dissertation slides

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