Workshop on Reliable Engineering Computing

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Workshop on Reliable Engineering Computing
UNCERTAINTY IN THERMAL BASIN MODELING - AN
INTERVAL FINITE ELEMENT APPROACH
Sebastião C. A. Pereira -
Petrobras RD Center - Brazil (System
Analyst Department of Geology and Geophysics)
Ulisses T. Mello -
Watson Research Center -IBM/USA Rafi L.
Muhanna - Georgia Institute of
Technology/USA Nelson F.F. Ebecken
- Federal University of Rio de Janeiro/Brazil
September 17, 2004 - Savannah, Georgia, USA
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UNCERTAINTY IN THERMAL BASIN MODELING
Contents

• Overview
• Formulation
• Implementation
• Application
• Conclusion

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Overview
Uncertainty is the only certainty in oil
exploration (J. H. Fang, 1990)

• Generation p1
• Migration p2
• Reservoir p3
• Geometry p4
• Retention p5
• Syncronism p6

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oil kitchen (kerogen)
(US)
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Overview
Heat Flux
f (lithosphere and crust stretching)
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Overview
(McKenzie, 1978)
IBV Initial Boundary Value (transient)
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Overview
Layer Sedimentation (can be Erosion) time scale
million of years
Well log
Mesh
1D Basin Modeling
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Thermal Calibration
vitrinite reflectance (laboratory) (maximum T
value along the time)
temperature (while drilling)
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Formulation
Diffusion Problem
Time dependence
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Thermal Diffusion
Thermal conductivity
Heat capacity
Convection coefficient
Galerkin discretization (implicit method)
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Porous media
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Transient Solution
deltt0.5 time step
(Ma) stime0 initial time
(Ma) ftime120.0 final time
(Ma) ntimefix((ftime-stime)/deltt) number of
times   kkcc/delttkk fsol0 for it1ntime
fnff(cc/deltt)fsol
compute effective column
kk,fnfeaplyc2(kk,fn,bcdof,bcval) apply
essential boundary fnfeaplyc2n(fn,bcdofn,bc
valn) apply natural boundary
fsolkk\fn
solve the interval system end
(mesh and properties are changing along the time)
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EBE with Elements Overlap
Element By Element
EBE static problems (Muhanna Mullen, 2001) EBE
overlap transient problems
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EBE with overlap
T1T5
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EBE com overlap
real
Interval diagonal matrix
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Implementation
Interval Linear System of Equations Library

• Preconditioning
• -Gauss Elimination
• -Gauss-Seidel
• Powell (optimization method), (Rao, 1989)
• Combinatorial

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Powell
achar X que minimize f(X) (Ax-b)0
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Combinatorial

• 10 interval numbers -gt

(binary representation with 10 bits
7001010111100 genetic algorithm )
for i11024 end
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Implementation

(S. Deodato, 1994) (ConfInt and FzNum classes)

• C (template, overload)

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How to use
//1 - S. NING and R. B. KEARFOTT (SIAM J.,
1997), A ex. 3.1 simMatDenseltConfIntgt
Aci(4,4) // ltdouble or FzNumgt
basArray1DNumericltConfIntgt bci(4)
basArray1DNumericltConfIntgt xci(4)  
Aci00 ConfInt (4.,6.) Aci01
ConfInt (-1.,1.) . bci0 ConfInt
(-2.,4.) bci1 ConfInt (1.,8.) .
// enum GE0, PGE, GS, PGS,
POWELL, COMBINATORIAL simLinearSolverltConfIn
t, simMatDenseltConfIntgt gt lsi lsi.solveLSE (
POWELL, Aci, bci, xci )   cout ltlt"solution
ex. 3.1 \n" cout ltlt xci ltlt endl
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How to use
Library source code is free
fuzzy numbers interval loop gt hull
simLinearSolverltFzNum, simMatBandltFzNumgt gt
lsi lsi.solveLSE ( POWELL, Af, bf, xf )
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GEOFEM
GEOFEM GEological applications Of the Finite
Element Method (1D - Research Software)
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Application
Outline

• Few elements
• Increase number of elements
• Increase the interval width
• Real data.

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Test 1
Few elements and narrow intervals
Thermal Conductivity (Watts/m.K)
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Results at node 3
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Increasing the number of elements
Test 2
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Increasing the intervals width
Test 3
Thermal Conductivity (Watts/m.K)
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Real Data
Test 4

• EBE
• Combinatorial
• Monte Carlo
• Crisp

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Results
4783 m
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Conclusions

• PGE, PGS, Powell do not work for large problems
• Combinatorial limited to many parameters
• Monte Carlo a lot of experiments
• IFE (EBE) state of the art .

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Next Steps

• More properties as interval heat capacity,
  density,
• - ( 100 parameters)
• Others modeling fluid flow, chemical modeling,
• 2D, 3D (millions of elements)

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Basin Modeling Softwares x Uncertainty

• Petromod IES-Integrated Exploration Systems
  (Germany)
• MC (Latin cube)
• Themis IFP - Institute Français Du Pétrole
  (France)
• Experimental Design, Response Surface Model , MC
• SimBR Petrobras IBM (Brazil)
• Interval Arithmetic (in the future)

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SimBR 3D Basin Simulator
Santos 170000 km2 Recôncavo
11000 km2 SimBR 15600 km2
SimBR
SimBR 435000, 565000 Leste
7125000, 7245000 Norte UTM, MC 45,
Datum Aratu
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SimBR 3D Basin Simulator
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SimBR 3D Basin Simulator
S A LT
Movimentação do Sal
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SimBR 3D Basin Simulator
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3D Thermal Modeling
R E C Ô N C A V O
7 days in a cluster with 64 nodes
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Aknowledgements
Dr. Rafi Muhanna
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Thank you !