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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020
FAULT NETWORK UNCERTAINTY ASSESSMENT WITH AGENERATIVE GRAPH-BASED ALGORITHM
–CURRENT STATUS AND PERSPECTIVES
Guillaume Caumon, Gabriel Godefroy, Paul Marchal
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 2
Fault association problem
[Modified from Freeman et al., First Break 1990]
"Find the simplest and most likely
explanation for your data"
- Anonymous teacher -
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 3
Fault association problem
Hydrothermal alteration
Faults
Stratigraphic horizon
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 4
Well 1Well 2
Fault uncertainty: Should we care ? • Depends on the model purpose
• Faults geometry / connectivity affect:• Structural geometry
• Hydrodynamic behavior
• Mechanical behavior
0 1Saturation @8000 daysSimplest Downscaled
[Julio et al, Interp 2015]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 5
Motivation
“In developing the multiple hypotheses, the effort is to bring up into view every rational explanation of the phenomenon in hand […]
The chief danger that springs from affection is counteracted”T.C. Chamberlin, The method of Multiple working hypotheses, 1897
Approaches [Wellmann & Caumon, Adv Geophys 2018]:
Data perturbation [Wellmann et al, 2010; Lindsay et al 2013; Giraud et al 2019]
Model perturbation [Abrahamsen 1992; Lecour et al, 2001]
Expert-based [Bond et al 2007; Schaaf & Bond, 2019]
Expert-based + Model perturbation [Suzuki et al, 2008; Scheidt et al, 2018]
Iterative fault simulation [Cherpeau et al, 2010; 2012; 2015; Julio et al, 2015]
Stochastic processes [Holden et al, 2003; Aydin and Caers, 2017]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 6
Fault uncertainty: components
1. Identifying evidence
2. Associating evidence
3. Modeling & QC
?
?
?
?
[Data: Taylor et al, GJI 2010]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 7
Joining the dots…
By whitney waller - connect-the-dots, CC BY-SA 2.0, https://commons.wikimedia.org/w/index.php?curid=45762268
Rule: increment number
Strategy : search locally
Fault evidence have no numbers, but they carry information!
Need for:
- Effective rules to associate fault evidence
- An effective sampling algorithm
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 8
Fault association as a graph creation problem
[Modified from Freeman et al.,
First Break, 1990]
[Godefroy et al,
Math Geosci, 2019]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 9
The possibility graph
Each node = piece of fault evidence
Point, line, surface
[+ location uncertainty]
Each edge :
Likelihood that the endpoints belong to the same fault surface
𝑛 = 11 nodes
11×10/2 = 55 edges
Max number of scenarios (Bell number):
𝐵 𝑛 = 678570
[Godefroy et al,
Math Geosci, 2019]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 10
The possibility graph
Each node = piece of fault evidence
Point, line, surface
[+ location uncertainty]
Family marks (inducing subgraphs)
Each edge :
Likelihood that the endpoints belong to the same fault surface
4 + 7 nodes
6 + 21 edges
Max number of scenarios:
𝐵 4 + 𝐵 7 = 15 + 877
[Godefroy et al,
Math Geosci, 2019]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 11
Simple association likelihood: distance[Godefroy et al, Math Geosci, 2019]
𝐿𝑑 𝐯1 ↔ 𝐯2 =𝑙𝑚𝑎𝑥 − 𝐯𝑖𝐯𝑗𝑙𝑚𝑎𝑥 − 𝑙𝑚𝑖𝑛
𝑝𝑑𝑓(𝑙)
𝑙𝑃𝑟(𝑙 ≥ 𝐯𝑖𝐯𝒋 )
𝑙𝑙𝑚𝑖𝑛 𝑙𝑚𝑎𝑥
1
Example: uniform distribution
𝐯𝑖𝐯𝑗
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 12
Simple association likelihood: orientation
Example: uniform distribution
around direction 𝐭
𝐿𝑜 𝐯𝑖 ↔ 𝐯𝑗 = ൞1 if acos
𝐯𝑖𝐯𝑗
𝐯𝑖𝐯𝑗⋅𝐭
𝐭< 𝜃
0 if not
𝐭
𝜃
𝐯𝑖
[Godefroy et al, submitted]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 13
Simple association likelihood: stratigraphic offset
𝐯𝑖
𝐯𝑗
Create a test surface from the fault data
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 14
Simple association likelihood: stratigraphic offset
𝐯𝑖
𝐯𝑗
Create a test surface from the fault data
Deactivate stratigraphic information
in a buffer zone
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 15
Simple association likelihood: stratigraphic offset
𝐯𝑖
𝐯𝑗
ℋ𝑐
Create a test surface from the fault data
Deactivate stratigraphic information
in a buffer zone
Compute a continuous horizon ℋ𝑑 by LS
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 16
Simple association likelihood: stratigraphic offset
Create a test surface from the fault data
Deactivate stratigraphic information
in a buffer zone ℋ𝑑
Compute a continuous horizon ℋ𝑑 by LS
Compute a discontinuous horizon ℋ𝑑 by LS
𝐿𝑜 𝐯𝑖 ↔ 𝐯𝑗 =ℋ𝑐 −ℋ𝑑
𝑅𝑀𝑆 ℋ𝑐 + 𝑅𝑀𝑆 ℋ𝑑
𝐯𝑖
𝐯𝑗
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 17
Simple association likelihood: stratigraphic offset
ℋ𝑐
Create a test surface from the fault data
Deactivate stratigraphic information
in a buffer zone
Compute a continuous horizon ℋ𝑑 by LS
Compute a discontinuous horizon ℋ𝑑 by LS
𝐿𝑜 𝐯𝑖 ↔ 𝐯𝑗 =ℋ𝑐 −ℋ𝑑
𝑅𝑀𝑆 ℋ𝑐 + 𝑅𝑀𝑆 ℋ𝑑
𝐯𝑖
𝐯𝑗
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 18
Aggregating association likelihoods
𝐿 𝐯𝑖 ↔ 𝐯𝑗 = ෑ
𝛼
𝐿𝛼 𝐯𝑖 ↔ 𝐯𝑗
Remove edges with zero (or very small) likelihood
from the possibility graph
[Godefroy et al, Math Geosci, 2019]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 19
Identify possible largest faults
Faults : cliques of the possibility graph
Large faults : maximal cliques of the possibility graph
[Bron Kerbosh 1979]
[Godefroy et al, Math Geosci, 2019]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 20
Sequential Fault Sampling
Faults : cliques of the possibility graph
Large faults : maximal cliques of the possibility graph
[Bron Kerbosh 1979]
Randomly sample a maximal clique
[Godefroy et al, submitted]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 21
Sequential Fault Sampling
Faults : cliques of the possibility graph
Large faults : maximal cliques of the possibility graph
[Bron Kerbosh 1979]
Randomly sample a maximal clique
Optional: downscale
[Godefroy et al, submitted]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 22
Sequential Fault Sampling
Faults : cliques of the possibility graph
Large faults : maximal cliques of the possibility graph
[Bron Kerbosh 1979]
Randomly sample a maximal clique
Optional: downscale
[Godefroy et al, submitted]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 23
Sequential Fault Sampling
Faults : cliques of the possibility graph
Large faults : maximal cliques of the possibility graph
[Bron Kerbosh 1979]
Randomly sample a maximal clique
Optional: downscale
[Godefroy et al, submitted]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 24
Sequential Fault Sampling
Faults : cliques of the possibility graph
Large faults : maximal cliques of the possibility graph
[Bron Kerbosh 1979]
Randomly sample a maximal clique
Optional: downscale
[Godefroy et al, submitted]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 25
Sequential Fault Sampling
Faults : cliques of the possibility graph
Large faults : maximal cliques of the possibility graph
[Bron Kerbosh 1979]
Randomly sample a maximal clique
Optional: downscale
[Godefroy et al, submitted]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 26
Sequential Fault Sampling
Faults : cliques of the possibility graph
Large faults : maximal cliques of the possibility graph
[Bron Kerbosh 1979]
Randomly sample a maximal clique
Optional: downscale
Update possibility graph and cliques and start again
[Godefroy et al, submitted]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 27
From fault graphs to geometric structures
For each association scenario :• For each fault
• Determine outline
• Create surface by least-squares interpolation
Example obtained with SKUA
structure & stratigraphy
[Godefroy et al, submitted]
[e.g., Wellmann & Caumon 2018, Section 3]
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 28
Example: Santos BasinData: doi:10.17605/OSF.IO/MP97W
Visualization: https://gabrielgodefroy.github.io/StochasticInterpData/README.html
[Godefroy et al, submitted]
Seismic data courtesy of PGS
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 29
Example: Santos BasinData: doi:10.17605/OSF.IO/MP97W
Visualization: https://gabrielgodefroy.github.io/StochasticInterpData/README.html
[Godefroy et al, submitted]
• SFS is consistent: As more informative rules are used,• Average models gets closer to the reference
• Uncertainty decreases
• Adding data makes the problem harder !
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 30
Wrapping up• Sequential fault sampling
• Interesting level of abstraction (AI-compatible)
• Parsimonious
• Easily extensible to classify realizations
• Simple two-point rules• Not objective, but reproducible
• Need for multiple-point interactions
• Global a posteriori realism assessment [Caumon, 2010; de la Varga & Wellmann, 2018]
• Philosophical questions • Can we harness the curse of dimensionality (visualization, processing)?
• Can we hope to capture interpretive process in mathematical terms?
• Ethical use of technology
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G Caumon, G Godefroy & P Marchal EGU 2020 May 6th, 2020 31
Thanks
Consortium www.ring-team.org
Software: Emerson-Paradigm (SKUA), Schlumberger (Eclipse)
+ 138 universities
and surveys
Data: Equinor, CEREGE
Funding:
Collaborators: Nicolas Cherpeau, Jef Caers, Florian Wellmann,