fekete associates inc - ihsmarkit.com · in well testing” cipc 2003 fekete associates inc. well...
TRANSCRIPT
Louis Mattar
“Analytical Solutions
in Well Testing”
CIPC 2003
Fekete Associates Inc
Well Test Equation
tP
yP
xP
∂∂
=∂∂
+∂∂
α1
2
2
2
2
Solutions
•Analytical •Semi-Analytical
•Numerical - Finite Difference•Numerical - Finite Element
•Boundary Element?•Analytical Element?
•Complexity of Model / Solution
•Diffusity
•Inverse Problem
Issues
Issue
• Complexity
Definitions
• Analytical Models solve theproblem directly (transforms,substitutions, calculus)
• Solution is continuous acrossdomain.
Definitions• Numerical Simulation is the process
of dividing the reservoir intodiscrete blocks, having differentreservoir properties, in order todeal with complex reservoirproblems.
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
Numerical Vs Analytical
•Analytical :
-Whole reservoir-Single phase-Exact-Homogeneous *-Regular Boundaries**
Irregular Shapes
Irregular Boundaries
Modified Analytical ElementsA
k1
k3
k2
Rubik’s Cube (Integral Transforms)
• Better Understanding…• Cause and effect• Physics of the process•Groups that control response
kh/qµ or k/φµc
Advantages of AnalyticalSolutions
Numerical
•Numerical- Discretize Reservoir (1+ Million Cells)- Complex Reservoir- Heterogeneous- Computationally intensive- Multiphase**
Complexity?
or
Simplicity?
Occam's Razor
If you have two theorieswhich both explain theobserved facts then you
should use the simplest untilmore evidence comes along
•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
Issue
• Inverse Problem
Well test interpretation isessentially an Inverseproblem, and in general,is better suited toAnalytical Solutions
Inverse Problem
“Direct” problemversus
“Inverse” problem
“Direct” Problem: 2 + 3 = ?Answer : 5
“Inverse” Problem:
The Answer is 5.What are the two numbers ?
• non-unique solutions
Characteristics ofInverse Problems
K1
K2
2-Boundaries or Composite?
A good looking history matchis not a good enough answer.
xx
xx
x
x
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
Analytical models allow usto focus on the main issuesCreate a conceptual analysis.
Much better than numericalsimulator
•Pattern recognition
•Judgement
•Consistency checks
Issue
• Diffusivity
12
5
7
9
36
84
12
57
9
368
4
1
25
7 936
8
4
Diffusive Nature of PTA
Homogeneous
5.2
Diffusive Nature of PTA
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”
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”
Hole WellObs
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
Modeling can mean 2 things:
a) Model recognition from a set ofdata - Well Testing
b) using a model to forecast futureperformance - Numerical
Conclusions:
K eep
I t
S imple
S tupid
Einstein
Everything should be made as simple as possible,
but not simpler
Great Servant(Analytical)
•Terrible Master(Numerical)
Mathematics:Mathematics:
I would rather be
vaguely right(Analytical)
than precisely wrong(Numerical)
Justification for Numerical
•Detailed Geological Descriptionof “complex” reservoir geometryavailable in advance
•Multi-phase fluid flow, where gasis not fluid of interest
•Water cut matching