The document discusses using pressure pulse testing to characterize heterogeneous reservoirs. It investigates how frequency response represents heterogeneity and formulates periodic steady-state solutions for radial and vertical permeability distributions. The overall procedure is to apply pressure pulses at multiple frequencies, analyze the attenuation and phase shift response, and determine the permeability distribution by solving the inverse problem. The goal is to provide a new method for characterizing heterogeneous reservoirs using frequency-domain analysis of pressure pulse test data.

1. Pressure Pulse Testing
in Heterogeneous Reservoirs
Sanghui Sandy Ahn
Advisor: Roland N. Horne
Department of Energy Resources Engineering
Stanford University
Jan 26, 2012

2. Pressure Pulse Testing Technique
• Apply periodic pressure pulses from an active well and
measure at an observation point to estimate the
heterogeneous permeability.
• Several cycles by alternating flow and shut-in period
• Data: time-series pressure signals pinj(t), pobs (t)
q(t)
pobs(t)
pinj(t) k?
2

3. Challenges for Estimating Permeability Distribution
and Opportunities for Pressure Pulse Technique
• Limited measurements
– Square pulses have spectrum of frequencies
– The lower the frequency, the longer the distance of
cyclic influence (Rosa, 1991).
• History matching is dependent on flow rate data
– Attenuation and phase shift information does not
require flow rate data.
• The pressure time series data can be large
– Attenuation and phase shift information reduces the
size of the data being analyzed.
3

4. Overview
• Background
• Two Reservoir Models
– Characterization by Multiple Frequency Data
• Detrending
– Transient Reconstruction
– Detrending on Injection & Observation Pressure
– Effect of Number and Position of Pulses
– Effect of Sampling Frequency and Noise
• Inverse Problem Formulation
• Synthetic Data- Permeability Estimation Results
– Sinusoidal Frequencies
– Harmonic Frequencies from Square Pulses
– Comparison between the Three Methods
– Sensitivity to Perturbation in Frequencies
– Storage and Skin Effects
• Real Data - Permeability Estimation Results
– Quantization Noise
– Pressure Matching Results
4

5. Previous Approach
for Estimating Average Permeability in Time Domain
• Used to estimate average permeability and
porosity by:
– Transmissivity
kh pD
141.2qB Amplitude reduction
p
– Storativity
Time lag
kh t
ct h 0.0002637 2
r 2 t D / rD
(Ryuzo, 1991) 5

6. Previous Approach
for Estimating Average Permeability in Time Domain
• Cross-plot of attenuation and phase shift at dominant
frequency
– Periodic steady-state solution in homogeneous radial system
2
1 p p 1 p
D
t r2 r r
K0 ( r ) ct
p( r ,t ) p0 exp(i t ) , i
K 0 ( rw ) k
i K 0 ( re )
x e
Attenuation K 0 ( rw )
Phase shift
(Bernabe, 2005)
6

7. Pressure Data to Reveal Heterogeneity
• Extracting heterogeneous permeability distribution from a
single well
sqrt(t)K(rD,tD) with t D =102
0.4
0.35
0.3
0.25
0.2
0.15
K1(rD,tD)
0.1
0.05
0
0 1 2
10 10 10
rD
kref (Oliver, 1992)
1 1
pwD (tD ) K1 (rD , tD )(1 )drD
2 21 k (rD )
Known
To estimate 7

8. Sourcing Multiple Frequencies by Square Pulses
1 1
3 3
+ 5 .. + 5 ..
8

9. Solvability Condition for Inverse Problem:
What Multiple Frequencies Can Do with Limited Spatial Measurements
Permeability estimation problem
i sin i t
{ pinj(t), pobs(t) }
Spectrum of frequencies
+ pinj(t)
pobs(t)
(Rosa, 1991 )
k1 k
2
…
? kn
Careful frequency selection is
Different frequency carries required for successful extraction.
different effective propagation length
9

10. Attenuation & Phase Shift
= Frequency Response
pinj(t) h(t) pobs(t)
Time domain
pinj(t) * h(t) = pobs(t)
FT FT FT
Pinj(ω) ∙ H(ω) = Pobs(ω)
Frequency domain
Pinj (ω) H(ω) Pobs(ω)
Pobs ( )
H( ) x ( )e i ( )
Pinj ( )
H ( ) : frequency response
Attenuation x( ) | H ( ) | x : attenuation
: phase shift
Phase shift ( ) arg(H ( ))
: frequency
10

11. Visualization of Attenuation and Phase Shift
c d
Attenuation
• Amplitude ratio a
=a/b
Output pressure
Time Shift (~ Phase Shift ) b
• Delay in cycle
Input pressure
=c/d
11

12. Visualization of Attenuation and Phase Shift
Attenuation Phase Shift
• Amplitude ratio • Delay normalized in cycle
pobs ( ) obs ( ) inj ( )
x( ) ( )
pinj ( ) 2
1 1
3 3
5 .. 5 ..
12

13. Objectives
Characterize heterogeneous reservoir models using
analysis of multiple frequencies:
• Investigate how a frequency response represents
heterogeneity.
• Formulate the periodic steady-state solutions for
radial and vertical permeability distributions.
• Provide a new method that utilizes attenuation and
phase shift information at multiple frequencies to
determine the permeability distribution.
• Provide the desirable pulsing conditions for using the
frequency method.
13

14. Overall Procedure
14

15. Overview
• Background
• Two Reservoir Models
– Characterization by Multiple Frequency Data
• Detrending
– Transient Reconstruction
– Detrending on Injection & Observation Pressure
– Effect of Number and Position of Pulses
– Effect of Sampling Frequency and Noise
• Inverse Problem Formulation
• Synthetic Data- Permeability Estimation Results
– Sinusoidal Frequencies
– Harmonic Frequencies from Square Pulses
– Comparison between the Three Methods
– Sensitivity to Perturbation in Frequencies
– Storage and Skin Effects
• Real Data - Permeability Estimation Results
– Quantization Noise
– Pressure Matching Results
15

16. Radial Heterogeneity Inspection
using Pressure Pulse Testing Technique
Pinj (ω) x (ω) kr(r)
Pobs(ω) θ (ω)
600 600 600
Radial permeability, kr, md 500 Model 1 500 Model 2 500 Model 3
400 400 400
300 300 300
200 200 200
100 100 100
0 0 0
0 200 400 600 0 200 400 600 0 200 400 600
Radial distance, r, ft 16

17. Vertical Heterogeneity Inspection
using Pressure Pulse Testing Technique
• Partially penetrating well with cross flow
Pinj (ω) x (ω) kv(h)
Pobs(ω) θ (ω)
Model 4 Model 5 Model 6
2 2 2
4 4 4
6 6 6
Depth, h, ft
8 8 8
10 10 10
12 12 12
14 14 14
16 16 16
18 18 18
0 10 20 0 10 20 0 10 20
Vertical permeability, kv, md 17

18. Frequency Response
for Radial Ring Model
• Periodic steady-state solution at multiple frequencies
• Using conditions: inner/outer boundary, continuity
• Attenuation and phase shift are obtained directly without
time information
→ Diffusivity equation
→ Steady state assumption
→ Pressure solution
→ Attenuation and phase shift
18

19. Frequency Response
for Multilayered Model
• Periodic steady-state solution at multiple frequencies
• Using conditions: inner/outer boundary
• Attenuation and phase shift are obtained directly without
time information
→ Diffusivity equation
→ Steady state assumption
→ Pressure solution
→ Attenuation and phase shift
19

20. Frequency Response and Permeability Distribution
• Attenuation and phase shift information at varying frequencies forms a
differentiating characteristic for heterogeneity.
• H ( k , i ) H (k ,1 i )
Radial Ring model Multilayered model, kv/kr =0.1
High High
frequency frequency
Low Low
frequency frequency
20

21. Appropriate Sourcing Frequency Range with kr
Over one cycle,
too high frequency
21

22. Appropriate Sourcing Frequency Range with kv
Over one cycle,
too high frequency
22

23. Extension to Heterogeneous Permeability Distribution
Not only an option
but a necessary step
23

24. Overview
• Background
• Two Reservoir Models
– Characterization by Multiple Frequency Data
• Detrending
– Transient Reconstruction
– Detrending on Injection & Observation Pressure
– Effect of Number and Position of Pulses
– Effect of Sampling Frequency and Noise
• Inverse Problem Formulation
• Synthetic Data- Permeability Estimation Results
– Sinusoidal Frequencies
– Harmonic Frequencies from Square Pulses
– Comparison between the Three Methods
– Sensitivity to Perturbation in Frequencies
– Storage and Skin Effects
• Real Data - Permeability Estimation Results
– Quantization Noise
– Pressure Matching Results
24

25. Detrending
• To eliminate the pressure transient and obtain frequency data
at periodically steady-state
• Challenge: flow rate is unknown
q0 2q0 1
q(t ) sin t sin3 t ...
2 3
(Different weight based on duty cycle)
120
50
40
100
Injection
Pressure change, psi
Pressure change, psi
Upward trend 30
Observation
80 Injection 20
Observation 10
True transient, injection
60 0
True transient, observation
Reconstructed transients -10
Removing
40 -20
transient
-30
20
-40
-50
5 10 15 20 25 5 10 15 20 25
Time, hr Time, hr 25

26. Transient Reconstruction
• A good reconstruction of the first transient is obtained by
using the periodicity
– The first transient curvature till its maximum peak
– Pivot points per period:
Linearly interpolate between pivots
For unequal pulses,
at least at every pivots αTp :
Iteratively compute 26

27. Detrending on Injection Pressure
50%
(square pulses)
75% duty cycle
25% duty cycle
27

28. Detrending on Observation Pressure
50%
(square pulses)
75% duty cycle
25% duty cycle
28

29. Effect of Detrending on Square Pulses
No dc component
Change in the decomposition
at high frequencies
29

30. Accurate Frequency Data Retrieval
by Detrending, Square Pulses Case
• Frequency attributes from the
detrended pressure matches
better to the sinusoidal space.
• The higher the sourcing
frequency, the more
discrepancies are shown
between the square pulse and
analytical sinusoidal case.
30

31. Effect of Number and Position of Pulses
Accurate frequency data with
- Larger number of pulses
- Pulses at later time
31

32. Effect of Sampling Frequency
MAE Summary of
10 realizations with
With sampling rate of 22.6, 5.7, and 1.4 sec
1% Normal pressure noise
N
i
xnoise x x
i 1
N
with
noise
N i
noise
i 1
N
Accurate frequency data with
- Higher sampling frequency 32

33. Effect of Sampling Frequency with Noise
Pressure Pulses with 128 pts /cycle
100 80
With noise With noise
70
80 Injection No noise Observation No noise
60
Magnitude (dB)
Magnitude (dB)
50
60
40
40 30
0.8 20
20
10
0.7 0
0
-10
0.6
-20 -20
0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500
Phase shift
Frequency, rad/hr Frequency, rad/hr
0.1 0.5 1
No noise No noise
0.09 0.9
0.08
0.4 0.8
0.07 0.7
Attenuation
0.3
Phase shift
0.06 0.6
0.05
10 realizations 0.5
0.04
with 1% Gaussian noise
0.2
0.4
0.03 0.3
0.1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0.02 0.2
0.01
Attenuation
0.1
0 0
0 50 100 150 200 250 0 50 100 150 200 250 33
Frequency, rad/hr Frequency, rad/hr

34. Overview
• Background
• Two Reservoir Models
– Characterization by Multiple Frequency Data
• Detrending
– Transient Reconstruction
– Detrending on Injection & Observation Pressure
– Effect of Number and Position of Pulses
– Effect of Sampling Frequency and Noise
• Inverse Problem Formulation
• Synthetic Data- Permeability Estimation Results
– Sinusoidal Frequencies
– Harmonic Frequencies from Square Pulses
– Comparison between the Three Methods
– Sensitivity to Perturbation in Frequencies
– Storage and Skin Effects
• Real Data - Permeability Estimation Results
– Quantization Noise
– Pressure Matching Results
34

35. Inverse Problem Formulation and Performance
BFGS Quasi-Newton method with a cubic line search
• Matching attenuation and phase shift at multiple frequencies
– Computation: O(2Nw), with Nw frequencies
2 2 2 2
min x x(k, 1) ... x x(k, n) (k, 1) ... (k, n)
k 1 2 n 2 1 2 n 2
• Pressure history matching
2 2
min pt1 p(k, t1 ) ... ptm p(k, t m )
k 2 2
– Computation: O(2Nt*Ns), with Nt time series & Ns Stehfest coefficients
• Wavelet thresholding
2 2
min wt1 w(k, t1 ) ... wtl w(k, tl )
k 2 2
– Computation: similar to pressure history matching
35

36. Pressure Reconstruction
• Reconstruction of pressure by varying number
of wavelets
36

37. Computational Effort Comparison
• Convergence over iterations
Example of computational effort
1. History matching and Wavelet: ~ 30 mins
- Time points: 5000
- Stehfest: 8
2. Frequency information: ~ 30 secs
- Frequency points: 10
37

38. Overview
• Background
• Two Reservoir Models
– Characterization by Multiple Frequency Data
• Detrending
– Transient Reconstruction
– Detrending on Injection & Observation Pressure
– Effect of Number and Position of Pulses
– Effect of Sampling Frequency and Noise
• Inverse Problem Formulation
• Synthetic Data- Permeability Estimation Results
– Sinusoidal Frequencies
– Harmonic Frequencies from Square Pulses
– Comparison between the Three Methods
– Sensitivity to Perturbation in Frequencies
– Storage and Skin Effects
• Real Data - Permeability Estimation Results
– Quantization Noise
– Pressure Matching Results
38

39. Parameter Estimation Result for Radial Ring Model
Using Multiple Sinusoidal Frequencies
3
Model 1
4
2
1
5: rDc inf j
1.1 1 Dj (radius of cyclic influence)
Model 2 Model 3
39

40. Parameter Estimation Result for Multilayered Model
Using Multiple Sinusoidal Frequencies
3
Model 4
4
2
1
Model 6
Model 5
40

41. Parameter Estimation Result for Radial Ring Model
Using Varying Number of Sinusoidal Frequencies
Model 1
• Estimation with three or more
frequency components resulted in a
good match with the true distribution.
Model 3
Model 2
41

42. Parameter Estimation Result for Multilayered Model
Using Varying Number of Sinusoidal Frequencies
Model 4
• Estimation with three or more
frequency components resulted in a
good match with the true distribution.
Model 6
Model 5
42

43. Parameter Estimation Result for Radial Ring Model
Using Harmonic Frequencies from Square Pulses
• Model 2, comparison between three methods
No noise With 1% Gaussian noise in pressure
43

44. Parameter Estimation Result for Multilayered Model
Using Harmonic Frequencies from Square Pulses
• Model 6, comparison between three methods
No noise With 1% Gaussian noise in pressure
44

45. Robustness Check on Radial Ring Model
by Perturbation in Frequency Space
Model 1
Model 2 Model 3
45

46. Robustness Check on Multilayered Model
by Perturbation in Frequency Space
Model 4
Model 5 Model 6
46

47. Storage Effect
Periodic steady-state space remains the same
47

48. Skin Effect
With skin factor in the injection well:
• Injection pressure changes → periodic steady-state space changes
48

49. Combined Effect of Storage and Skin
Multiple distributions are possible
with unknown skin factor
• The larger the CD and skin, the more discrepancy with a steady state
model is observed
• Only a few low frequency points are reliable in steady state space.
cf. Sinusoidal model remains unchanged with varying CD
49

50. Storage and Skin Estimation
• Estimate from a constant rate pressure
response with a permeability estimation
– Storage:
– Skin (assuming that the skin effect is small)
50

51. Overview
• Background
• Two Reservoir Models
– Characterization by Multiple Frequency Data
• Detrending
– Transient Reconstruction
– Detrending on Injection & Observation Pressure
– Effect of Number and Position of Pulses
– Effect of Sampling Frequency and Noise
• Inverse Problem Formulation
• Synthetic Data- Permeability Estimation Results
– Sinusoidal Frequencies
– Harmonic Frequencies from Square Pulses
– Comparison between the Three Methods
– Sensitivity to Perturbation in Frequencies
– Storage and Skin Effects
• Real Data - Permeability Estimation Results
– Quantization Noise
– Pressure Matching Results
51

52. Quantization Noise on Pressure
5 5
Pressure change, injection (psi)
Pressure change, injection (psi)
4.5 4.5
Original Original
4 4
Discretized Discretized
3.5 Quantization error 3.5 Quantization error
3 3
2.5 2.5
2 2
1.5 1.5
1 1
0.5 0.5
0 0
0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
Time, hr Time, hr
• Discretization in time • Discretization in pressure amplitude
- Finite precision to record in time - Finite bit-representation for
magnitudes
52

53. Quantization Noise
Aliasing effect
Quantization noise in time
in time
White noise
in pressure White noise
amplitude Quantization noise
in pressure amplitude
53

54. Field Data 1
Transient Extraction
54

55. Field Data 1
Detrending and Spectrum Analysis
55

56. Field Data 1
Radial Permeability Estimate by the Frequency Method
56

57. Field Data 1
Radial Permeability Estimate in Comparison with History Matching
CD = 10000
s = 0.2
57

58. Conclusions
• Developed framework for estimating permeability distribution using
frequency attributes
– Periodic steady-state solutions for radial and mutilayered models
– Detrending is established without flow information, which brings a
clearer periodicity in the pressure data
– Utilization of harmonic frequency contents
• Conditions for accurate frequency attributes to periodically steady
state:
– Sufficient attenuation and phase shift data pairs
– Greater number of pulses
– Higher sampling rate
– Pulses at later time
– Beyond wellbore storage and skin effects: tD > CD(60 + 3.5s)
• Compared to history matching and wavelet thresholding:
– No need to know the flow information
– Less computational effort
– Can perform as good as history matching
58

59. Limitations of the Frequency Method
• Storage and skin should be determined separately from the
frequency method.
• Only several harmonics are useful from real pulsing data due
to noise.
• The available frequency components may not be enough to
cover the whole distance range.
59

60. Acknowledgements
• Prof. Roland Horne, Lou Durlofsky, Jef Caers,
Tapan Mukerji, and Michael Saunders
• Department of Energy Resources Engineering
Faculty, Staff, and Students
• Shell
• SUPRI-D members
60

61. Pressure Pulse Testing
in Heterogeneous Reservoirs
Thank you!
Q&A
Sanghui Sandy Ahn
Energy Resources Engineering, Stanford University
61

62. Supplementary slides
62

63. Abstraction of Pressure Transmission (1)
[Model 1]
[Model 2] [Model 3]
63

64. Abstraction of Pressure Transmission (2)
• By attenuation and phase shift
• General trend from the injection well:
– Decreasing attenuation and increasing phase shift
• Distinctive heterogeneity appearing as different slopes
64

65. Decomposition by Pulse Shapes
• Odd multiples of the sourcing frequencies are available.

66. Sensitivity to Boundary Conditions
lim p jD 0
rD Infinite reservoir
p jD
0 No flow
rD rD re D
p jD (reD , t D ) 0 Constant pressure
Multilayered Model
Radial Ring Model
66

67. Parameter Estimation Result for Radial Ring Model
Using Harmonic Frequencies from Square Pulses
• Model 1, comparison between three methods
No noise With 1% Gaussian noise in pressure
67

68. Parameter Estimation Result for Radial Ring Model
Using Harmonic Frequencies from Square Pulses
• Model 3, comparison between three methods
No noise With 1% Gaussian noise in pressure
68

69. Parameter Estimation Result for Multilayered Model
Using Harmonic Frequencies from Square Pulses
• Model 4, comparison between three methods
No noise With 1% Gaussian noise in pressure
69

70. Parameter Estimation Result for Multilayered Model
Using Harmonic Frequencies from Square Pulses
• Model 5, comparison between three methods
No noise With 1% Gaussian noise in pressure
70

71. Wavelet Thresholding
71

72. Future Work
More examples to apply the frequency method
• Incorporating horizontal well configuration, fractured
reservoirs, etc.
• Water and oil relative permeabilities estimation
72