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

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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