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Louis Mattar “Analytical Solutions in Well Testing” CIPC 2003 Fekete Associates Inc
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### Transcript of Presentación Louis Mattar Testing - SPE

CIPC 2003

Louis MattarFekete Associates Inc Analytical Solutions in Well Testing

Well Test Equation

P P 1 P + 2 = 2 y t x2 2

Solutions Analytical Semi-Analytical

Numerical - Finite Difference Numerical - Finite Element Boundary Element? Analytical Element?

IssuesComplexity of Model / Solution Inverse Problem Diffusity

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 EquationsAnalytical Vs Numerical

P P 1 P + 2 = 2 t y x2 2

P

1 n+ 1 i,j

2P +P x

1 n+ i,j 2

1 n+ 1 i+,j

P +

1 n+ 1 i,j

2P +P y

1 n+ i,j 2

1 n+ 1 i+,j

1 P P = t

1 n+ i,j

n i,j

Numerical Vs Analytical

Analytical : -Whole reservoir -Single phase -Exact -Homogeneous * -Regular Boundaries**

Irregular Shapes

Irregular Boundaries

Modified Analytical ElementsA k1 k3

k2

Rubiks Cube (Integral Transforms)

Advantages of Analytical Solutions Better Understanding Cause and effect Physics of the process Groups that control response kh/q or k/c

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

Inverse Problem

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

Direct problem versus Inverse problemDirect Problem: 2 + 3 = ? Answer : 5 Inverse Problem: The Answer is 5. What are the two numbers ?

Characteristics of Inverse Problems non-unique solutions

K2 K1

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 appropriateNumerical models are too complex-too many Degrees of

FreedomReservoir 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.Pattern recognition Judgement Consistency checks

Much better than numerical simulator

Issue Diffusivity

Diffusive Nature of PTA 7 8 2 5 9 3 6

9 2 4 7 1 2 1 8 9 8 5 4 4 7 6 3 6 1 5 3

Diffusive Nature of PTA Homogeneous

5.2

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 distancebetween a production well and an observation well, without

any discernible differencebeing evident in interference test

Obs

Hole

Well

Linking Analytical & Numerical Obtain Analytical Solution Use geostatistical model to Generate Permeability Field Populate Numerical SimulatorExample Naturally Fractured Reservoir Determine Fractal Parameters Generate Permeability Field

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

Mathematics:Great Servant(Analytical)

Terrible Master(Numerical)

I would rather be vaguely right(Analytical)

than precisely wrong(Numerical)

Justification for NumericalDetailed Geological Description of complex reservoir geometry available in advance Multi-phase fluid flow, where gas is not fluid of interest Water cut matching