dissertation slides
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My past 4 yrs of research - a new method for reservoir characterization by multiple frequenciesTRANSCRIPT
Pressure Pulse Testing in Heterogeneous Reservoirs
Sanghui Sandy AhnAdvisor: Roland N. Horne
Department of Energy Resources EngineeringStanford University
Jan 26, 2012
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)
2
k ?pinj(t)
pobs(t)
q(t)
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
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
Previous Approach for Estimating Average Permeability in Time Domain
• Used to estimate average permeability and porosity by:
– Transmissivity
– Storativity
5
p
pqB
kh D2.141
22 /0002637.0
DD
trtr
tkhhc
Amplitude reduction
Time lag
(Ryuzo, 1991)
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
6
)(
)(
0
0
w
ei
rK
rKex
)(
)()exp(
0
00),(
w
trrK
rKtipp i
k
ct,
r
p
rr
p
t
pD
12
21
Phase shiftAttenuation
(Bernabe, 2005)
Pressure Data to Reveal Heterogeneity
• Extracting heterogeneous permeability distribution from a single well
7
(Oliver, 1992)
100
101
102
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
rD
sqrt(t)K(rD,tD) with tD =102
D
D
ref
DDDwD drrk
ktrKtp )
)(1)(,(
2
1
2
1)(
11
K1(rD,tD)
KnownTo estimate
Sourcing Multiple Frequencies by Square Pulses
8
+ +
1
3
..5
1
3
..5
Solvability Condition for Inverse Problem:What Multiple Frequencies Can Do with Limited Spatial Measurements
9
(Rosa, 1991 )
Spectrum of frequencies
Different frequency carries different effective propagation length
+
tii sin
…kn
k2k1
pinj(t)pobs(t)
{ pinj(t), pobs(t) }
Permeability estimation problem
Careful frequency selection is required for successful extraction.
?
Attenuation & Phase Shift = Frequency Response
10
pinj(t) pobs(t)h(t)
Pinj (ω) Pobs(ω)H(ω)
Time domain
Frequency domain
FT FT FT
pinj(t) * h(t) = pobs(t)
Pinj(ω) ∙ H(ω) = Pobs(ω)
Attenuation
Phase shift
|)(|)( Hx
))(arg()( H
:)(H frequency response
:x attenuation
phase shift:
: frequency
)()()(
)()( i
inj
obs exP
PH
Input pressure
Output pressure
b
a
c d
Visualization of Attenuation and Phase Shift
Attenuation
• Amplitude ratio= a / b
Time Shift (~ Phase Shift )
• Delay in cycle= c / d
11
Visualization of Attenuation and Phase Shift
12
Attenuation• Amplitude ratio
)(
)()(
inj
obs
p
px
Phase Shift• Delay normalized in cycle
2
)()()(
injobs
1
3..5
1
3..5
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
Overall Procedure
14
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
Radial Heterogeneity Inspection using Pressure Pulse Testing Technique
16
kr(r)x (ω)θ (ω)
Pinj (ω)Pobs(ω)
0 200 400 6000
100
200
300
400
500
600
Radia
l perm
eabili
ty, k
r, md
0 200 400 6000
100
200
300
400
500
600
Radial distance, r, ft0 200 400 600
0
100
200
300
400
500
600
Model 1 Model 2 Model 3
Vertical Heterogeneity Inspection using Pressure Pulse Testing Technique
• Partially penetrating well with cross flow
17
x (ω)θ (ω)
Pinj (ω)Pobs(ω)
kv(h)
0 10 20
2
4
6
8
10
12
14
16
18
Depth
, h, ft
0 10 20
2
4
6
8
10
12
14
16
18
Vertical permeability, kv, md
0 10 20
2
4
6
8
10
12
14
16
18
Model 4 Model 5 Model 6
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
18
→ Attenuation and phase shift
→ Pressure solution
→ Steady state assumption
→ Diffusivity equation
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
19
→ Attenuation and phase shift
→ Pressure solution
→ Steady state assumption
→ Diffusivity equation
Frequency Response and Permeability Distribution
• Attenuation and phase shift information at varying frequencies forms a differentiating characteristic for heterogeneity.
•
20
Radial Ring model Multilayered model, kv/kr =0.1
)1,(),( iHiH kk
Low frequency
Highfrequency
Low frequency
Highfrequency
21
Over one cycle,too high frequency
Appropriate Sourcing Frequency Range with kr
Appropriate Sourcing Frequency Range with kv
22
Over one cycle,too high frequency
Extension to Heterogeneous Permeability Distribution
23
Not only an option but a necessary step
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
Detrending• To eliminate the pressure transient and obtain frequency data
at periodically steady-state
• Challenge: flow rate is unknown
255 10 15 20 25
20
40
60
80
100
120
Time, hr
Pre
ssure
change, psi
Injection
Observation
True transient, injection
True transient, observation
Reconstructed transients
5 10 15 20 25
-50
-40
-30
-20
-10
0
10
20
30
40
50
Time, hr
Pre
ssure
change, psi
Injection
Observation
...3sin3
1sin
2
2)( 00 tt
qqtq
Removing transient
Upward trend
(Different weight based on duty cycle)
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:
26
For unequal pulses, at least at every pivots αTp :
Linearly interpolate between pivots
Iteratively compute
27
50%(square pulses)
25% duty cycle
75% duty cycle
Detrending on Injection Pressure
28
50%(square pulses)
25% duty cycle
75% duty cycle
Detrending on Observation Pressure
29
No dc component
Change in the decomposition at high frequencies
Effect of Detrending on Square Pulses
• 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
Accurate Frequency Data Retrieval by Detrending, Square Pulses Case
Effect of Number and Position of Pulses
31
Accurate frequency data with- Larger number of pulses - Pulses at later time
Effect of Sampling Frequency
32
Accurate frequency data with- Higher sampling frequency
with noise
MAE Summary of10 realizations with1% Normal pressure noise
N
xxxN
i
i
noise1
N
N
i
inoise
1
With sampling rate of 22.6, 5.7, and 1.4 sec
0 50 100 150 200 250 300 350 400 450 500-20
0
20
40
60
80
100
Magnitude (
dB
)
Frequency, rad/hr
With noise
No noise
0 50 100 150 200 2500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Frequency, rad/hr
Attenuation
No noise
0 50 100 150 200 250 300 350 400 450 500-20
-10
0
10
20
30
40
50
60
70
80
Magnitude (
dB
)
Frequency, rad/hr
With noise
No noise
0 50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency, rad/hr
Phase s
hift
No noise
Effect of Sampling Frequency with Noise Pressure Pulses with 128 pts /cycle
33
10 realizations with 1% Gaussian noise
Injection Observation
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Attenuation
Phase s
hift
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
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
• Pressure history matching
– Computation: O(2Nt*Ns), with Nt time series & Ns Stehfest coefficients
• Wavelet thresholding
– Computation: similar to pressure history matching
35
2
2
2
21 ),(...),(min
1 mtt tpptppm
kkk
2
2
2
21
2
2
2
21 ),(...),(),(...),(min
11nn nn
xxxx kkkkk
2
2
2
21 ),(...),(min
1 ltt twwtwwl
kkk
Pressure Reconstruction
• Reconstruction of pressure by varying number of wavelets
36
Computational Effort Comparison
• Convergence over iterations
37
Example of computational effort1. History matching and Wavelet: ~ 30 mins
- Time points: 5000 - Stehfest: 8
2. Frequency information: ~ 30 secs- Frequency points: 10
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
1
2
3
4
jj DDcr 11.1inf5:
Parameter Estimation Result for Radial Ring ModelUsing Multiple Sinusoidal Frequencies
(radius of cyclic influence)
Model 1
Model 3Model 2
40
1
2
3
4Model 4
Model 6
Model 5
Parameter Estimation Result for Multilayered ModelUsing Multiple Sinusoidal Frequencies
• Estimation with three or more frequency components resulted in a good match with the true distribution.
41
Parameter Estimation Result for Radial Ring ModelUsing Varying Number of Sinusoidal Frequencies
Model 1
Model 3
Model 2
• Estimation with three or more frequency components resulted in a good match with the true distribution.
42
Model 4
Model 6
Model 5
Parameter Estimation Result for Multilayered ModelUsing Varying Number of Sinusoidal Frequencies
Parameter Estimation Result for Radial Ring ModelUsing Harmonic Frequencies from Square Pulses
• Model 2, comparison between three methods
43
No noise With 1% Gaussian noise in pressure
Parameter Estimation Result for Multilayered ModelUsing Harmonic Frequencies from Square Pulses
• Model 6, comparison between three methods
44
No noise With 1% Gaussian noise in pressure
45
Robustness Check on Radial Ring Modelby Perturbation in Frequency Space
Model 1
Model 3Model 2
46
Robustness Check on Multilayered Modelby Perturbation in Frequency Space
Model 4
Model 6Model 5
47
Storage Effect
Periodic steady-state space remains the same
Skin EffectWith skin factor in the injection well:
• Injection pressure changes → periodic steady-state space changes
48
Combined Effect of Storage and Skin
49
• 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
Multiple distributions are possiblewith unknown skin factor
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
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
Quantization Noise on Pressure
52
0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Pre
ssure
change, in
jection (
psi)
Time, hr
Original
Discretized
Quantization error
0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Pre
ssure
change, in
jection (
psi)
Time, hr
Original
Discretized
Quantization error
• Discretization in time
- Finite precision to record in time
• Discretization in pressure amplitude
- Finite bit-representation for magnitudes
Quantization Noise
53
in time
in pressureamplitude
Aliasing effect
White noise
Quantization noise in time
Quantization noisein pressure amplitude
White noise
54
Field Data 1Transient Extraction
55
Field Data 1Detrending and Spectrum Analysis
56
Field Data 1Radial Permeability Estimate by the Frequency Method
57
CD = 10000s = 0.2
Field Data 1Radial Permeability Estimate in Comparison with History Matching
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
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
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
Thank you!
Q & A
Sanghui Sandy Ahn
Energy Resources Engineering, Stanford University
61
Pressure Pulse Testing in Heterogeneous Reservoirs
Supplementary slides
62
Abstraction of Pressure Transmission (1)
63
[Model 1]
[Model 2] [Model 3]
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
Decomposition by Pulse Shapes• Odd multiples of the sourcing frequencies are available.
Sensitivity to Boundary Conditions
66
0lim jDr
pD
0
eDD rrD
jD
r
p
0),( DeDjD trp
Infinite reservoir
No flow
Constant pressure
Radial Ring Model
Multilayered Model
Parameter Estimation Result for Radial Ring ModelUsing Harmonic Frequencies from Square Pulses
• Model 1, comparison between three methods
67
No noise With 1% Gaussian noise in pressure
Parameter Estimation Result for Radial Ring ModelUsing Harmonic Frequencies from Square Pulses
• Model 3, comparison between three methods
68
No noise With 1% Gaussian noise in pressure
Parameter Estimation Result for Multilayered ModelUsing Harmonic Frequencies from Square Pulses
• Model 4, comparison between three methods
69
No noise With 1% Gaussian noise in pressure
Parameter Estimation Result for Multilayered ModelUsing Harmonic Frequencies from Square Pulses
• Model 5, comparison between three methods
70
No noise With 1% Gaussian noise in pressure
Wavelet Thresholding
71
Future Work
More examples to apply the frequency method
• Incorporating horizontal well configuration, fractured reservoirs, etc.
• Water and oil relative permeabilities estimation
72