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Estimating rockmass properties based on DFN methods
1ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
Caroline Darcel, Romain Le Goc, Etienne Lavoine, Diane DoolaegheItasca Consultants, FrancePhilippe Davy, Univ Rennes, CNRS, FranceDiego Mas Ivars, SKB, KTH, Sweden
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β’ Anisotropyβ’ Scale effectβ’ Integration from cm to km
Objective: Introduce Discrete Fracture Network methods for rockmass modelling
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DFN modeling
Modeling
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Relevant complexity
β’ Stress, strain, hydraulic, transport, HM, thermalβ’ Available dataβ’ Modeling as DEM, discrete or equivalent continuous
modeling
Modeling purpose
Define which metric of a DFN is the controlling factor of rockmass elastic properties
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Classical approach: fractured rock as a block assembly
(π π π π π π , π π π π π π , π π , πΊπΊπΊπΊπΊπΊ) are adapted to heavily jointed rock masses β’ Assembly of blocksβ’ Several sets of potentially infinite fractures (isotropy)β’ Fractured system Scale = spacingβ’ Applicable when model resolution >> spacing
But lack of β’ Density and scale (size distribution of fractures)β’ 3D representationβ’ Anisotropy β’ Quantitative description
3ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
(Hoek and Brown, 2018)
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Concepts for fractured rock
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population of individual fracturesBlock Assembly
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Fractured rock β block Assembly vs Network of fracture
5ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
population of individual fractures
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Fractured rock β block Assembly vs Network of fracture
6ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
population of individual fractures
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Fractured rock β block Assembly vs Network of fracture
7ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
population of individual fractures
Need to assess the density vs size
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Description of the fractured rock with DFNOriginally applied to Crystalline rocks Potential host for nuclear waste storage Connectivity, flow and transport modeling
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Observation and mapping
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β’ Mapping conditions: Resolution and Censoring
β’ Physical boundaries of the distribution, min and maxsizes, not directly accessible
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A Fracture, what is it
A fracture in the model is an ensemble of fracture segments that define a consistent plane (mechanical coherence).
10 m
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A Fracture, what is it
Roughly Planar discontinuity
Resulting from rock failure controlled by physical processes and field conditions
Can be cracks, joints, faults, shear zones, bedding planes
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Lateral dimensions >> thickness
2D planar object
Position, size, orientation, shape
Flow, transport, mechanical properties
β¦ to the fracture population (Discrete Fracture Network)
East
North
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Build the fracture size density distribution ππ(ππ)
β’ Logarithmic binning to count the number of fracture traces whose size πππ is in the range ππ; ππ + ππππ
β’ Normalized by map area and bin size
ππ2π·π· ππ = 1ππππππππ
ππ ππ<ππβ²<ππ+ππππ,πΏπΏππππ
β’ Physical boundaries of the distribution, minand max sizes, not directly accessible
β’ Count orientations
β’ Variability β¦
β’ Stereological analysis required for a 3D model
12ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
0.4 0.6 0.8 2 4 61
0.1
1
y
ππ2π·π· ππ
Fracture trace size ππ[SKB report P-04-35]
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Example β 2D trace size distribution
0.1 1 10 1001E-5
1E-4
0.001
0.01
0.1
1
10
100
1000 Model fit ASM000208
n(l)
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ππ2π·π· ππ
Fracture trace size ππ
β’ Power-law trend may be identified
β’ from a limited observation range
β’ parameters independent fromobservation range
ππ ππ = πΌπΌ οΏ½ ππβππ
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From data to size distributions β Laxemar site
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Power-law model for ππ(ππ) consistent with observations
Single or double power-law model
Power-law model is a good proxy to model density variation with scale
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10-1 101 103 10510-16
10-13
10-10
10-7
10-4
10-1
102
n 2d(l
)
fracture trace length (m)
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DFN in the Rockmass
15ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
β’ Multiscale DFNβ’ No a priori REV
β’ ππ32 (ππ2/ππ3) dominated by small fractures
β’ Connectivity related to large fractures
β’ Mechanical properties potentially related to size and orientations
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Scale, resolution, size, density
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Resolution decrease
Resolution increase
System dimension increase
Start by one single fracture
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Predicting equivalent elastic properties From single fracture to fracture population (DFN)
DFN: any set of disc-shaped planar fractures (multi-oriented, multi-scale)
Elastic conditions (no damage)
Rock matrix: isotropic elastic, Youngβs modulus πΈπΈππ and Poisson ratio ππππ Fracture mechanical model
Coulomb slip , cohesion (ππ), friction (angle ππ), normal (ππππ) and shear stiffness (πππ π )
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Mechanical model - single fracture
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ππππ = ππππ ππ π‘π‘π‘π‘ ππ + πΆπΆ
ππππ
πππ π
π‘π‘
ππππππππ
ππππ = πππ π οΏ½ π‘π‘ if ππππ < ππππ
ππππ = ππππ if ππ > ππππ
ππππ = 0 if Ο = 0 or πππ π = 0Frictionless fracture
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Single fracture isolated
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ππβ
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~ πΈπΈππππ
ππππ = 3ππ8β 1β βππm 2
1βππππ2β πΈπΈm
ππ
Frictionless isolated fracture
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0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
4
5
disp
lace
men
t (x
10-7)
distance to disk centresheardisp.
ππβ
Remote stressππ shear stressIntact rockππππ Poisson ratioπΈπΈππ Modulus
Fractureππ sizeπ‘π‘ average shear displacement
ππππ equivalent matrix to fracture stiffness
shear stressππ
displ. π‘π‘
Stress and displacement at fracture Displacement profile
size ππ ππ/2
ππ = ππππππ
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shear disp.
Frictional isolated fracture
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stress ππβshear stress
ππ
displ. π‘π‘
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
10
disp
lace
men
t (x1
0-7)
distance to disk centre
ks=1*109
ks=5*109
ks=1*1010
ks=5*1010
Stress and displacement at fracture Displacement profile
[Davy et al, 2018]
ππ = ππππ + πππππ‘π‘ = π‘π‘ππ = π‘π‘ππ
Fracture friction, cohesion and stiffness terms
πππ π increase
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ππ = ππππππ+πππ π
Remote stressππ shear stressIntact rockππππ Poisson ratioπΈπΈππ Modulus
Fractureππ sizeπ‘π‘ average shear displacement
ππππ ~ πΈπΈππππ
equivalent matrix to fracture stiffness
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Stress perturbation around a fracture
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Increasing πππ π relatively to ππππdecreases the stress perturbation
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πππ π ππππ
β0 1
Remote stressππ shear stressIntact rockππππ Poisson ratioπΈπΈππ Modulus
Fractureππ sizeπ‘π‘ average shear displacement
ππππ equivalent matrix to fracture stiffness~πΈπΈππππ
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with πππ π = πΈπΈπππππ π
ππ βͺ πππ π π‘π‘ = πππππ π +ππππ
β ππππππ
β ππ
ππ β« πππ π π‘π‘ = πππππ π +ππππ
β πππππ π
Two regimes for the shear displacement
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If πππ π is negligible, fracture size defines the shear displacement
If πππ π is dominant, shear displacement is independent from fracture size
Fracture sizes are critical
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Shear vs slipping regime for ππ βͺ πππ π
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ππππβ = πππ π +πππππππ π
οΏ½ ππππ
πππ π + ππππππππ
ππππ
πππ π
ππ
π‘π‘
Shear regime Slip regimeCritically stressed fractures
Difference related to ππ/πππ π
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Remote stressππ shear stressIntact rockππππ Poisson ratioπΈπΈππ Modulus
Fractureππ sizeπ‘π‘ average shear displacement
ππππ ~ πΈπΈππππ
equivalent matrix to fracture stiffness
Fracture sizes are critical
Critically stress regime threshold is size dependent
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From single fracture to DFN and rock mass
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ππππππ ππ=πΊπΊππ π§π§ππ. π°π°
πΊπΊπΌπΌβ π‘π‘ππ π¬π¬ππ. π±π±πΏπΏπ§π§
=πΊπΊπππ‘π‘ππππ β π§π§. π°π° π¬π¬. π±π±
Fracture ππ contribution to rock mass strain
π°π°
π±π±ππππ
π π ππ
Sample Surface πΊπΊπΌπΌ
πΏπΏπ§π§
DFN contribution to rock mass strain tensor ΜΏππ :Sum the contribution of each fracture ππ and intact rock ππ
Rock mass with DFN
ππππππ = οΏ½ππππππππ ππ
+ (ππππππ)ππ
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Fracture network to rock mass strain
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Derive effective compliance tensor components πΆπΆππππππππ:
General case conditions :β’ Shear displacement (ππππ)β’ Effective theory to account for fracture interactions for large densitiesβ’ Change of regime for critically stressed fractures (slipping, dilation) β’ Normal displacement (ππππ)
ππππππ = οΏ½ππππππππ ππ
+ (ππππππ)ππ
DFN contribution to rock mass strain tensor ΜΏππ :Sum the contribution of each fracture ππ and intact rock ππ
ππππππ = πΆπΆππππππππππππππ
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Davy et al., 2018, Elastic properties of fractured rock masses with frictional properties and powerβlaw fracture size distributions: JGR, v. 123, p. 6521 β 6539.
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Comparison to numerical simulations
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0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0 ks/km 0 [0.1; 0.4] [0.6; 2.4]
effective theory
E / Em
E (effective theory) / Em
πππ π Numerical simulations
Theoretical calculations
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Predicting πΈπΈππππππ with analytical solutions for simple casesππππ β« πππ π =0
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0 5 10 15 200
10
20
30
40
50
60
E (G
Pa)
percolation parameter, pΞΈ
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In this case, the DFN percolationparameter ππ(ππ) is the controlling factorof the rockmass effective elastic modulus
πΈπΈππππππ = πΈπΈ0 exp βππ οΏ½ ππ ππ
ππ ππ =1πποΏ½ππ
ππππππ cos2 ππππ sin2 ππππ
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Over DFN range ππ β« ππππ
ππ32 total fracture surface per unit volume
Predicting πΈπΈππππππ with analytical solutions for simple cases - ππππ β« πππ π constant
Over DFN range ππ βͺ ππππ
ππ β so called percolation parameter
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πΈπΈππππππ = πΈπΈππ exp βππ οΏ½ ππ ππ
ππ ππ =1πποΏ½
ππππππππ cos2 ππππ sin2 ππππ
πΈπΈππππππ =πππ π
ππ32 ππ + πππ π /πΈπΈππ
ππ32(ππ)~ 1ππβππ ππππππ cos2 ππππ sin2 ππππ
Potential size effect since ππis scale dependent
No size effect since ππ32 is scale independent
Application to realistic multiscale DFN
ππππ(ππ)
Domain size πΏπΏ
πππ π =πΈπΈπππππ π
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Application to site conditions β Forsmark case Input : generated DFN Input : Intact rock properties Input : stress state Output: Compliance tensor Scale effect Level of anisotropy
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Application - DFN and rock conditionsSKB Forsmark site, Sweden
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πππΊπΊ
π»π»π»π»
πππΈπΈ
ππππ
πΈπΈππ
Intact RockπΈπΈππ = 76 πΊπΊπππΊπΊππππ = 0.23
Fracturesπππ π ππππ = 46.55 Γ ππππ0.4039 Γ 106ππππ > 100πππ π
ππππ πππ₯π₯πππ¦π¦πππ§π§πππ¦π¦π§π§πππ₯π₯π§π§πππ₯π₯π¦π¦
=
1πΈπΈπ₯π₯π₯π₯
βπππ¦π¦π₯π₯πΈπΈπ¦π¦
βπππ§π§π₯π₯πΈπΈπ§π§
0 0 0
βπππ₯π₯π¦π¦πΈπΈπ₯π₯
1πΈπΈπ¦π¦
βπππ§π§π¦π¦πΈπΈπ§π§
0 0 0
βπππ₯π₯π§π§πΈπΈπ₯π₯
βπππ¦π¦π§π§πΈπΈπ¦π¦
1πΈπΈπ§π§
0 0 0
0 0 01
2πΊπΊπ¦π¦π§π§0 0
0 0 0 01
2πΊπΊπ₯π₯π§π§0
0 0 0 0 01
2πΊπΊπ₯π₯π¦π¦
Γ
πππ₯π₯πππ¦π¦πππ§π§πππ¦π¦π§π§πππ₯π₯π§π§πππ₯π₯π¦π¦
Mechanical properties
πππ»π»
ππβπππ£π£
DFN (FFM01 unit)
ππππππππ = 0.1πΏπΏ = 20
No critically stressed fractures
5 10 15 20 25 ππππCompliance tensor οΏ½πͺπͺ
ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
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Evolution of πΈπΈππππ with πΏπΏ
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Given the DFN conditions:
β’ If πππ π such that ππ βͺ πππ π β maximise the scaling effect
ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
Increasing domain size πΏπΏ tend to put more large fractures without significantly changing ππ32
πΈπΈπ§π§π§π§
πΈπΈππππ
0 10 20 30 4005
1015202530354045505560657075
Exx Eyy Ezz Gxy Gyz Gzx
------------------ Ezz if ks=0 Eyy if ks=0 Em
[GPa
]
L
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Evolution of πΈπΈππππ with πΏπΏ
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Given the DFN conditions:
β’ If πππ π such that ππ βͺ ππππ β maximise the scaling effect
With current mechanical properties
β’ πππ π = 3.4ππππ GPa. mβ1
β’ 1.5 m β€ πππ π β€ 3.5 mβ’ Decrease of πΈπΈππππ with πΏπΏ up to ~10m.
β’ πΈπΈπ₯π₯π₯π₯ decrease from 76 Gpa to about 62 GPa, i.e. about 25%. *
ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
0 10 20 30 4005
1015202530354045505560657075
Exx Eyy Ezz Gxy Gyz Gzx
------------------ Ezz if ks=0 Eyy if ks=0
[GPa
]
L
πΈπΈπ§π§π§π§
πΈπΈππππ
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Evolution of πΈπΈππππ - Anisotropy
34
1 DFNππππππππ = 0.1πΏπΏ = 20 Ori = FFM01πππ π (ππππ)ππππ β β
β’ πΈπΈππππ variations : 60 to 70 Gpa(about 12%)
β’ πΈπΈπ§π§π§π§ less affected by fractures than horizontal πΈπΈβπ§π§
β’ (Horizontal directional πΈπΈβπ§π§consistent with fracture sets NE and NW, less affected by fracture shearing are at trend 45Β° )
ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
0 10 20 30 4005
1015202530354045505560657075
Exx Eyy Ezz Gxy Gyz Gzx
------------------ Ezz if ks=0 Eyy if ks=0
[GPa
]
L
20-25%12%
πΈπΈπ§π§π§π§
πΈπΈππππ
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SUMMARY
DFN representation of rockmass help to integrate multiscale fracture distribution
Rockmass effective properties can be derived and controlling factors β as a combination between mechanical, geometrical and scale -identified
Extent of scale effect and anisotropy can be quantified
Tool (DFN.lab) to integrate DFN in geomechanical models
35ITASCA SYMPOSIUM, VIENNA 17-21 FEBRUARY 2020
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DFN.lab Numerical platform for modelling fractured media
Statistical analysis
Data
Flow
Generationmodel
Mechanics
Expertise
Expertise