Fekete Associates Inc - ihsmarkit.com · in Well Testing” CIPC 2003 Fekete Associates Inc. Well...

44
Louis Mattar “Analytical Solutions in Well Testing” CIPC 2003 Fekete Associates Inc

Transcript of Fekete Associates Inc - ihsmarkit.com · in Well Testing” CIPC 2003 Fekete Associates Inc. Well...

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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