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Transcript of Fekete Associates Inc - in Well Testing¢â‚¬â€Œ CIPC 2003...

  • Louis Mattar

    “Analytical Solutions

    in Well Testing”

    CIPC 2003

    Fekete Associates Inc

  • Well Test Equation

    t P

    y P

    x P

    ∂ ∂

    = ∂ ∂

    + ∂ ∂

    α 1

    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 the problem directly (transforms, substitutions, calculus)

    • Solution is continuous across domain.

  • Definitions • Numerical Simulation is the process

    of dividing the reservoir into discrete blocks, having different reservoir properties, in order to deal with complex reservoir problems.

  • Well Test Equations Analytical Vs Numerical

    t P

    y P

    x P

    ∂ ∂

    = ∂ ∂

    + ∂ ∂

    α 1

    2

    2

    2

    2

    t PP

    y PPP

    x PPP jinjin

    n ji

    n ji

    n ji

    n ji

    n ji

    n ji

    ∂ −

    = ∆

    +− +

    ∆ +− +++

    ++ −

    + +

    ++ − ,,

    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 Analytical Solutions

  • Numerical

  • •Numerical - Discretize Reservoir (1+ Million Cells) - Complex Reservoir - Heterogeneous - Computationally intensive - Multiphase**

  • Complexity?

    or

    Simplicity?

  • Occam's Razor

    If you have two theories which both explain the observed facts then you

    should use the simplest until more evidence comes along

  • •Numerical Simulation may be justified when reservoir complexities are known.

    •But how often, or how well, do we know these reservoir complexities, in advance?

    •Often these reservoir complexities are only discovered through testing

  • Issue

    • Inverse Problem

  • Well test interpretation is essentially an Inverse problem, and in general, is better suited to Analytical Solutions

    Inverse Problem

  • “Direct” problem versus

    “Inverse” problem “Direct” Problem: 2 + 3 = ? Answer : 5

    “Inverse” Problem:

    The Answer is 5. What are the two numbers ?

  • • non-unique solutions

    Characteristics of Inverse Problems

  • K1

    K2

  • 2-Boundaries or Composite?

  • A good looking history match is not a good enough answer.

    x x

    x x

    x

    x

  • The selected MODEL must be appropriate •Numerical models are too complex-too many Degrees of Freedom •Reservoir Complexities are often unknown a priory •Cannot see forest for trees

  • Analytical models allow us to focus on the main issues Create a conceptual analysis.

    Much better than numerical simulator

    •Pattern recognition

    •Judgement

    •Consistency checks

  • Issue

    • Diffusivity

  • 12

    5

    7

    9

    3 6

    8 4

    1 2

    5 7

    9

    36 8

    4

    1

    2 5

    7 9 3 6

    8

    4

    Diffusive Nature of PTA

  • Homogeneous

    5.2

    Diffusive Nature of PTA

  • The question is, how much information is contained about the spatial distribution of permeability in the well-testing response in a heterogeneous formation?

    “Average” permeability in a region Not

    Permeability at a “fixed radius”

  • Sageev and Horne (SPEFE 1988, 428 – 437)

    “It is possible to have a hole in the reservoir as large as half the distance between a production well and an observation well, without any discernible difference being 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 of data - Well Testing

    b) using a model to forecast future performance - 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 Description of “complex” reservoir geometry available in advance

    •Multi-phase fluid flow, where gas is not fluid of interest

    •Water cut matching

    Complexity of Model / Solution Issue Definitions Definitions Better Understanding… Issue Issue Justification for Numerical