presentación louis mattar testing - spe
TRANSCRIPT
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Louis Mattar
“Analytical Solutions
in Well Testing”
CIPC 2003
Fekete Associates Inc
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Well Test Equation
tP
yP
xP
∂∂
=∂∂
+∂∂
α1
2
2
2
2
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Solutions
•Analytical •Semi-Analytical
•Numerical - Finite Difference•Numerical - Finite Element
•Boundary Element?•Analytical Element?
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•Complexity of Model / Solution
•Diffusity
•Inverse Problem
Issues
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Issue
• Complexity
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Definitions
• Analytical Models solve theproblem directly (transforms,substitutions, calculus)
• Solution is continuous acrossdomain.
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Definitions• Numerical Simulation is the process
of dividing the reservoir intodiscrete blocks, having differentreservoir properties, in order todeal with complex reservoirproblems.
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Well Test EquationsAnalytical Vs Numerical
tP
yP
xP
∂∂
=∂∂
+∂∂
α1
2
2
2
2
tPP
yPPP
xPPP ji
nji
nnji
nji
nji
nji
nji
nji
∂−
=∆
+−+
∆+− ++
+++
−+
+++
− ,,1
2
1,1
1,
1,1
2
1,1
1,
1,1 122
α
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Numerical Vs Analytical
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•Analytical :
-Whole reservoir-Single phase-Exact-Homogeneous *-Regular Boundaries**
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Irregular Shapes
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Irregular Boundaries
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Modified Analytical ElementsA
k1
k3
k2
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Rubik’s Cube (Integral Transforms)
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• Better Understanding…• Cause and effect• Physics of the process•Groups that control response
kh/qµ or k/φµc
Advantages of AnalyticalSolutions
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Numerical
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•Numerical- Discretize Reservoir (1+ Million Cells)- Complex Reservoir- Heterogeneous- Computationally intensive- Multiphase**
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Complexity?
or
Simplicity?
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Occam's Razor
If you have two theorieswhich both explain theobserved facts then you
should use the simplest untilmore evidence comes along
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•Numerical Simulation may bejustified when reservoir complexitiesare known.
•But how often, or how well, do weknow these reservoir complexities, inadvance?
•Often these reservoir complexitiesare only discovered through testing
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Issue
• Inverse Problem
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Well test interpretation isessentially an Inverseproblem, and in general,is better suited toAnalytical Solutions
Inverse Problem
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“Direct” problemversus
“Inverse” problem
“Direct” Problem: 2 + 3 = ?Answer : 5
“Inverse” Problem:
The Answer is 5.What are the two numbers ?
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• non-unique solutions
Characteristics ofInverse Problems
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K1
K2
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2-Boundaries or Composite?
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A good looking history matchis not a good enough answer.
xx
xx
x
x
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The selected MODEL must be appropriate•Numerical models are toocomplex-too many Degrees ofFreedom•Reservoir Complexities areoften unknown a priory•Cannot see forest for trees
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Analytical models allow usto focus on the main issuesCreate a conceptual analysis.
Much better than numericalsimulator
•Pattern recognition
•Judgement
•Consistency checks
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Issue
• Diffusivity
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12
5
7
9
36
84
12
57
9
368
4
1
25
7 936
8
4
Diffusive Nature of PTA
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Homogeneous
5.2
Diffusive Nature of PTA
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The question is, how muchinformation is contained aboutthe spatial distribution ofpermeability in the well-testingresponse in a heterogeneousformation?
“Average” permeability in a regionNot
Permeability at a “fixed radius”
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Sageev and Horne (SPEFE 1988, 428 – 437)
“It is possible to have a hole in thereservoiras large as half the distancebetween a production well and anobservation well, withoutany discernible differencebeing evident in interference test”
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Hole WellObs
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Linking Analytical & Numerical
Example• Naturally Fractured Reservoir• Determine Fractal Parameters• Generate Permeability Field
•Obtain Analytical Solution•Use geostatistical model to
Generate Permeability Field• Populate Numerical Simulator
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Modeling can mean 2 things:
a) Model recognition from a set ofdata - Well Testing
b) using a model to forecast futureperformance - Numerical
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Conclusions:
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K eep
I t
S imple
S tupid
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Einstein
Everything should be made as simple as possible,
but not simpler
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Great Servant(Analytical)
•Terrible Master(Numerical)
Mathematics:Mathematics:
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I would rather be
vaguely right(Analytical)
than precisely wrong(Numerical)
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Justification for Numerical
•Detailed Geological Descriptionof “complex” reservoir geometryavailable in advance
•Multi-phase fluid flow, where gasis not fluid of interest
•Water cut matching