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Shell Exploration & Production3/1
/06
File
Title
Copyright: S
hell
Explo
ration &
Pro
duction L
td.
A Algebraic Computations
on Noisy, Measured Data
Daniel Heldt, Sebastian Pokutta, Hennie Poulisse
Based on Joint On-Going Work:Based on Joint On-Going Work:
Daniel Daniel HeldtHeldt, Martin , Martin KreuzerKreuzer, Sebastian , Sebastian PokuttaPokutta, , Hennie PoulisseHennie Poulisse
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Shell Exploration & Production
ContentsContents
•• Industrial Application of Computer Algebra Industrial Application of Computer Algebra
•• ApVI ApVI CalculatedCalculated
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Shell Exploration & Production
The ‘Champions Field’, South-Chinese Sea, offshore Brunei
Going out into the Field ! Going out into the Field !
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Shell Exploration & Production
(test- and bulk) separatorsheaders
transportation tubing well heads
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Shell Exploration & Production
Selected Well on Test Test Separator
(3 Phase)
Gas
Oil
Water
WELL MEASUREMENTS
THP, DHP, THT,
FLP, LGF etc. TEST SEPARATOR
MEASUREMENTS
imitating production imitating production
circumstances: circumstances:
‘‘deliberatedeliberate’’ disturbances disturbances
Surface – and Sub-surface
‘‘inputsinputs’’ ‘‘outputsoutputs’’
output = function of inputsoutput = function of inputs
output =output =
inputinput
physical relation between inputsphysical relation between inputs
variables of polynomial functionvariables of polynomial function
polynomial functionpolynomial function
Production Relation for an Oil Well Production Relation for an Oil Well
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Shell Exploration & Production
example of a polynomial well productionexample of a polynomial well production
‘‘lowlow’’ degree, degree, ‘‘largelarge’’ number of number of indeterminatesindeterminates
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Shell Exploration & Production
Motivating ExampleMotivating ExampleGas
Oil
Water
transportationtransportation
LineLine
production well Aproduction well A
production well Bproduction well B
production well Cproduction well C
production well Dproduction well D
Bulk SeparatorBulk Separator
(3 Phase)(3 Phase)
total total
productionproduction
constructed polynomialsconstructed polynomials
measuredmeasured
construct polynomialconstruct polynomial
polynomials: interactions between productionspolynomials: interactions between productions
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Shell Exploration & Production
construction of well production polynomialconstruction of well production polynomial
indeterminatesindeterminates
production polynomial production polynomial f f has to be fitted to a set of points:has to be fitted to a set of points:
evaluations of evaluations of indeterminates indeterminates at first data pointat first data point
number of points in the order of thousands not uncommonnumber of points in the order of thousands not uncommon
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Shell Exploration & Production
important observation:important observation:
•• different polynomials (supports, degrees) different polynomials (supports, degrees)
•• evaluations close together evaluations close together
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Shell Exploration & Production
relations in the data, among the relations in the data, among the indeterminates indeterminates
transitivity fails transitivity fails
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Shell Exploration & Production
Small Polynomials:Small Polynomials:
Vanishing IdealVanishing Ideal
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Shell Exploration & Production
exampleexample
univariate univariate polynomialpolynomial
perturb pointsperturb points
polynomial of degree 5 vanishing on pointspolynomial of degree 5 vanishing on points
•• remove the remove the purturbations purturbations first before constructing the polynomialsfirst before constructing the polynomials
•• construct polynomial allowing it to pass construct polynomial allowing it to pass ‘‘close byclose by’’ prescribed points prescribed points
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Shell Exploration & Production
delta-Approximatedelta-Approximate Vanishing Ideal Vanishing Ideal
of course we still have:of course we still have:
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Shell Exploration & Production
Empirical Ring:Empirical Ring:
Empirical Polynomial:Empirical Polynomial:
Best thing one can do:Best thing one can do:
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Shell Exploration & Production
HOW TO CALCULATE THE HOW TO CALCULATE THE
ApVI ApVI
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Shell Exploration & Production
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Shell Exploration & Production
Classical Classical VersionVersion
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Shell Exploration & Production
Drawbacks:
• The found relations are far away from vanishing on the points:
• numerical error prevent accurate calculation
• exact solution would result in an O with #O = #P (1000 here!!)
• The exact relations (for the perturbed data) are usually of very high degree
How to overcome these problems:
• Do not force the relations to pass through every point, but demand passing „close
by“
• Allow points which lie “close together” to be “melt together”
• “Divide the good ones from the bad ones”
• Process blocks rather than single elements to (hopefully) prevent sub-optimal
solutions and speed-up computations
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Shell Exploration & Production
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Shell Exploration & Production
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Shell Exploration & Production
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Shell Exploration & Production
Approximate Vanishing Approximate Vanishing IdealIdeal
with with Singular Singular Value DecompositionValue Decomposition
Truncate below
eps = 0.001
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Shell Exploration & Production
Approximate Vanishing Approximate Vanishing IdealIdeal
with with Singular Singular Value DecompositionValue Decomposition
Truncate below
eps = 0.01
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Shell Exploration & Production
Approximate Vanishing Approximate Vanishing IdealIdeal
with with Singular Singular Value DecompositionValue Decomposition
Truncate below
eps = 0.1
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Shell Exploration & Production
Approximate Vanishing Approximate Vanishing IdealIdeal
with with Singular Singular Value DecompositionValue Decomposition
Truncate below
eps = 0.5
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Shell Exploration & Production
Another Another ((shortshort) ) exampleexample......
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Shell Exploration & Production
Approximate Vanishing Approximate Vanishing IdealIdeal
with with Singular Singular Value DecompositionValue Decomposition
Truncate below
eps = 0.7
Classical Version
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Shell Exploration & Production
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Shell Exploration & Production
Timings (GB-Version)Timings (GB-Version)
(2024 (2024 pointspoints, 9 , 9 indetsindets))
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Shell Exploration & Production
Timings (BB-Version)Timings (BB-Version)
(2024 (2024 pointspoints, 9 , 9 indetsindets))
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Shell Exploration & Production
Application within Application within ShellShell
Calculate relations for the productions...Typical datasize: 2000-5000 points in up to 15 indets
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Shell Exploration & Production
Application within steel industriesApplication within steel industries
Application fields:• predict production quality
• automated quality assurance
Typical datasize: 10000-15000 points in 10-15 indets
• solution time < 238.73 s
• #G = 464, #O = 391
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Shell Exploration & Production
AcknowledgementAcknowledgement::
C. Fassino and J. Abbott worked in parallel on an algorithm which also computes almost
vanishing ideals, but with a different approach... See:
C. Fassino. An Approximation to the Gröbner Basis of Ideals of Perturbed Point. Preprint (2006)
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Shell Exploration & Production
Thank you! !Thank you! !