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