,lf)

a-open, T=cte

a-open, T=lf

KFM08A - 200-400 m

1/Ls

Channeling

indicator m-1


DFN framework, overview

DFN is basically a combination of statistics and deterministic

objects, that aims to extrapolate information with sound

models

DFN (medium description as a population of discrete

‘idealized’ fractures) allows for an ‘easy’ integration of data

DFN is also a tool to find the critical parameters or length

scales that should be measured to improve prediction

DFN models can be used for flow and mechanical

applications

Not all DFN models are equivalent

Issues

Issue 1. Scaling is a critical component of the DFN framework

Issue 2. Any prior knowledge, theoretical or empirical, is

welcome

Issue 3. A DFN is a stochastic model, which contains intrinsic

variability and extrinsic controls

Issue 4. Calibrating DFN model is not enough, validating

models is a prerequisite

76

DFN

DFN

Data

Conceptual

models

Medium

model

Prediction for

applications

Prior

knowledge

ECM


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