CIPC Louis Mattar. Fekete Associates Inc. Analytical Solutions in Well Testing

Transcription

1 CIPC 2003 Louis Mattar Fekete Associates Inc Analytical Solutions in Well Testing

2 Well Test Equation 2 P 2 P 1 P + = x 2 y 2 α t

3 Solutions Analytical Semi-Analytical Numerical - Finite Difference Numerical - Finite Element Boundary Element? Analytical Element?

4 Issues Complexity of Model / Solution Inverse Problem Diffusity

5 Complexity Issue

6 Definitions Analytical Models solve the problem directly (transforms, substitutions, calculus) Solution is continuous across domain.

7 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.

8 Well Test Equations Analytical Vs Numerical t P y P x P = + α t P P y P P P x P P P j i n j i n n j i n j i n j i n j i n j i n j i = ,, , 1, 1 1, 2 1 1, 1, 1, α

9 Numerical Vs Analytical

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

11 Irregular Shapes

12 Irregular Boundaries

13 Modified Analytical ElementsA k1 k2 k3

14 Rubik s Cube (Integral Transforms)

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

16 Numerical

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

18 Complexity? or Simplicity?

19 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

20 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

21 Issue Inverse Problem

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

23 Direct problem versus Inverse problem Direct Problem: =? Answer : 5 Inverse Problem: The Answer is 5. What are the two numbers?

24 Characteristics of Inverse Problems non-unique solutions

25 K1 K2

26 2-Boundaries or Composite?

27 A good looking history match is not a good enough answer. x x x x x x

28 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

29 Analytical models allow us to focus on the main issues Create a conceptual analysis. Pattern recognition Judgement Consistency checks Much better than numerical simulator

30 Diffusivity Issue

31 Diffusive Nature of PTA

32 Diffusive Nature of PTA Homogeneous 5.2

33 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

34 Sageev and Horne (SPEFE 1988, ) 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

35 Obs Hole Well

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

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38 Modeling can mean 2 things: a) Model recognition from a set of data - Well Testing b) using a model to forecast future performance - Numerical

39 Conclusions:

40 K eep I t S imple S tupid

41 Einstein Everything should be made as simple as possible, but not simpler

42 Mathematics: Great Servant (Analytical) Terrible Master (Numerical)

43 I would rather be vaguely right (Analytical) than precisely wrong (Numerical)

44 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