Pressure Pulse Testing in Heterogeneous Reservoirs

PRESSURE PULSE TESTING

IN HETEROGENEOUS RESERVOIRS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Sanghui Ahn

February 2012

http://creativecommons.org/licenses/by/3.0/us/

This dissertation is online at: http://purl.stanford.edu/rx603hm3283

© 2012 by Sang Hui Ahn. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-3.0 United States License.

ii



http://purl.stanford.edu/rx603hm3283

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Roland Horne, Primary Adviser


Jef Caers


Louis Durlofsky

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

Most oil and gas reservoirs are naturally heterogeneous. The description of a reser-

voir is challenging because the measurement data are limited spatially. Various

studies have been conducted to extract the heterogeneous permeability from well

test pressure data. A periodic pumping test is one category of well test that has

been utilized to estimate reservoir permeability. So far the technique has proven

useful in estimating average permeability by utilizing the dominant sourcing fre-

quency. In terms of describing a more comprehensive heterogeneous permeability

distribution, heavy computational effort in matching the entirety of the pressure

history data was required. This also requires us to know the flow rates. It is of

much interest how to best design and control the pressure pulsing technique to

reveal the heterogeneous nature of reservoirs efficiently.

We propose a new inverse framework for obtaining the permeability distribu-

tion by utilizing frequency contents. This work investigated how to utilize mul-

tiple frequency components of pressure pulse testing data in estimating the hor-

izontal permeability and vertical permeability distributions between an injection

point and an observation point. Models of a radial multicomposite reservoir and

a partially penetrating well in a multilayered reservoir with crossflow were ex-

amined. Attenuation and phase shift information acquired from pressure pulse

tests at multiple frequencies was used to estimate the permeability distribution of

reservoirs. A semianalytical solution was derived for the two reservoir models for

single-phase flow in a periodic steady state.

Nonlinear optimization was used to infer the permeability distribution that

iv

satisfies the given frequency response data. Quasi-Newton line search optimiza-

tion with gradient information was used for both methods. Synthetic homoge-

neous and heterogeneous reservoir cases were examined under periodic steady-

state conditions with multiple sinusoidal inputs and multiple frequencies con-

tained in square pulses. The estimation performance of the frequency method was

investigated and compared to straightforward pressure history matching and the

wavelet compression method.

The study also examined the benefits and limitations of using multiple frequen-

cies in estimating the permeability distribution. In addition, the sensitivity of the

permeability estimation to perturbation in both pressure data and frequency at-

tributes was investigated. A heuristic method for detrending was devised which

helps with obtaining accurate attenuation and phase shift information. Cases with

different values of storage, skin, and boundary conditions were considered. The

impact of varying the number of periods and the sampling rate was analyzed to

determine the sensitivity of Fourier transformation to these factors.

By matching attenuation and phase shift at various harmonic frequencies, which

are the multiples of the fundamental sourcing frequency, the results were found

to be in good agreement with the actual permeability distribution trend. Attenua-

tion and phase shift provide an ‘indicator characteristic’ which can reveal reservoir

heterogeneity. The pressure pulse testing with multiple frequencies is useful in

describing the heterogeneity of the reservoir parameters quantitatively, when the

radius of cyclic influence is covered by the sourced frequencies. By processing the

time series pressure data effectively, the amount of both the time and frequency

conditioning data is reduced, and there is no need to utilize the flow rate data.

Thus the frequency method proved more efficient computationally than matching

the full history pressure data. However, the accurate extraction of the frequency

parameters is essential for determining the permeability distribution. The suc-

cessful pulse test design relies heavily on the choice of sourcing frequency which

depends on the factors such as permeability range of inspection, the distance be-

tween the two points, and the mechanic precision of the measurement device. The

v

quality of the permeability estimate improves with more pulses, an increased sam-

pling rate, and processing pulses from later time.

vi

Acknowledgements

Foremost, I would like to express my sincere gratitude to my advisor Prof. Roland

Horne for the continuous support of my Ph.D. study and research, his patience, in-

spiration, enthusiasm, and immense knowledge. His guidance helped me through-

out every step of my research.

I would also like to thank the rest of my thesis committee members: Prof. Louis

Durlofsky, Prof. Jef Caers, Prof. Tapan Mukerji and Prof. Michael Saunders, for

their feedback and insightful comments.

My sincere thanks also goes to my former colleagues in the Shell Oil Company:

Jean-Charles Ginestra, and George Stegemeier for offering me the internship op-

portunities which inspired me to leverage my Electrical Engineering background

and mature it into my PhD research. I tremendously appreciate them for providing

the field data used in this study.

I have been fortunate to have many friends who cherish me. I am indebted to

my many colleagues in school for providing a stimulating and fun environment in

which to learn and grow. My time at Stanford was made enjoyable in large part

due to the many friends and groups that became a part of my life. I thank my

colleagues in well testing group, Obinna Duru, Aysegul Dastan, Priscila Ribeiro,

Yang Liu, Zhe Wang, and Abeeb Awotunde for their friendship and technical ad-

vice. I am especially grateful to my beloved Korean colleagues in the department:

Kwangwon Park, Cheolkyun Jeong, and Hyungki Kim who shared their enthusi-

ast and insights in the petroleum industy. I also thank my friends: Ariel Espos-

ito, Miranda Lee, Joohwa Lee, Whitney Sargent, Daniel Pivonka, Addy Satija, and

Meeyoung Park for providing support and friendship.

vii

Lastly, I would like to thank my parents for all their unwavering love and en-

couragement during all stages of my life. You give me strength to focus on the

positive and be grateful in all circumstances.

viii

Contents

Abstract iv

Acknowledgements vii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Study Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Frequency Method and Reservoir Heterogeneity 11

2.1 Overview of Pressure Pulse Processing . . . . . . . . . . . . . . . . . . 11

2.2 Analysis in Frequency Domain . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Frequency Domain Representation of Pressure: Magnitude

and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Frequency Analysis of Periodic Waveforms . . . . . . . . . . . 15

2.2.3 Frequency Response of Pressure Pulse Testing . . . . . . . . . 19

2.2.4 Attenuation and Phase Shift . . . . . . . . . . . . . . . . . . . . 20

2.2.5 Reservoir Description by Frequency Method . . . . . . . . . . 21

2.3 Multicomposite Radial Ring Model . . . . . . . . . . . . . . . . . . . . 23

2.3.1 Formulation of the Steadily Periodic Solution . . . . . . . . . . 24

2.3.2 Frequency Response of Radial Ring Model . . . . . . . . . . . 29

2.4 Partially Penetrating Well with Cross Flow in Multilayered Model . . 38

2.4.1 Formulation of the Steadily Periodic Solution . . . . . . . . . . 39

ix

2.4.2 Frequency Response of Multilayered Model . . . . . . . . . . 45

2.5 Discussion on Extension to Heterogeneous Permeability Distribution 54

2.6 Generation of Square Pulses . . . . . . . . . . . . . . . . . . . . . . . . 57

2.7 Radius or Depth of Cyclic Influence . . . . . . . . . . . . . . . . . . . 57

2.8 Discussion of Relationship between Attenuation and Phase Shift . . . 59

2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3 Pressure Data Preprocessing 66

3.1 Quantification of Pressure Transient from Pulses . . . . . . . . . . . . 67

3.2 Heuristic Detrending Method . . . . . . . . . . . . . . . . . . . . . . . 67

3.2.1 Heuristic Detrending for Square Pulses . . . . . . . . . . . . . 68

3.2.2 Heuristic Detrending for Unequal Pulses . . . . . . . . . . . . 70

3.3 Detrending on Radial Ring Model . . . . . . . . . . . . . . . . . . . . 74

3.4 Detrending on Multilayered Model . . . . . . . . . . . . . . . . . . . . 83

3.5 Determinants for Accuracy of Frequency Attributes . . . . . . . . . . 91

3.5.1 Effect of Number and Position of Pulses . . . . . . . . . . . . . 91

3.5.2 Effect of Sampling Frequency . . . . . . . . . . . . . . . . . . . 95

3.5.3 Effect of Pressure Noise . . . . . . . . . . . . . . . . . . . . . . 97

3.6 Preprocessing on Field Data . . . . . . . . . . . . . . . . . . . . . . . . 100

3.6.1 Quantization Noise in Field Data . . . . . . . . . . . . . . . . . 100

3.6.2 Field Data 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.6.3 Field Data 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.6.4 Field Data 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.6.5 Field Data 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4 Inverse Problem Frameworks 125

4.1 Frequency Method with Attenuation and Phase Shift . . . . . . . . . 126

4.2 Two Other Methods for Comparison . . . . . . . . . . . . . . . . . . . 128

4.2.1 Pressure History Matching . . . . . . . . . . . . . . . . . . . . 128

4.2.2 Wavelet Thresholding . . . . . . . . . . . . . . . . . . . . . . . 128

4.3 BFGS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

x

4.4 Convergence Performance Comparison . . . . . . . . . . . . . . . . . 135

4.5 Reconstructed Pressure by Three Methods . . . . . . . . . . . . . . . . 136

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5 Permeability Estimation on Radial Ring Model 139

5.1 Using Multiple Sinusoidal Frequencies . . . . . . . . . . . . . . . . . . 140

5.1.1 Homogeneous Radial Ring - Model 1 . . . . . . . . . . . . . . 141

5.1.2 Heterogeneous Radial Ring - Model 2 and 3 . . . . . . . . . . 142

5.2 Using Harmonic Frequencies from Square Pulses . . . . . . . . . . . . 147

5.2.1 Homogeneous Radial Ring - Model 1 . . . . . . . . . . . . . . 147

5.2.2 Heterogeneous Radial Ring - Model 2 and 3 . . . . . . . . . . 149

5.3 Permeability Estimation with Added Pressure Noise . . . . . . . . . . 150

5.4 Perturbation in Frequency Space . . . . . . . . . . . . . . . . . . . . . 154

5.5 Effects of Storage and Skin . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.6 Effect of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 167

5.7 Application to Field Data . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.7.1 Field Data 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.7.2 Field Data 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6 Permeability Estimation on Multilayered Model 178

6.1 Using Multiple Sinusoidal Frequencies . . . . . . . . . . . . . . . . . . 178

6.1.1 Homogeneous Multilayered System - Model 4 . . . . . . . . . 179

6.1.2 Heterogeneous Multilayered System - Model 5 and 6 . . . . . 180

6.2 Using Harmonic Frequencies from Square Pulses . . . . . . . . . . . . 184

6.2.1 Homogeneous Multilayered System - Model 4 . . . . . . . . . 184

6.2.2 Heterogeneous Multilayered System - Model 5 and 6 . . . . . 185

6.3 Permeability Estimation with Added Pressure Noise . . . . . . . . . . 185

6.4 Perturbation in Frequency Space . . . . . . . . . . . . . . . . . . . . . 190

6.5 Effects of Storage and Skin . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.6 Effect of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 194

6.7 Application to Field Data . . . . . . . . . . . . . . . . . . . . . . . . . 200

xi

6.7.1 Field Data 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6.7.2 Field Data 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7 Conclusions and Future Work 209

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

7.2 Recommendation for Practical Pulse Test Design and Analysis . . . . 212

7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

8 Nomenclature 215

A Frequency Range and Permeability Distribution 217

B Tridiagonal Matrix Algorithm (TDMA) 219

C Different Boundary Conditions 221

C.1 Radial Composite Model . . . . . . . . . . . . . . . . . . . . . . . . . . 221

C.2 Multilayered Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

xii

List of Tables

3.1 Performance of heuristic detrending on radial ring model . . . . . . . 77

3.2 Performance of heuristic detrending on multilayered model . . . . . 85

5.1 Radial permeability estimation error with varying number of fre-

quencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.2 Radial permeability estimation error with different termination cri-

teria and number of frequencies for Model 1 . . . . . . . . . . . . . . 149

5.3 Model 1 - Radial permeability estimation error . . . . . . . . . . . . . 152



5.6 Radial permeability error with perturbation to frequency data . . . . 154

5.7 Storage effect on radial ring model: mismatch of frequency attributes 162

5.8 Skin effect: mismatch of frequency attributes . . . . . . . . . . . . . . 162

5.9 Radial permeability and dimensionless storage estimation error . . . 162

5.10 Performance of frequency method in comparison with history match-

ing for Field data 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.1 Vertical permeability estimation error with varying number of fre-

quencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.2 Model 4 - Vertical permeability estimation error . . . . . . . . . . . . 188



6.5 Vertical permeability error with perturbation to frequency data . . . 191

xiii

6.6 Storage effect on multilayered model: mismatch of frequency at-

tributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.7 Performance of frequency method in comparison with history match-

ing for Field data 3 and 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 208

xiv

List of Figures

1.1 Parallel tangent technique in measuring attenuation and phase shift

(Kamal and Brigham, 1976). . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Cross-plot of attenuation (A) and phase shift (θ) with lines of con-

stant dimensionless permeability (η) and dimensionless storativity

(ξ) with linear flow model (Bernabe et al., 2005). . . . . . . . . . . . . 5

1.3 Estimated permeability from single well drawdown pressure data

in heterogeneous reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Vertical interference and pules testing configuration . . . . . . . . . . 8

2.1 Overall procedure for estimating permeability distribution from pres-

sure measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Pressure data (top) and Fourier domain magnitude (bottom left) and

phase (bottom right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Three types of periodic pulse shape: square (left), rectangular (cen-

ter) and triangular (right). . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Square wave decomposition in time domain (left) and Fourier coef-

ficients for three pulse shapes (right). . . . . . . . . . . . . . . . . . . . 18

2.5 Square pulsing can be better utilized with an analysis of a set of

multiple sinusoidal pulses sourced (left) at the injection point and

received (right) at the observation point. . . . . . . . . . . . . . . . . . 22

2.6 Schematics of a multicomposite radial ring system for estimating

radial permeability distribution. . . . . . . . . . . . . . . . . . . . . . 23

2.7 Frequency attributes for radial ring models . . . . . . . . . . . . . . . 31

xv

2.8 Sinusoidal pressure at multiple observation points Model 1 (left)

and Model 2 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.9 Frequency response of homogeneous radial model: attenuation (left)

and phase shift (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.10 Sensitivity of frequency data to dimensionless radial distance for

homogeneous radial model . . . . . . . . . . . . . . . . . . . . . . . . 34

2.11 Attenuation vs. phase shift cross plot of radial ring models at mul-

tiple frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.12 Schematics of a partially penetrating well with cross flow system for

estimating vertical permeability in a multilayered reservoir. . . . . . 39

2.13 Analysis on multilayered models with kr = 100 md . . . . . . . . . . 47

2.14 Multilayered models with kv/kr = 0.1: attenuation (left) and phase

shift (right) over vertical distance . . . . . . . . . . . . . . . . . . . . . 48

2.15 Frequency response of multilayered model: attenuation (left) and

phase shift (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.16 Sensitivity of frequency data to depth for multilayered model . . . . 51

2.17 Attenuation vs. phase shift cross plot of multilayered models . . . . 53

2.18 Illustration of frequency method in comparison with previous method 56

2.19 Characteristic of system’s response for pressure pulse testing envi-

ronment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.20 Estimation of phase shift from attenuation using Hilbert transform . 64

3.1 Illustration of transient reconstruction . . . . . . . . . . . . . . . . . . 69

3.2 Detrending on various pulse shapes . . . . . . . . . . . . . . . . . . . 73

3.3 Detrending on radial model with square pulses . . . . . . . . . . . . . 78

3.4 Detrending on radial model with 25% duty cycle pulses . . . . . . . . 79


3.6 Misfit for frequency attributes on radial model with three pulse shapes 81

3.7 Attenuation vs. phase shift for three pulse shapes on radial model . . 82

3.8 Detrending on multilayered model with square pulses . . . . . . . . 86

3.9 Detrending on multilayered model with 25% duty cycle pulses . . . . 87

xvi


3.11 Misfit for frequency attributes on multilayered model with three

pulse shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.12 Attenuation vs. phase shift for three pulse shapes on multilayered

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.13 Number of pulses and location of windows used for sensitivity check. 91

3.14 Accuracy of frequency attributes with varying number of pulses and

window position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.15 Frequency components for a pulse located at the third pulse. . . . . . 93

3.16 Fourier analysis when a first pulse is included . . . . . . . . . . . . . 94

3.17 Accuracy of frequency attributes with varying sampling frequency . 96

3.18 Fourier magnitude plots with added Gaussian noise in pressure at

injection point (left) and observation point (right). . . . . . . . . . . . 97

3.19 Attenuation (left) and phase shift (right) of ten realizations of noisy

pressure with 128 points per cycle. . . . . . . . . . . . . . . . . . . . . 98

3.20 MAE summary for attenuation (left) and phase shift (right). . . . . . 99

3.21 Illustration of quantization error (in green) caused by a limited pre-

cision in amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.22 Effect of quantization of time in frequency domain . . . . . . . . . . . 103

3.23 Effect of smoothing on frequency data when quantized in time . . . . 104

3.24 Effect of quantization of pressure amplitude on Fourier magnitude

spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.25 Effect of smoothing on frequency data when quantized in pressure

amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.26 Preprocessing of Field data 1 in time domain . . . . . . . . . . . . . . 110

3.27 Detrending of Field data 1 . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.28 Effect of smoothing, Field data 1 . . . . . . . . . . . . . . . . . . . . . 112

3.29 Frequency attributes by varying number of pulses, Field data 1 . . . 113

3.30 Field data 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.31 Field data 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.32 Field data 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xvii

3.33 Field data 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.34 Field data 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.35 Improved signal decomposition by removing outliers from observa-

tion pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.36 Field data 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.1 Input data for frequency method . . . . . . . . . . . . . . . . . . . . . 127

4.2 Wavelet decomposition process . . . . . . . . . . . . . . . . . . . . . . 130

4.3 Wavelet decomposition approximation (cA6) and detail coefficients

(cD1,· · · ,cD6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.4 Reconstruction of pressure by varying number of wavelet coefficients.133

4.5 Comparison of convergence curves (objective function versus itera-

tions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4.6 Reconstructed pressure for multilayered model . . . . . . . . . . . . . 137

5.1 Radial permeability estimation by sinusoidal frequencies . . . . . . . 143

5.2 Permeability estimation result (right) with frequencies (left) that reach

far beyond the observation point for Model 1. . . . . . . . . . . . . . 144

5.3 Radial permeability estimate with varying number of sinusoidal fre-

quencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.4 Effect of termination criteria on permeability estimation performance 149

5.5 Radial permeability estimation from square pulses . . . . . . . . . . . 151

5.6 Radial permeability estimate with perturbation in frequency space . 155

5.7 Storage effect on radial ring model . . . . . . . . . . . . . . . . . . . . 157

5.8 Skin effect on radial ring model with CD = 0 . . . . . . . . . . . . . . 158

5.9 Skin effect on radial ring model with CD = 100 . . . . . . . . . . . . . 163

5.10 Effect of skin factors on attenuation and phase shift with CD = 100 . 163

5.11 Cross-plot of attenuation and phase shift with storage and skin (sum-

mary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.12 Attenuation and phase shift of three models with skin factors. . . . . 164

5.13 Constant rate pressure response with storage and skin effects . . . . 165

5.14 Radial permeability estimate with CD = 100 . . . . . . . . . . . . . . 166

xviii

5.15 Sensitivity of attenuation and phase shift with different boundary

conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.16 Field data 1- Radial permeability estimates by the frequency method 171

5.17 Field data 1- Reconstruction of pressure data in comparison with

history matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.18 Field data 2- Radial permeability estimates by the frequency method 175



6.1 Vertical permeability estimation by sinusoidal frequencies . . . . . . 181

6.2 Vertical permeability estimate with varying number of sinusoidal

frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.3 Vertical permeability estimation from square pulses . . . . . . . . . . 187

6.4 Vertical permeabilty estimate with perturbation in frequency space . 190

6.5 Storage effect on multilayered model . . . . . . . . . . . . . . . . . . . 195

6.6 Skin effect on multilayered model with CD = 0 . . . . . . . . . . . . . 196

6.7 Skin effect on multilayered model with CD = 100 . . . . . . . . . . . 197

6.8 Effect of skin factors on attenuation and phase shift with CD = 100 . 197

6.9 Cross-plot of attenuation and phase shift with storage and skin (sum-

mary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

6.10 Attenuation and phase shift of three models with skin factors . . . . 198

6.11 Sensitivity of attenuation and phase shift with different boundary

conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.12 Field data 3- Vertical permeability estimates by the frequency method 202



6.14 Field data 4- Vertical permeability estimates by the frequency method 206



C.1 Frequency attributes over distance with different boundary condi-

tions for radial ring model . . . . . . . . . . . . . . . . . . . . . . . . . 223

xix

C.2 Frequency attributes over depth with different boundary conditions

for multilayered model . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

xx

Chapter 1

Introduction

1.1 Motivation

Pulse testing has been used for more than forty years to analyze the reservoir per-

meability by using flow rate and pressure history (Johnson et al., 1966). Since then,

many authors have discussed, expanded on, and applied the method in practice

to investigate the connectivity between wells. The pulse test is a procedure where

the cyclic flow rate or pressure from an active well is transmitted to a monitoring

well through the reservoir. Sensitive gauges are used for measuring the change

in pressure. Recent cyclic testing methods have been successful in estimating the

average permeability (Renner and Messar, 2006). Currently only the dominant

(or, fundamental) frequency component of the sourced square pulse signal is used

to diagnose the average permeability between the pulsing well and the observa-

tion well. As the square pulse contains odd-integer multiples of the fundamental

frequency component of the signal, it is of much interest to see if the additional

information on top of the fundamental frequency can contribute to the quality of

the permeability distribution estimate for a reservoir.

Utilization of specific frequencies has proven to be useful in characterizing the

reservoir in various ways. Datta-Gupta et al. (1995) analyzed two-well tracer tests

in a heterogeneous medium by matching their transfer function in the frequency

1

CHAPTER 1. INTRODUCTION 2

domain. Pan (1999) concluded that the perturbation frequency and the investiga-

tion range can be used to infer the macroscopic behavior of the reservoir. Hollaen-

der et al. (2002) summarized the harmonic behavior for various flow regimes in

well testing. Rosa (1991) demonstrated that the radii of cyclic influence decrease

with increasing frequencies in pulse testing. To be more specific, he demonstrated

that a pressure signal with low frequency penetrates further into the reservoir than

one with high frequency. He characterized the areal heterogeneous permeability

by history matching the flow rate and pressure data from the active well and the

observation well. From an inverse framework point of view, the full history match-

ing of pressure is a time-consuming algorithm in estimating the permeability dis-

tribution. The correlation between the radius of cyclic influence and the frequency

of a sinusoidal flow rate was not actively used as a set of inputs for estimating the

permeability distribution in any other studies.

One advantage in using the frequency method for pressure pulse testing is that

the knowledge of flow rate data is not required. This is due to the fact that the

utilized frequency information, which is a pair of attenuation and phase shift, is

a ratio between the sourced and observed pressure signal. The benefit is that in

practice the flow rate measurement is less accurate than the pressure measurement.

Attenuation and phase shift parameters are two components that fully describe

the frequency response of the system in general for any linear time invariant (LTI)

system. In our case, this is a reservoir model. The attenuation indicates how much

the observed signal is reduced from the sourced signal. The phase shift indicates

how much delay the signal experiences in passing through the medium. If the

attenuation and phase shift parameters convey similar information close to what

the original pressure signal carries over time, it could be an effective way to com-

press the pressure data. It is important to investigate the frequency method for

interpreting and estimating reservoir parameters.

It is also important to investigate under what conditions the pulse testing tech-

nique is most beneficial in estimating reservoir properties. Najurieta and Bridas

(1993) argued that greater precision in pressure measurements enhances the accu-

racy of the calculations for the analysis of the influence of local heterogeneities.


1.2 Literature Review

Pulse testing was described by Johnson et al. (1966) and used in practice in indus-

try to check the communication details between wells. Pulse testing is most often

used for estimating average permeability and has been used in practice for more

than forty years. In pulse testing, the flow rate or pressure pulse from an active

well or a pulsing well is transmitted through porous media to a monitoring well.

Pulse testing falls into the class of multiwell testing defined as having two mea-

surement data sets which are gathered from locations a distance apart. The pulses

are generated in several cycles through alternating the flow and shut-in period.

The flow rate is relatively harder to measure. For this reason the measurements

often record only pressure. The monitoring well records the pressure change from

the pulsing well with a bottomhole pressure gauge. The method was developed

to analyze reservoir transmissivity or transmissibility (kh/

µ) and storativity (φhct),

which can describe permeability and porosity, respectively.

The flow rate input usually has a rectangular shape which generates rich fre-

quency contents in the pressure measurements. Another similar type of testing

is harmonic testing, which has a sinusoidal signal and is ideal for experiments

but harder to generate in practice. Other terminologies such as periodic inter-

ference testing, and periodic pumping testing (in hydrology studies) have been

introduced.

The advantage of pulse testing is that the time required to obtain a diagnostic

pressure response is short compared to conventional well testing such as buildup

or drawdown tests (Kuo, 1972). The data are less affected by noise or wellbore

effects such as multiphase flow or phase redistribution. The disadvantage of har-

monic testing is that for the same radius of investigation the duration of the test

needs to be longer than the conventional test. However, the interpretation of pres-

sure pulse test data can be done in a similar manner as in conventional well testing

(Hollaender et al., 2002).

Previous studies for average permeability estimation are quite well established.

To obtain an equivalent permeability for a homogeneous model, a parallel tangent


technique is often used (Figure 1.1). Transmissivity khµ = 141.2qB pD

∆p is calculated

from the amplitude reduction (or, attenuation) and storativity φcth = 0.0002637 khµ

∆tr2tD/r2

Dis calculated from the time lag (or, phase shift) between the injected flow rate data

and output pressure data. However, relying on such time domain analysis for

pulse testing suffers from the limitation that it is difficult to determine the exact

peak points from measurement data.

Figure 1.1: Parallel tangent technique in measuring attenuation and phase shift(Kamal and Brigham, 1976).

A variation of the reservoir model and its effect on pulse testing has been stud-

ied extensively. Kamal and Brigham (1976) studied the pulse shape with unequal

duty cycle. Reservoir parameter extraction in homogeneous or anisotropic perme-

ability cases was analyzed by Kamal (1975). Accounting for wellbore storage and

skin effects were discussed by many authors (Dinges and Ogbe, 1988; Winston,

1983; Ogbe and Brigham, 1987, 1989). Prats and Scott (1975) showed that wellbore

storage causes a delay in the time lag, which decreases as the interwell distance

increases and as the storage effect decreases. Ogbe and Brigham (1987) developed

an iterative procedure for obtaining wellbore storage and skin based on a diagnos-

tic chart that typically converged to a set of estimated values within 3% with three

iterations.


Bernabe et al. (2005) and Renner and Messar (2006) established radial and lin-

ear flow models (Figure 1.2) and calculated diffusivity from attenuation and phase

shift. The radial flow model is more prevalent for the pressure pulse testing ap-

plication. In this case the path starts with the wellbore storage effect followed by

radial flow as explained in detail in their discussion of the radius of investiga-

tion for pressure transient tests (Kuchuk, 2009). The linear model involves more

complicated inverse optimization with two parameters (Bernabe et al., 2005). This

method requires a careful selection of the weighted objective function to model

because the scale of the two parameters is different.

Figure 1.2: Cross-plot of attenuation (A) and phase shift (θ) with lines of constantdimensionless permeability (η) and dimensionless storativity (ξ) with linear flowmodel (Bernabe et al., 2005).

Stegemeier (1982) provided an analytical method to interpret pulse tests by us-

ing exponential integral functions in describing the pressure response to periodic

rate change.

Pulse testing in heterogeneous media and its area of influence was studied by

Vela and McKinley (1970). The study revealed that the pulse test is only affected

by heterogeneities within an influence area. The correction factor is devised such

that it accommodates the weight of different transmissivity or storativity ranges.


Kuo (1972) pointed out the possibility of using frequency analysis and its ability

to reveal heterogeneity within the reservoir. He argued that by placing multiple

wells to investigate the average permeability between each well pair, the distri-

bution can be possibly be determined. Permeability estimation in multicomposite

radial reservoirs was studied by Rosa (1991) and implemented in practice using

nonlinear regression on pressure data with the Gauss-Marquardt technique. The

inverse problem was solved where the inner and outer radii are specified. Rosa

noted that the multirate test was more sensitive to reservoir heterogeneities and

that the inverse problem for determining the permeability is not a unique solution

problem (Feitosa et al., 1994).

Determining the heterogeneous permeability distribution from pressure has

been studied by several authors (Oliver, 1990, 1992; Feitosa et al., 1994) in the con-

text of multiple well tests, which is the general category to which the pulse testing

technique belongs. Oliver used a perturbation technique that defines how much

permeability changes from the known reference average value. How to define

the weight spatially has been studied based on well-test pressure curves. With

the data kernel G(r, t), the pressure response caused by the absolute permeability

variation can be written as p(t) =∫

G(r, t)F(r)dr where F(r) is the inverse perme-

ability variation F(r) = 1− 1/

kD(r) and kD(r) = k(r)/k. The data kernel G(r, t)

is large near the wellbore and decreases until it reaches the radius of investigation.

Feitosa et al. (1994) extended the approach by Oliver and showed a more robust

scheme that can be applied for large variations of the parameters (Figure 1.3). The

complexities in the general approximate pressure solution for the heterogeneous

case are discussed by Habashy and Torres-Verdin (1996). Similarly, a more general

heterogeneous radial and linear model for well testing was established by Levitan

(2002).

Although less common than horizontal or radial permeability inspection, pulse

testing has been used for multilayered reservoirs in an effort to obtain vertical per-

meability (Kaneda et al., 1991). Estimating a vertical permeability distribution is

important because of the impact of the layer permeability values on the primary


Figure 1.3: Estimated permeability from single well drawdown pressure data inheterogeneous reservoir (Feitosa et al., 1994).

and secondary recovery processes; an estimate of the vertical permeability dis-

tribution also helps with predicting well performance and planning the effective

depletion strategy of a reservoir (Johnson et al., 1966; Kaneda et al., 1991). Ayan

and Kuchuk (1995) showed that both horizontal and vertical permeability can be

obtained using a wireline formation tester. They formulated convolution with re-

spect to the sourcing flow rate and extracted the effective average permeability.

To accomplish this, the vertical probe is displaced a short distance from the sink

probe. A similar design for pulse testing was proposed by Proett and Chin (2000)

who generated sinusoidal displacements at frequency range of 0.1 - 10 Hz and

determined horizontal and vertical average permeability from the measured time

delay between the vertical probes.

Layer pulse testing was described by Saeedi and Standen (1987) with two Re-

peat Formation Tester (RFT) surveys, and the horizontal and vertical permeabili-

ties were obtained using a numerical simulator and a real data case. Earlougher

(1980) presented analysis and design of pulse testing between two perforations in

estimating vertical permeabilities near the well (Figure 1.4).


Figure 1.4: Vertical interference and pules testing configuration (Earlougher, 1980).

The retrieved pressure data from well tests can be integrated with complex ge-

ological models to provide a comprehensive description of reservoirs. Techniques

for conditioning permeability fields to well test data have been presented by many

authors. Li et al. (2010) investigated that the uncertainties in reservoir models can

be reduced by conditioning a geological model directly to pressure data. Simi-

larly, Deutsch (1999) used pressure data to infer the effective permeability around

the wellbore and constrained the spatial distributions modeled from complex ge-

ological patterns. Srinivasan (2000) generated multiple geological realizations by

applying Oliver’s radial kernel weights (Oliver, 1990) to constrain to well test data.

Kim et al. (2009) calibrated geological models using pressure data from permanent

downhole gauges. In analyzing two dimensional heterogeneous reservoirs, the

pressure data are measured from multiple observation points and decomposed to

a series of single well tests.

Similar to Fourier analysis, wavelet analysis deals with the expansion of func-

tions in terms of a set of basis functions. Dastan (2010) applied the wavelet-based

nonlinear regression method to cyclic pressure transient data in analyzing reser-

voir parameters such as permeability, storage and skin. Awotunde (2010) used


wavelet methods to perform an extensive analysis of time series data in estimating

spatial features in the reservoir.

The periodic pulse test has been developed and used in practice in other fields

such as hydrology and geothermal wells. A pumping test, the term used for pulse

testing in hydrology, is conducted to evaluate an aquifer by stimulating the aquifer

through constant pumping, and observing the aquifer’s response in observation

wells. The pumping test helps with characterizing a system of aquifers, aquitards

and flow system boundaries (Dawson and Istok, 1991). For geothermal appli-

cations, Becker and Guiltinan (2010) conducted field experiments in a fractured

sandstone reservoir to estimate hydraulic properties during active production of

geothermal wells.

The application of multiple frequency information was further extended to

other fields where it has enhanced the quality of estimation. Simulation results

for reconstructing two small objects showed improvement because of the incorpo-

ration of multiple frequency data in the tomography imaging test (Milstein et al.,

2004).

1.3 Study Objectives

This study aimed to characterize heterogeneous permeability distributions for reser-

voir models using analysis of multiple frequencies, and to aid in a better design

and control for the pulse testing technique. The objectives of this study are out-

lined below:

1. Formulate the periodic steady-state solutions in two different pulse testing

reservoir models for both the radial and vertical permeability distribution.

2. Provide a new method that utilizes attenuation and phase shift information

at multiple frequencies to determine the interwell permeability spatial distri-

butions.

3. Investigate and visualize how a frequency response represents the level of

heterogeneity.


4. Investigate the desirable frequency conditions in estimating permeability dis-

tribution for the inverse problem framework.

5. Compare the frequency method with other existing methods such as history

matching and wavelet.

6. Investigate the advantages and disadvantages of the method under various

testing conditions.

1.4 Dissertation Outline

In Chapter 2, the mathematical background of steady-state solutions for two types

of reservoir models is described in detail. In Chapter 3, the necessity of pre-

processing pressure pulses is discussed and the heuristic detrending of pressure

pulses is introduced. Chapter 4 outlines the inverse framework using multiple fre-

quency attributes in estimating permeability distributions. The newly established

frequency method is compared with two other methods, pressure history match-

ing and wavelet on the estimation. In Chapters 5 and 6, the inverse problem of

estimating permeability distribution using frequency information is demonstrated

with two idealized forms of heterogeneity such as radially heterogeneous reser-

voirs and multilayered reservoirs. The performance of the frequency method is

compared with history matching and wavelet method. In addition, the robust-

ness of the method is examined in the presence of noise in the time and frequency

domain. Various outer boundary conditions and storage and skin effect are also

examined. Finally, Chapter 7 summarizes the results and discusses possible future

work on this topic.

Chapter 2

Frequency Method and Reservoir

Heterogeneity

This chapter describes the principles behind the frequency method that uses at-

tenuation and phase shift data at multiple frequencies in characterizing reservoir

models from pressure pulse testing. Two reservoir models are investigated in de-

tail as examples of relating frequency attributes to horizontal and vertical perme-

ability distributions. The sinusoidal steady-state space forms the ideal basis for

parameter estimation with periodic pulses. The mathematical formulation is de-

rived for two reservoir models: a radial ring model and a multilayered reservoir

with partial perforation.

2.1 Overview of Pressure Pulse Processing

The flow chart in Figure 2.1 illustrates the overall procedure of using frequency

information contained in pulse test data to estimate spatial distributions of perme-

ability. Given the pressure pairs measured from the active and monitoring wells

for a radial model or from a perforated layer and observation layer, we estimate the

permeability distribution as a function of distance. The procedure is largely com-

posed of two parts: (1) pressure data processing to extract attenuation and phase

shift and (2) inverse problem formulation in matching attenuation and phase shift

11

CHAPTER 2. FREQUENCY METHOD AND RESERVOIR HETEROGENEITY 12

data at multiple harmonic frequencies. The former part deals with effectively ex-

tracting attenuation and phase shift from the pulse sets; the latter part estimates

the permeability distribution by least-squares parameter estimation. The inverse

problem is addressed to estimate the equivalent permeability distribution between

two locations. As square pulses can be decomposed into multiple frequency com-

ponents, the reservoir model is excited as if multiple sinusoidal pressure data were

sourced. The sinusoidal steady space results provide a framework for analyzing

the square pulses. Detrending constitutes a preprocessing of data before applying

the Fourier transform technique. The Fourier transform extracts information re-

garding the magnitude and phase at harmonics of the fundamental frequency. The

main variable is the permeability distribution and all the rest of the parameters are

assumed to be known and constant values.

Figure 2.1: Overall procedure for estimating permeability distribution from pres-sure measurements.

The frequency method in estimating permeability is conducted according to the

following procedure.

1. Quantify a transient trend in pressure. From given pressure pairs, obtain the

constant rate pressure response.


2. Remove the transient trend by an appropriate weight depending on the duty

cycle of the pressure signal. The pressure data are now almost steadily peri-

odic.

3. Obtain the frequency attributes (attenuation and phase shift) at the harmonic

frequencies from the pressure pulses. Typically the harmonic frequencies are

odd multiples of the dominant sourcing frequency.

4. Find permeability distribution as a function of distance based on the fre-

quency data at multiple frequencies.

2.2 Analysis in Frequency Domain

2.2.1 Frequency Domain Representation of Pressure: Magnitude

and Phase

Frequency is the rate of change with respect to time. If a signal does not change

at all, its frequency is zero. If a signal changes instantaneously, its frequency is

infinite. The Fourier transform is a mathematical operation which decomposes a

signal into a sine wave of different frequencies. The Fourier transform extracts

the information of magnitude and phase in the frequency domain. The magnitude

tells how much of a certain frequency component is present and the phase tells

where the frequency component is in the time signal. These attributes represent

the average frequency content of the signal over the entire time that the signal is

acquired.

Many of the previous pulse testing methods measured the two components of

pressure in time; the tangent technique measures in time the peak-to-peak ampli-

tude, which is the difference between the maximum and the minimum amplitudes

of a waveform, for the magnitude and the time difference between the drawdown

or buildup response for the time delay (Stegemeier, 1982; Kamal and Brigham,

1976). However, the time domain analysis suffers from the ambiguity in defining

the exact magnitude and time delay. It is difficult to define these values because


the observed pressure waveform is warped as the signal goes through the medium

in such a way that its shape does not resemble the original pressure. When square

flow rate is sourced at the well, for instance for a radial reservoir, the shape of the

injected pressure at the wellbore is in between square and sawtooth. However,

the shape of the observed pressure is triangular with upward transient. Frequency

domain diagnosis of the magnitude and phase summarizes the average content

according to each frequency and shows the frequency components other than the

fundamental frequency which are not captured in the time domain.

The pressure magnitude and phase are explained in more detail as follows.

Figure 2.2 is an example of an injection pressure and observation pressure pair

resulting from a square pulse flow rate train. Figure 2.2 shows the translation

of the pressure time signal contents in frequency domain. The Fourier magni-

tude is defined as the absolute value of the Fourier transform polar represen-

tation, whereas the Fourier phase is the argument. The magnitude is usually

recorded in decibels abbreviated as dB, which is 20 times the logarithm of pres-

sure data. In most applications power is proportional to the square of amplitude,

and it is best for the two decibel formulations to give the same result in such typ-

ical cases. The following Fourier magnitude is shown for a pressure signal p(t):

10 log10P(ω)2 = 20 log10P(ω). A complete sine wave in the time domain trans-

lates to one single delta function in the frequency domain. The frequency domain

spans from zero to the highest frequency which is slightly less than, not equal

to, fs/2 = 1/2∆t. The first sample (Pinj(0) or Pobs(0)) of the transformed series

is the DC component, more commonly known as the average of the input series.

The Fourier transform of a real data series results in a symmetric series about the

Nyquist frequency which is the highest observable frequency.

For the phase, the Fourier domain yields the angle relative to the start of the

time domain signal. The unit for phase is typically radians, with 2π corresponding

to one cycle.


0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

Time (hr)

Pre

ssur

e ch

ange

(ps

i)

InjectionObservation

0 50 100 150 200 250 300

0

10

20

30

40

50

60

70

80

Pre

ssur

e m

agni

tude

(dB

)

Frequency, rad/hr


0 50 100 150 200 250 300

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3P

hase

(ra

d)

Frequency, rad/hr


Figure 2.2: Pressure data (top) and Fourier domain magnitude (bottom left) andphase (bottom right).

2.2.2 Frequency Analysis of Periodic Waveforms

To analyze how the pressure data translate to the frequency components shown in

Figure 2.2, the underlying principle should be understood. The periodic pressure

signals pinj(t) and pobs(t) are derived from the periodic flow rate q(t), where the

shape of the signals are modified from the flow rate by the diffusive nature of

reservoirs. Let f (t) represent those periodic signals in general and let Tp be its

period, then:

f (t + Tp) = f (t) , for all t (2.1)


For simplicity, the pulses are assumed to be centered on t = 0. A periodic

signal has a Fourier series representation with Fourier coefficients. Any shape

of periodic signal is decomposed as in Equation 2.2, with ω denoting the fun-

damental frequency which is equivalent to 2πTp

, with Tp denoting the periodicity.

There are harmonic frequency components in a periodic signal corresponding to

ω, 2ω, 3ω, · · · , nω. By definition, a harmonic of a waveform is a frequency com-

ponent of the signal that is an integer multiple of the fundamental frequency. Each

scaled cosine waves with frequencies nω are held in the amplitudes an as below.

f (t) = a0 +∞

∑n=1

an cos(nωt) (2.2)

The value a0 represents a nonperiodic trend, and it is the average value of the

signal.

a0 =1T

∫ T/2

−T/2f (t)dt (2.3)

The Fourier coefficients an are obtained by integrals over the period:

an =2T

T/2∫−T/2

f (t) cos(

2πtnT

)dt (2.4)

It is essential to understand how strongly the nth harmonic component is em-

bedded in the waveform. Three types of shape are considered: a perfect square, a

rectangular and a triangular shape (Figure 2.3). These are the signal shapes of flow

rate and pressure that are often observed in pressure testing. The coefficients can

be expressed in a closed form when the shape of the flow rate is known. First for

the case of square pulses with peak to peak amplitude A in time:

an =2Anπ

sin(nπ

2) (2.5)

Note that in Equation 2.5 the even nth harmonics yield zero values for square


5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time

Am

plitu

de

Square

5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time

Rectangular

5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time

Triangular

Figure 2.3: Three types of periodic pulse shape: square (left), rectangular (center)and triangular (right).

pulses. Hence, the square pulses always have odd harmonic frequency compo-

nents of 1ω, 3ω, 5ω, · · · .Although square pulses are prevalent in practice, a more general form of puls-

ing is rectangular pulsing. Consider pulses with a duty cycle of k/Tp, which means

having nonzero values in an active state for the duration of k out of Tp. The fre-

quency component at nth harmonic frequency is calculated as follows with a time

domain amplitude of A (Smith, 1997):

an =2Anπ

sin(nπk

Tp) (2.6)

Another useful form to consider is triangular pulses, which oftentimes resem-

ble observed pressure signals. For triangular pulses, the Fourier coefficients are:

an =4A

(nπ)2 (2.7)

One can obtain an approximation to a square pulse by taking a fundamental


and only a few harmonics into the summation. As the number of harmonics in the

summation increases, the constructed waveform becomes a better representative

of a square wave (Figure 2.4 left graph). The magnitude of the frequency compo-

nents an for the first 20 harmonics in the frequency domain for three pulse shapes

is demonstrated in the right side of Figure 2.4, when A = 1. It is important to note

that square and triangular pulses show zero magnitude for even harmonics. In

addition, the dominant frequency component records the highest magnitude fol-

lowed by the odd multiples of the dominant harmonic frequency components at

3ω, 5ω, 7ω, .... The harmonic frequency components decreases as the frequency in-

creases as suggested by Equation 2.6 - Equation 2.7. The similar trend is observed

in Figure 2.2 over the frequency range.

The shape of the signal decomposition does not stay the same for the injection

and observation pressure pair; the injection pressure looks like a cross between a

square and a sawtooth shape whereas the observation pressure is similar to a trian-

gular shape. As the observed signal shows faster decay of high frequency compo-

nents there exists a limitation in how many harmonic frequency components can

be used for permeability estimation. This limitation translates to less reliability in

using higher frequency contents.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Harmonic number

Abs

olut

e F

ourie

r m

agni

tude

Square25% duty cycleTriangular

Figure 2.4: Square wave decomposition in time domain (left) and Fourier coeffi-cients for three pulse shapes (right).


2.2.3 Frequency Response of Pressure Pulse Testing

A way to interpret the given pressure data in the frequency domain has been de-

scribed thus far. The two important components, magnitude and phase, reveal

the information that is carried in pressure. The magnitude and phase values from

pressure pulses appear not only at the fundamental sourcing frequency, but also

at the harmonic multiples of the fundamental frequency. All of these frequencies

can be used for permeability estimation. The last step is to link the frequency in-

formation from the input and output pressure pair with the reservoir where the

permeability distribution is unknown.

The underlying principle for analyzing the relationship between the injected

and observed signals is a convolution relation in the context of multiwell testing.

In general, such a convolution relationship holds for linear, time-invariant (LTI)

systems. The reservoir satisfies the linear condition, meaning that when the input

flow rate signal is applied a scalar number of times then the output signal is also

multiplied by the same scalar. The reservoir is also time-invariant because when

an input is applied t seconds later, the output is identical except for a time delay of

t seconds. The convolution relation has been used for many time series analyses

and interpretation of multiwell tests because the intrinsic response of the reservoir

is the key to understanding the reservoir behavior, for example, Levitan (2002);

von Schroeter and Gringarten (2009).

The pressure, pinj(t) is measured at the injection point, and pobs(t) is measured

at some distance away. For a radial ring model, the former is at an injected well,

and the latter is at an observation well. For a multilayered case, the former is

sourced at a perforated layer, the latter is measured at an observation layer at a

different depth. With the impulse response function denoted as h(t) which rep-

resents the character of the reservoir medium, the convolution is written with a

convolution operator ∗ as:

pobs(t) = pinj(t) ∗ h(t) =t∫

0

pinj(τ)h(t− τ)dτ (2.8)


The function h(t) is the response that would occur when pinj(t) is applied as

a unit-impulse pressure input. The convolution theorem simply states that the

total response is the superposition of these impulse responses. In general, the con-

volution relation can be restated in the frequency (ω) domain by a Fourier trans-

form. Let Pinj(ω) and Pobs(ω) be the Fourier transformed frequency components

from pinj(t) and pobs(t), respectively. That is, Pobs(ω) = F{pobs(t)}, Pinj(ω) =

F{pinj(t)}, and H(ω) = F{h(t)}. Then the Equation 2.8 translates in the Fourier

domain:

Pobs(ω) = Pinj(ω)H(ω) (2.9)

H(ω) is called the frequency response that describes the specific reservoir prop-

erties. The reservoir is now described in terms of how each frequency component

is transferred through the system. This method completely replaces the interpre-

tation with respect to time.

2.2.4 Attenuation and Phase Shift

There are two necessary components to describe the frequency response H(ω)

fully: the gain function, which is referred to as the “attenuation”, is the magni-

tude of the frequency response (x(ω) = |H(ω)|), and the angle which is referred

to as the “phase shift”, is the angle of the frequency response (θ(ω) = ∠H(ω)).

To construct a complex number H(ω) = A(ω) + iB(ω), with A the real part and

B the imaginary part, the attenuation is given as√

A(ω)2 + B(ω)2, and the phase

shift is given as tan−1 (B(ω)/

A(ω)). The relative magnitude and phase between

the injected and observed pressure signals, which are attenuation and phase shift,

describe the frequency response of the reservoir. In general, when relating the at-

tenuation and phase shift parameters to the pressure signal from the active and the

observation well, the following holds:

pinj(t) = p(r = rw, t) = ginj(r, ω)eiωt = |ginj(r)|ei(ωt+∠ginj(r,ω)) (2.10)


pobs(t) = p(r, t) = gobs(r, ω)eiωt = |gobs(r)|ei(ωt+∠gobs(r,ω)) (2.11)

x(ω) = |H(ω)| = |Pobs(ω)

Pinj(ω)| = |gobs(r, ω)

ginj(r, ω)| (2.12)

θ(ω) = ∠H(ω) = ∠Pobs(ω)−∠Pinj(ω) = ∠gobs(r, ω)−∠ginj(r, ω) (2.13)

In this study, the phase shift value is normalized by 2π so that both attenuation

and phase shift lie between (0, 1). When the observed signal is delayed by more

than a cycle, the recorded magnitude of the signal is also near zero; this case will

not be considered for the study.

For periodic testing, it is much easier to quantify H(ω) by attenuation and

phase shift than obtaining h(t) because the frequency components are available.

This is not true for nonperiodic tests such as drawdown or buildup tests, because

the division of the measurement signals for calculating the frequency response in-

troduces more nonlinearity when interpreting the reservoir.

2.2.5 Reservoir Description by Frequency Method

To describe the reservoir system completely, parameters in the frequency domain

need to replace what pressure data carries in the time domain. This is required to

describe H(ω) as a function of frequencies ω. The frequency response is found by

exciting the reservoir with sinusoidal signals of different frequencies. In practice,

it is hard to generate perfect sinusoidal pressure signals. The square or rectangu-

lar pulse already includes harmonics of dominant frequencies, although of lesser

magnitude for the higher frequency signal (Figure 2.5). Therefore, the pressure

pulses effectively send multiple sinusoidal pulses at all harmonics at once. If the

harmonic components are strong and span sufficiently over the frequency range,

then the frequency response is described successfully. This means that the char-

acteristics of the reservoir can be revealed by the multiple frequency components.


Figure 2.5: Square pulsing can be better utilized with an analysis of a set of multi-ple sinusoidal pulses sourced (left) at the injection point and received (right) at theobservation point.

To model the pressure signal in terms of each frequency component, the sinu-

soidal steady-state space assumption is applied to flow rate and pressure as both

being periodic such as p(r, t) = g(r, ω)eiωt, where the pressure response can be

described separately by the multiplication of the distance part and the time part. It

is important to note that the exponential function is easier to handle for derivation

of steady-state solutions than using sin or cos functions directly.

Therefore the frequency method works based on the following premises. First,

it is recognized that the frequency information represented as a frequency response

H(ω) for a range of ω ∈ (ωsourcing, ωsampling) carries the same amount of informa-

tion as the pressure time series signal. In other words, when the square pulse

is sourced, the corresponding frequency information in each harmonic frequency


component will represent the reservoir characteristics in the same way as the time

series pressure data because the square pulse is a sum of odd multiples of har-

monic frequency components. The second premise is that the analytical reservoir

model is based on steady-state periodic assumptions. The frequency information

can be represented in a closed form without the time component, and this forms

the basis for permeability estimation with extracted attenuation and phase shift

information.

2.3 Multicomposite Radial Ring Model

The assumptions for the radial reservoir model are as follows: horizontal, single-

phase flow, slightly compressible fluid of constant viscosity and compressibility,

negligible gravitational force, the reservoir is initially at equilibrium, and no tem-

poral variations during pulse tests. Figure 2.6 illustrates the model for which the

analytical solution is derived.

Figure 2.6: Schematics of a multicomposite radial ring system for estimating radialpermeability distribution.


2.3.1 Formulation of the Steadily Periodic Solution

Here pinj(t) denotes the pressure at the wellbore where a certain amount of flow

is injected (or produced). Therefore pinj(t) = p(r = rw, t). The pressure shows

the maximum amplitude and decreases as it passes through the porous media.

pobs(t) denotes the observed pressure some distance away from the source, that is,

pobs(t) = p(r, t).

The following diffusivity equation with boundary conditions is used. Each

equation is discretized with the dimensionless parameters which are described

next. The discretization applies to the ring with numbers j = 1, .., N + 1.

∇ · (k(r)∇p) = φµct∂p∂t→ 1

rD

∂

∂rD(rD

∂pDj

∂rD) =

1ηDj

∂pDj

∂tD(2.14)

- Inner boundary condition:

2πhµ

[ k(r)r∂p∂r

]r=rw = qB → [∂pD1

∂rD]rD=1 = −eiωDtD (2.15)

- Outer boundary for an infinite reservoir:

limr→∞

p(r, t) = pinit → limrD→∞

pDN+1(rD) = 0 (2.16)

- Two continuity conditions between zones (j ∈ {1, ..., N}):

pDj(rDj+1 , tD) = pDj+1(rDj+1 , tD), kDj

∂pDj

∂rDj+1

= kDj+1

∂pDj+1

∂rDj+1

(2.17)

The dimensionless parameters are the following. Note that for the dimension-

less frequency, only the first block permeability (k1) is used for this formulation.

- Dimensionless permeability:

kD =k j

k1(2.18)


- Dimensionless time:

tD = 2.637 · 10−4 k1tφ1µctr2

w(2.19)

- Dimensionless radial distance:

rD = r/rw (2.20)

- Dimensionless frequency of periodic pulses:

ωD =φ1µctr2

wω

2.637 · 10−4k1, with ω =

2π

Tp(2.21)

- Dimensionless hydraulic diffusivity constant:

ηDj =ηj

η1, with ηj =

k j

φjµct(2.22)

- Dimensionless mobility:

λDj =λj

λ1, with λj =

k j

µ(2.23)

- Dimensionless pressure:

pDj =k1h

141.2qBµ(pinit − pj) (2.24)

- Dimensionless storage:

CD =5.615C

2πφcthr2w

(2.25)

With periodic steady-state assumption, the distance and time part of the equa-

tion can be divided as pD(rD, tD) = gD(rD, ωD)eiωDtD . The exponential function

is used for convenience for the derivation. The actual sinusoidal pressure then

corresponds to imaginary part of pD(rD, tD). Equation 2.14 - Equation 2.17 are


arranged in terms of gD(rD):

1rD

∂

∂rD(rD

∂gDj

∂rD) =

iωD

ηDj

gDj (2.26)

- Inner boundary condition:

[∂gD1

∂rD]rD=1 = −1 (2.27)


limrD→∞

gDN+1(rD, ωD) = 0 (2.28)

- Continuity conditions between zones:

gDj(rDj+1 , ωD) = gDj+1(rDj+1 , ωD), kDj

∂gDj

∂rDj+1

= kDj+1

∂gDj+1

∂rDj+1

(2.29)

The pressure solution format is expressed with coefficients Cj, j ∈ {1, .., 2N +

2}. In terms of jth block pressure (j ∈ {1, ..., N + 1}),

gDj(rD, ωD) = C2j−1K0(

√iωD

ηDj

rD) + C2j I0(

√iωD

ηDj

rD) (2.30)

where I0(· · · ) and K0(· · · ) are modified Bessel functions of the first and second

kind, respectively, and of zero order. C2j−1 and C2j are two constants which are

unknown at each ring.

Substituting Equation 2.30 to the inner boundary condition in Equation 2.27:

−√

iωD

ηD1

C1K1(

√iωD

ηD1

) +

√iωD

ηD1

C2 I1(

√iωD

ηD1

) = −1 (2.31)


The outer boundary condition in Equation 2.28 requires that:

C2N+2 = 0 (2.32)

The set of continuity equations brings the following ( j ∈ {1, ..., N}):

C2j−1K0(√

iωDηDj

rDj+1) + C2j I0(√

iωDηDj

rDj+1)

−C2j+1K0(√

iωDηDj+1

rDj+1)− C2j+2 I0(√

iωDηDj+1

rDj+1) = 0(2.33)

−kDj

√iωDηDj

C2j−1K1(√

iωDηDj

rDj+1) + kDj

√iωDηDj

C2j I1(√

iωDηDj

rDj+1)

+kDj+1

√iωD

ηDj+1C2j+1K1(

√iωD

ηDj+1rDj+1)− kDj+1

√iωD

ηDj+1C2j+2 I1(

√iωD

ηDj+1rDj+1) = 0

(2.34)

In calculating the coefficients for Cj ( j ∈ {1, ..., 2N + 1}), the matrix Ac = d is

formed from Equation 2.31 - Equation 2.34, with the matrix A of size (2N + 1)×(2N + 1) of Bessel functions, with vectors c and d of size (2N + 1) of the following:

c =[

C1 C2 ... C2N+1

]T, d =

[−1 0 ... 0

]T(2.35)

The matrix A can be further transformed to a tridiagonal matrix B. The vector

d remains unchanged because its entries for j ∈ {2, ..., 2N + 1} are zeros. The

derivation is similar to the one stated by Rosa (1991), except this study focuses

on the steady-state behavior. The Thomas algorithm (William et al., 2007) is used

which does not require the direct inversion of B for calculating Bc = d. See Ap-

pendix B for more details.

Finally, when a set of coefficients Cj are solved, the attenuation x(ωD) and

phase shift θ(ωD) at jth ring can be obtained from the pressure solution as the


following:

xj(ωD)eiθj(ωD) =gobs(rDj , ωD)

ginj(rD = 1, ωD)=

C2j−1K0(√

iωDηDj

rDj) + C2j I0(√

iωDηDj

rDj)

C1K0(√

iωDηD1

) + C2 I0(√

iωDηD1

)(2.36)

The wellbore storage and skin effect are incorporated by modifying the follow-

ing pressure equation and the inner boundary condition.

−C∂pw

∂t+

2πhµ

(k(r)r

∂p∂r

)r=rw

= qB

→ −CD∂pwD

∂tD+

(rD

∂pD

∂rD

)rD=1

= −eωDtD

→ −iωDCDgwD +

(rD

∂gD

∂rD

)rD=1

= −1 (2.37)

pw(t) = p(rw, t)− s(

r∂p∂r

)r=rw

→ pwD(tD) = pD(rD = 1, tD)− s(

rD∂pD

∂rD

)rD=1

→ gwD(ωD) = gD(rD = 1, ωD)− s(

rD∂gD

∂rD

)rD=1

(2.38)

By combining Equation 2.37 and Equation 2.38, the following holds. A and B

have to be modified accordingly.

− 1 = −iωDCDgD1 + s iωDCD∂gD1

∂rD+

∂gD1

drD(2.39)

= −iωDCD

(C1K0(

√iωD

ηD1

) + C2 I0(

√iωD

ηD1

)

)...

+(siωDCD + 1)

(−√

iωD

ηD1

C1K1(

√iωD

ηD1

) +

√iωD

ηD1

C2 I1(

√iωD

ηD1

)

)


Hence, the attenuation and phase shift are obtained with storage and skin ef-

fects as the following.

xj(ωD)eiθj(ωD) =gobs(rDj , ωD)

gwD(1, ωD)(2.40)

=

C2j−1K0(√

iωDηDj

rDj) + C2j I0(√

iωDηDj

rDj)

C1K0(√

iωDηD1

) + C2 I0(√

iωDηD1

)− s[−√

iωDηD1

C1K1(√

iωDηD1

) +√

iωDηD1

C2 I1(√

iωDηD1

)]

For the porosity distribution which varies in relation with permeability, the sys-

tem of equations can be modified to solve for porosity also. A functional relation

often exists between the permeability and porosity, such as the Carman-Kozeny

type equation (k j = α exp(βφj)). The dimensionless diffusivity constant and time

can be changed accordingly.

2.3.2 Frequency Response of Radial Ring Model

Transmission of Pressure in Radial Ring Model and Representation by Frequency

Attributes

First we examine the pressure behavior at the center of each radial ring for homo-

geneous and heterogeneous radial permeabilities. The main point of investigation

is how well attenuation and phase shift can reflect the pressure behavior for dif-

ferent permeability distributions. Of course, multiple measurements between the

injection and observation point are not available in practice; instead, by sourcing

multiple frequencies, it can be demonstrated that the attenuation and phase shift

data between the injection and observation point are representative of a certain

permeability distribution.

As shown in Figure 2.7 (a), the three different radial permeability distribution

models are examined. The block pressures have different spacing over radial dis-

tance, each of which reflects a level of heterogeneity (Figure 2.7 (b, c, d)).

The heterogeneity information becomes more apparent by plotting attenuation

and phase shift at a sourcing frequency at each ring. In general, as the pressure


signal passes through the medium the further away from the injection well, the

weaker the magnitude of signals and the more delayed their arrival (Figure 2.7 (e,

f)). Such information is shown by attenuation and phase shift in a way that they

have different slope distinctively for each region of uniform permeability. Attenua-

tion and phase shift provides a way to identify different permeability distributions

in a concise manner.

Figure 2.8 demonstrates a series of sinusoidal pressure for two permeability dis-

tribution models in dimensionless pressure magnitude and time. The transmission

of pressure is apparently different for two models. This forms a basis for analyzing

odd multiple harmonics that would be obtained from square pulses. With respect

to a specific sourcing frequency, the time series information can be summarized

with a attenuation and a phase shift value at each measurement distance.


0 5000

100

200

300

400

500

600

Rad

ial p

erm

eabi

lity,

kr, m

d

Model 1

0 5000

100

200

300

400

500

600

Radial distance, r, ft

Model 2

0 5000

100

200

300

400

500

600

Model 3

0 2 4 6 8 10 120

1

2

3

4

5

6

7

Time (hr)

Pre

ssur

e ch

ange

(ps

i)

r = rw

Ring 1Ring 2Ring 3Ring 4Ring 5Ring 6Ring 7Ring 8Ring 9Ring 10

(a) (b)

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (hr)

Pre

ssur

e ch

ange

(ps

i)

r = rw


0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

Time (hr)

Pre

ssur

e ch

ange

(ps

i)

r = rw


(c) (d)

0 100 200 300 400 500 6000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Atte

nuat

ion

Model 1Model 2Model 3

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Pha

se s

hift


(e) (f)

Figure 2.7: Frequency attributes for radial ring models: (a) radial permeabilitydistribution for three models, (b) pressure at multiple radial rings for Model 1,(c) pressure at multiple radial rings for Model 2, (d) pressure at multiple radialrings for Model 3, (e) attenuation over radial distance, (f) phase shift over radialdistance.


0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 106

−5

−4

−3

−2

−1

0

1

2

3

4

5

Dim

ensi

onle

ss s

inus

oida

l pre

ssur

e

Dimensionless time

r = rw


0.5 1 1.5 2 2.5 3 3.5 4

x 106

−6

−4

−2

0

2

4

6D

imen

sion

less

sin

usoi

dal p

ress

ure

Dimensionless time

r = rw


Figure 2.8: Sinusoidal pressure at multiple observation points Model 1 (left) andModel 2 (right).


Consideration for Appropriate Sourcing Frequency

Figure 2.9 illustrates a frequency response of a homogeneous radial reservoir (Model

1) with three different permeabilities from 100 md to 300 md. The larger the per-

meability, the greater magnitude and smaller phase shift values are observed.

In practice, it is crucial to decide upon the sourcing periodicity (ω) in which

the pulse testing is operated. As noted by Figure 2.9, the desirable choice of the

frequency should be such that the accuracy of attenuation and phase shift to be

gathered are within (0, 1). Also the engineer should keep in mind that the attenu-

ation and phase shift values at the dominant frequency should not be too extreme

so that the measurement precision supports the higher frequency components as

well in order for multiple frequency analysis to take place. In essence, the sourc-

ing frequency should be determined depending on the overall magnitude of the

permeability. It is undesirable to have an observed signal that is highly attenuated

with almost zero magnitude or a phase shift which is delayed over a cycle. If that

happens, lowering the periodicity is recommended to obtain a clear observation

pressure signal. This is consistent with the finding by Huang et al. (1998) that for a

reservoir with low permeability, a longer testing time for a pulse test is required.

Figure 2.9: Frequency response of homogeneous radial model: attenuation (left)and phase shift (right).


Another factor to consider is the distance between the points of injection and

observation. The further the distance from an active well, the lower the frequency

excitation should be to have the same frequency attributes. A homogeneous reser-

voir of kr = 100 md was investigated as an example with varying dimensionless

frequencies over different observation distances, as shown in Figure 2.10. The re-

lationship of attenuation and phase shift data with sourcing frequencies and the

observation distance suggest that the suitable frequency attributes can be obtained

from pressure data when the distance is not too far and the sourcing frequency

is not too high. A good strength of observed signal is desirable for the following

reasons: it is convenient to analyze, the recording devices have limited precision,

and the reservoir response should not be masked by pressure noise. It is impor-

tant to note that at a high frequency range or at a long observation distance, the

attenuation on the order of 10−4 can be observed, which translates to small pres-

sure amplitudes for observed signals relative to injection signals. This is too small

a precision for the observation signal to be recorded in practice. In terms of phase

shift, the value can exceed more than one cycle if the sourced frequency is too high

or the observation distance is too far away.

0 10 20 30 40 50 60 70 8010

−10

10−8

10−6

10−4

10−2

100

Dimensionless radial distance

Atte

nuat

ion

wD=0.0001

wD=0.001

wD=0.01

wD=0.1

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dimensionless radial distance

Pha

se s

hift

wD=0.0001

wD=0.001

wD=0.01

wD=0.1

Figure 2.10: Sensitivity of frequency data to dimensionless radial distance for ho-mogeneous radial model: attenuation (left) and phase shift (right).

Therefore the possible permeability range of inspection, the distance between


the two points, and the mechanical precision of the measurement are the deci-

sive factors for designing pressure pulse testing with an appropriate sourcing fre-

quency. In other words, low periodicity of flow rate pulsing is desirable for a

reservoir that is presumed to be a low permeability field, and has a long dis-

tance between the injection and observation point. The pressure measurement

device should support certain precision so that even weak pressure signals can

be recorded.

In addition, the petrophysical properties bring difference to ωD in Equation 2.21

are also determining factors that engineers have to keep in mind.


Characterization of Heterogeneity by Frequency Data

Shown on the cross-plot in Figure 2.11 are the attenuation and phase shift data

which are extracted from multiple sinusoidal pressures with different sourcing

frequencies which are spaced logarithmically. Each data point represents atten-

uation (x-axis) against phase shift (y-axis) at a specific frequency. The observation

well is placed 570 ft away from the injection well. Each of three reservoir models

falls on their own line in the cross-plot independent of the permeability multiplier

within the model, suggesting that the multiple frequency data form a differen-

tiating characteristic for heterogeneity. The different levels of heterogeneity are

identified with different shapes of the curves on the cross-plot when excited with

multiple frequencies. The characterization by multiple sinusoidal frequencies is

expected to provide the same knowledge of the permeability distribution as in

Figure 2.8 where we have multiple observation points. Here we have one observa-

tion point but with different sourcing sinusoidal signals with different frequencies,

which have different propagation lengths.

The cross-plot also suggests that without knowing the exact frequency infor-

mation, the attenuation and phase shift pair can still indicate the degree of hetero-

geneity in the reservoir by giving answers to dimensionless permeability, though

not absolute values of the permeability distribution.

Frequency Data with Sourcing Frequency and Permeability Range

Multiplying all the permeabilities in the reservoir by a scalar (5, in this example

in Figure 2.11) does not change the relationship on the cross-plot of attenuation

vs. phase shift. In principle, the characteristic of attenuation and phase shift is

the same as long as the dimensionless values kD and ωD are the same. In Ap-

pendix A, the relationship between the sourcing frequency and the permeability

range is proved to be the following relationship in the frequency response, that is,

H(αk, ω) = H(k, 1/α·ω) (2.41)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Attenuation

Pha

se s

hift

Model 15*Model 1Model 25*Model 2Model 35*Model 3

Figure 2.11: Attenuation vs. phase shift cross plot of radial ring models at multiplefrequencies


Therefore, for attenuation: x(αk, ω) = x(k, 1/α·iω). The same applies for

phase shift: θ(αk, ω) = θ(k, 1/α·ω).

As the pressure goes through the reservoir in a diffusive manner, there may be

multiple reservoirs with different permeability distributions that show the same

frequency response as other reservoirs. But still the information on the cross-plot

is a summary of the frequency response and forms a good set of conditioning input

data for the inverse problem from which one can infer the permeability distribu-

tion.

2.4 Partially Penetrating Well with Cross Flow in Mul-

tilayered Model

When the open section of a well casing does not include the full thickness of the

aquifer it penetrates, the well is referred to as partially penetrating. Partially pen-

etrating wells are common. Under partial penetrating conditions, the flow around

the pumping well has an additional vertical flow component, reflected by the up-

ward inflection points in the time-drawdown response (Ehlig-Economides and

Joseph, 1969).

We investigated a pulse testing environment where the pressure measurements

create enough difference for the injected and observed location so that the inter-

layer vertical permeability can be estimated. The sourcing pressure is measured at

the perforated layer and the observed pressure is recorded at some depth where

the pressure magnitude is reduced and delayed. The attenuation and phase shift

data by the obtained pressure pair are used here for characterization of permeabil-

ity distributions. Figure 2.12 illustrates the model in which the analytical solution

is derived. The assumption for the multilayered model is similar to the radial case:

single-phase flow, slightly compressible fluid of constant viscosity and compress-

ibility, negligible gravitational force, the reservoir is initially at equilibrium, and

no temporal variations of properties during the tests.


Figure 2.12: Schematics of a partially penetrating well with cross flow system forestimating vertical permeability in a multilayered reservoir.

2.4.1 Formulation of the Steadily Periodic Solution

The analytical formation for a constant rate pressure response in the multilayered

model is based on previous studies by many others (Ehlig-Economides and Joseph,

1969; Clegg and Mills, 1969; Hatzignatiou et al., 1987; Park, 1989; Gomes and Am-

bastha, 1993). The partial penetration conditions are adapted from Bilhartz and

Ramey (1977) and the steadily periodic solution form is derived.

In this work, the vertical permeability is defined in terms of interlayer crossflow

as (kv)j+1/2 = 2(1/kv)j+(1/kv)j+1

for evenly spaced layers. Thus, the formulation has

(N − 1) unknown permeabilities to estimate for N layers.

Here pinj(t) denotes the pressure at the wellbore at a perforated layer where a

certain amount of flow is injected (or produced). For simplicity, the first layer is as-

sumed to be perforated in this study, that is, (pinj(t) = p1(r = rw, t)) where pj(r, t)

denotes the jth layer pressure located r distance away at time t. The pressure

shows the maximum amplitude for this layer. pobs(t) denotes the observed pres-

sure some distance away from the sourcing, or perforated layer. In practice, this


can be located at Nth layer, either at the wellbore, that is, pobs(t) = pN(r = rw, t) or

some radial distance away such as pobs(t) = pN(r, t). A set of continuity equations

is not used in this case because the pressure is defined distinctively for each layer.

A similar derivation is found by Ehlig-Economides and Joseph (1969); Park (1989)

except that the derivation is for steady state.

The following diffusivity equation ( Equation 2.42) with boundary conditions

(Equation 2.43 - Equation 2.46) is used. Each equation is discretized with dimen-

sionless parameters which are introduced next. The discretization applies to the

layer in j, j = 1, .., N:

1r

∂∂r

(rk j

∂pj∂r

)+ ∂

∂z

(kvj

∂pj∂z

)= (φµct)j

∂pj∂t

→ κj∇2pjD + λj+1/2(pj+1D − pjD) + λj−1/2(pj−1D − pjD) =∂pjD∂tD

(2.42)

- No flow at the wellbore for all other nonperforated layers, j = 2, .., N:

0 =∂pjD∂rD

∣∣∣∣rD=1

(2.43)

- Inner boundary condition for wellbore pressure at the perforated layer 1:

pw f =

(p1 − s1r

∂p1

∂r

)r=rw

→ pωD =

(p1D − s1

∂p1D

∂rD

)rD=1

(2.44)

- Inner boundary condition at the wellhead with a constant production q ap-

plied at the perforated layer 1 at 0 ≤ z ≤ hw:

q = C∂pw f

∂t− (kh)1rw

∂p1

∂r

∣∣∣∣r=rw

→ 1 = CD∂pωD

∂tD− κ1

∂p1D

∂rD

∣∣∣∣rD=1

(2.45)


limr→∞

pj(r, t) = pinit → limrD→∞

pjD(rD) = 0 (2.46)


The dimensionless parameters are the following, with (kh)t =n∑

j=1k jhj, (φh)t =

n∑

j=1φjhj and γj =

φjhjn∑

k=1φkhk

.

- Dimensionless permeability:

κj =(kh)j(kh)t

for radial permeability

λj =r2

w(kh)t

(kvh

)j+1/2

for vertical permeability(2.47)

- Dimensionless time:

tD = 2.637 · 10−4 (kh)tt(φh)tµctr2

w(2.48)

- Dimensionless radial distance:

rD = r/rw (2.49)

- Dimensionless frequency of periodic pulses:

ωD =(φh)tµctr2

wω

2.637 · 10−4(kh)t, with ω =

2π

Tp(2.50)

- Dimensionless pressure:

pjD =(kh)tt

141.2qBµ(pinit − pj) (2.51)

- Dimensionless wellbore storage:

CD =5.615C

2π(φh)tcthr2w

(2.52)

With the periodic steady-state assumption, the distance and time related part

can be divided as pjD(rD, tD) = gjD(rD, ωD)eiωDtD for jth layer. Equation 2.42


-Equation 2.46 are arranged in terms of gD(rD):

κj∇2gjD + λj+1/2(gj+1D − gjD) + λj−1/2(gj−1D − gjD) = iωDγjgjD (2.53)

- Inner boundary condition at the wellhead:

1 = iωCDgwD − κ1dg1D

drD

∣∣∣∣rD=1

(2.54)

- No flow at the wellbore for all other nonperforated layers, j = 2..N:

0 =dgjD

drD

∣∣∣∣rD=1

(2.55)

- Inner boundary condition at the wellbore:

gwD = g1D(rD = 1)− s1dg1D

drD

∣∣∣∣rD=1

(2.56)


limrD→∞

gjD(rD) = 0 (2.57)

The solution form for Equation 2.53 for each layer is the following:

gjD(rD, ωD) = Aj(ωD)K0(σ(ωD)rD) + Bj(ωD)I0(σ(ωD)rD) (2.58)

First, to obtain the σk(ωD) values, the solution form in Equation 2.58 is applied

to Equation 2.53. This is to solve N homogeneous equations of the form with

LX = 0. A is a symmetric tridiagonal matrix, where each element of the matrix is


denoted with lj,k as follows:

lj,k(ωD) =

λj−1/2 k = j− 1 (j > 1)

κjσ2 − iωDγj − λj−1/2 − λj+1/2 k = j

λj+1/2 k = j + 1 (j < n)

0 otherwise

(2.59)

and X is a vector form with respect to a fixed σ as follows:

X =

A1K0(σ(ωD)rD) + B1 I0(σ(ωD)rD)

· · ·AjK0(σ(ωD)rD) + Bj I0(σ(ωD)rD)

· · ·AnK0(σ(ωD)rD) + Bn I0(σ(ωD)rD)

(2.60)

The σ values can be calculated by solving det L(σ) = 0. An equivalent condi-

tion is to find eigenvalues of another symmetric matrix M, that is, σk = eig(M)

where the entries of M are listed as follows:

mj,k(ωD) =

− λj−1/2√κj−1κj

k = j− 1 (j > 1)iωDγj+λj+1/2+λj−1/2

κjk = j

− λj+1/2√κjκj+1

k = j + 1 (j < n)

0 otherwise

(2.61)

The matrix M has a total of N eigenvalues, σk for k = 1, ..., N. When all the

eigenvalues are found, the general solution in Equation 2.58 is now written as:

gjD(rD, ωD) =N

∑k=1

[Aj,k(ωD)K0(σk(ωD)rD) + Bj,k(ωD)I0(σk(ωD)rD)] (2.62)

Ehlig-Economides and Joseph (1969) showed that there is a relationship be-

tween Aj,k and Bj,k depending on the three types of outer boundary conditions as


Bj,k = bk Aj,k with the following bk:

bk =

0 infinite boundaryK1(σkreD)I1(σkreD)

no flow boundary

−K0(σkreD)I0(σkreD)

constant boundary

(2.63)

With the above relation, Equation 2.62 is modified as:

gjD(rD, ωD) =N

∑k=1

Aj,k[K0(σk(ωD)rD) + bk I0(σk(ωD)rD)] (2.64)

With respect to each eigenvalue σk, the relation L(σk)X = 0 further simplifies

the solution form for gjD. The matrix elements of L gives (N − 1) relationship in

terms of A1,k. A new set of coefficients αi,k (i = 2, · · · , N) is used. The coefficients

are defined iteratively.

A2,k = −l1,1l1,2

A1,k := α2,k A1,k

A3,k = −l2,1 A1,k+l2,2 A2,k

l2,3:= α3,k A1,k

· · · ,

An,k = −ln−1,n−2 An−2,k+ln−1,n−1 An−1,k

ln−1,n:= αn,k A1,k

(2.65)

Then the solution form in Equation 2.68 is updated again as:

gjD(rD, ωD) =N

∑k=1

αj,k A1,k[K0(σk(ωD)rD) + bk I0(σk(ωD)rD)] (2.66)

The unknown coefficients A1,k, (k = 1, · · · , N) can be solved from the inner

boundary conditions in Equation 2.54 and Equation 2.55.

The general frequency response for a multilayer case with a partial perfora-

tion is the pressure ratio between an injection pressure and observation pressure.

For the observation pressure part measured at Nth layer at a radial distance of

rD, gobs(rD, ωD) = gND(rD, ωD), and the injection pressure part measured at the


1st perforated layer, ginj(rD, ωD) = g1D(rD = 1, ωD), the following frequency re-

sponse is used to obtain a pair of attenuation phase shift at a dimensionless fre-

quency ωD .

H(ωD) =gND(rD, ωD)

gwD(1, ωD)=

N∑

k=1αN,k A1,k[K0(σk(ωD)rD) + bk I0(σk(ωD)rD)]

N∑

k=1α1,k A1,k[K0(σk(ωD)) + bk I0(σk(ωD))]

(2.67)

When a skin factor is involved, the denominator changes to the following based

on Equation 2.56:

gwD(rD = 1, ωD) =N

∑k=1

α1,k A1,k[K0(σk(ωD)) + bk I0(σk(ωD))] + · · · ,

s1

N

∑k=1

α1,k A1,k[K1(σk(ωD))− bk I1(σk(ωD))] (2.68)

2.4.2 Frequency Response of Multilayered Model

Transmission of Pressure in Multilayered Model and Representation by Fre-

quency Attributes

Figure 2.13 (a) shows three different vertical permeability distributions (kv) of mul-

tilayered models with a fixed radial permeability for all layers, kr = 100 md. Fig-

ure 2.13 (b, c, d) show magnitudes of square pressure pulses measured at the well-

bore over depth (pD(hj, rw), i = 1, .., 9, ∆h = 2 f t). The time series information can

be represented with attenuation and phase shift data at each layer for three differ-

ent models with different permeability distributions as shown in Figure 2.13 (e, f).

A monotonically decreasing trend in magnitude and an increasing trend in phase

shift are observed over depth. Each model with a different permeability distribu-

tion results in a distinguishing frequency attributes. For instance, the difference

of attenuation between Model 4 and Model 6 is small, but phase shift shows an

order of O(10−3) difference. Therefore the time-series pressure time data that are

hard to distinguish can be summarized concisely by attenuation and phase shift,


which provides a metric to distinguish different permeability distributions.

Figure 2.14 shows frequency attributes over depth for another common case

in which the kv/kr ratio is fixed, kr(h) varies in each layer according to the kv(h)

in each model. Each model with a different permeability distribution results in a

distinguishing frequency attributes as well.

If the pressure measurements at multiple depths as in Figure 2.13 (e, f) were

available, the permeability distribution can be obtained easily because the number

of unknown permeabilities corresponds to number of measurements. In practice

the measurement at multiple depths is not available, and only limited locations

can record pressures, meaning that the solvability condition becomes harder. This

study essentially investigated the capability of how well multiple frequencies can

reveal the heterogeneity between injection and observation point where the pres-

sure pulses are recorded. In other words, Figure 2.17 data are the ones that are

available from real pressure pulse tests by sourcing multiple frequencies embed-

ded in square pulses, and the unknown permeability distribution are inferred from

multiple frequency information where each frequency has an effective length of in-

fluence.


0 10 20

2

4

6

8

10

12

14

16

18

Dep

th, h

, ft

Model 4

0 10 20

2

4

6

8

10

12

14

16

18

Vertical permeability, kv, md

Model 5

0 10 20

2

4

6

8

10

12

14

16

18

Model 6

0 2 4 6 8 100

1

2

3

4

5

6

7

8

Time (hr)

Pre

ssur

e ch

ange

(ps

i)

Layer 1Layer 2Layer 3Layer 4Layer 5Layer 6Layer 7Layer 8Layer 9Layer 10

(a) (b)

0 2 4 6 8 100

1

2

3

4

5

6

7

Time (hr)

Pre

ssur

e ch

ange

(ps

i)


0 2 4 6 8 100

1

2

3

4

5

6

7

8

Time (hr)

Pre

ssur

e ch

ange

(ps

i)


(c) (d)

2 4 6 8 10 12 14 16 180.05

0.1

0.15

0.2

0.25

0.3

Depth, h, ft

Atte

nuat

ion


2 4 6 8 10 12 14 16 180.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

Depth, h, ft

Pha

se s

hift


(e) (f)

Figure 2.13: Analysis on multilayered models with kr = 100 md: (a) vertical per-meability distribution for three models, (b) pressure at multiple layers for Model4, (c) pressure at multiple layers for Model 5, (d) pressure at multiple layers forModel 6, (e) attenuation over vertical distance, (f) phase shift over vertical dis-tance


2 4 6 8 10 12 14 16 180.05

0.1

0.15

0.2

0.25

0.3

Depth, h, ft

Atte

nuat

ion


2 4 6 8 10 12 14 16 180.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

Depth, h, ft

Pha

se s

hift


Figure 2.14: Multilayered models with kv/kr = 0.1: attenuation (left) and phaseshift (right) over vertical distance


Consideration for Appropriate Sourcing Frequency

Figure 2.15 demonstrates the frequency information with two different homoge-

neous vertical permeabilities at multiple frequencies for a multilayered reservoir

(Model 4). The larger the vertical permeability, the more attenuation and less

phase shift are observed, which is a similar frequency relationship applied for the

radial reservoir model in Figure 2.9. The first case is a reference case where kv is

10 md and kr is 100 md for all layers. The second case is when the vertical perme-

ability is the same while the kv/kr ratio is changed. The third case is when each

permeability value is as twice large as the first case. The last case is when the ver-

tical permeability is twice the reference case but the kv/kr ratio is different from

the second case. With the different kv/kr ratio, the horizontal permeability creates

about 10% difference in attenuation whereas the difference in phase shift is not

significant. The phase shift is nondiscriminatory to the radial permeability value,

which means it is difficult to obtain kv and kr at the same time by using attenu-

ation and phase shift values. In theory, kv/kr introduces another nonlinearity to

be estimated as well. As relying on a small difference in attenuation for the radial

permeability estimation on top of the vertical permeability estimation is risky, for

simplicity in vertical permeability investigation, the radial permeabilities are as-

sumed known for all cases. Radial permeability can be found from a conventional

well test.

Figure 2.15 is also suggestive of the appropriate sourcing frequency range de-

pending on the scale of vertical permeability, because it is undesirable to have an

observed signal that is highly attenuated with almost zero magnitude or the phase

shift which is delayed over a cycle. The precision of the device in measuring pres-

sure data would be another limiting factor. Lowering the periodicity in that case is

recommended to obtain a clear observation pressure signal.

Another factor to consider is the distance between the point of injection and

observation. The further the depth from an injection point, the lower the fre-

quency excitation should be to have the same frequency attributes. A homoge-

neous reservoir of kv = 10 and kr = 100 md was investigated as an example with


Figure 2.15: Frequency response of multilayered model: attenuation (left) andphase shift (right)

varying dimensionless frequencies over different observation depth at a wellbore

(Figure 2.16). The similar relationship which applies to a radial model also applies

to a multilayered model. The relationship of attenuation and phase shift data with

sourcing frequencies and the observation depth suggest that the suitable frequency

attributes can be obtained from pressure data when the observation distance is not

too far and the sourcing frequency is not too high. It is important to note that at

a high frequency range or at a long observation distance, the attenuation on the

order of 10−4 can be observed, which translates to small pressure amplitudes for

observed signals relative to injection signals. This is too small a precision for the

observation signal to be recorded in practice. In terms of phase shift, the value can

exceed more than one cycle if the sourced frequency is too high or the observation

distance is too far away.


0 5 10 15 20 25 3010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Depth, h

Atte

nuat

ion

wD=0.00001

wD=0.0001

wD=0.001

wD=0.01

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Depth, h

Pha

se s

hift

wD=0.00001

wD=0.0001

wD=0.001

wD=0.01

Figure 2.16: Sensitivity of frequency data to depth for multilayered model: attenu-ation (left) and phase shift (right).


Characterization of Heterogeneity by Frequency Data

Multiple frequency attributes of attenuation and phase shift measured at 18 ft

away from the layer further distinguishes the difference between Model 4, 5, and 6

as shown in Figure 2.17. The attenuation and phase shift cross-plots are shown for

two cases gathered from a wide range of logarithmically spaced frequencies. The

discrepancies of the frequency data are apparent for radial permeabilities with a

ratio of kv/kr = 0.1, whereas a good precision is required for the latter case with

kr = 100 md. The permeability estimation performance is examined further for

those models in Chapter 6.

Frequency Data with Sourcing Frequency and Permeability Range

Figure 2.17 shows that for a reservoir with a ratio of kv/kr = 0.1, the charac-

teristic of attenuation and phase shift is the same as long as the dimensionless

values for radial and vertical permeability kD and frequencies ωD are the same, as

in the radial ring case. The relationship for the frequency response H(αk, iω) =

H(k, 1/α·iω) with scalar multiplication by α is also maintained for vertical perme-

ability as long as the kv/kr ratio is the same; when radial permeability kr is fixed,

then the relationship no longer holds.


0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

Attenuation

Pha

se s

hift


0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

Attenuation

Pha

se s

hift


Figure 2.17: Attenuation vs. phase shift cross plot of multilayered models withkr = 100 md (left) and kv/kr = 0.1 (right).


2.5 Discussion on Extension to Heterogeneous Perme-

ability Distribution

The following discussion is to illustrate how the permeability estimation process

can be benefited by extending the unknown permeability distribution to be het-

erogeneous. A radial heterogeneous model with a permeability distribution Fig-

ure 2.18 (a) is used to demonstrate. The previous method by Renner and Messar

(2006) that is discussed here is limited to the case when a pair of attenuation and

phase shift values is used to estimate an average permeability.

From the measured injection and observation pressure, a pair of attenuation

and phase shift values is obtained at a dominant sourcing frequency. With the

estimated frequency data, the previous approach confines the solution space to be

homogeneous and attempts to estimate the average permeability. In estimating the

average permeability, the pair of attenuation and phase shift values was used such

that the orthogonal distance from the measured data and the analytical solution

space becomes the shortest. The red line in Figure 2.18 (c) represents a homoge-

neous radial reservoir. The average permeability of the reservoir ranges from 10

md to 1000 md. Then the average permeability is estimated to the closest point in

the least-square sense, which is projected as 323.7 md in this example. The true

average permeability from Figure 2.18 (a) is 102.3 md. The previous method does

not estimate an average permeability well by confining the permeability distribu-

tion to be homogeneous. The average permeability estimation by the previous

approach can be misleading.

The given pair of attenuation and phase shift values belongs to the green line

in Figure 2.18 (b). This line is constituted by many frequency excitations which

represent the actual reservoir heterogeneity. One pair of attenuation and phase

shift is not enough to reveal a reservoir characteristic as at least two data points

are required to constitute a line. By extracting more attenuation and phase shift

data at multiple harmonic frequencies that are available from the pressure pulses,

the frequency data points have the ability to represent the reservoir as illustrated

in Figure 2.18 (d). The number of degrees of freedom corresponds to the number of


grids or blocks used to describe the reservoir. A user of the frequency method has

to be careful that the harmonic frequencies might not span the frequency points

that are necessary to identify the reservoir.

Permeability heterogeneity is one of the most important reservoir parameters

to be identified for further field developments. By allowing the permeability dis-

tribution to be heterogeneous, an accurate assessment of the permeabilities can be

obtained. When the heterogeneity is described by pressure pulse tests, the optimal

placement of oil, gas or water wells can be designed for maximum oil recovery.

The optimization of recovery is crucially dependent on the quality of the reservoir

where the recovery factor is very sensitive to reservoir heterogeneity. Therefore,

an accurate knowledge of horizontal and vertical permeability distribution is es-

sential.


3 7 11 15 19 23 27 31 35 390

20

40

60

80

100

120

140

160

180

200

Dimensionless distance

Rad

ial p

erm

eabi

lity,

kr, m

d

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Attenuation

Pha

se s

hift

Heterogeneous solutionHomogeneous solutionAt dominant frequency

(a) (b)

0.1 0.15 0.2 0.25 0.3

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

Attenuation

Pha

se s

hift

At dominant frequencyAnalytical homogeneous solution

323.7 md ( 1000 md )

( 10 md )

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Attenuation

Pha

se s

hift

Periodically steady−state solutionAt harmonic frequenciesAt dominant frequency

(c) (d)

Figure 2.18: Illustration of frequency method in comparison with previousmethod: (a) permeability distribution, (b) one dominant frequency point (in blue),characteristic by restricting to homogeneous model (in red), and characteristic byoriginal heterogeneous model (in green), (c) erroneous estimate of average perme-ability, (d) description of reservoir by harmonic frequency data.


2.6 Generation of Square Pulses

In generating square pulses analytically, the superposition of the constant rate

pressure solution is applied. With Stehfest coefficients Vi and Laplace transformed

dimensionless constant response pressure response pD(s), the pressure is calcu-

lated from the formulation in the Laplace domain as a function of Laplace time

s. The equations are solved in the Laplace transform domain and then inverted

back into the time domain by the Stehfest inversion formula developed by Stehfest

(1970).

pD(tD) =ln 2tD

N

∑i=1

Vi pD

(ln 2tD

i)

(2.69)

The Laplace transformation of a constant rate response function is expressed

by the infinite integral as pD(s) =∫ ∞

0 e−st pDdt.

By alternately producing and shutting the active well, the pressure response is

created by superposition of Equation 2.69.

pD(tD) = p(t1D) +N−1

∑m=1

(−1)m p(t1D − tmD) (2.70)

2.7 Radius or Depth of Cyclic Influence

Investigation in Radial Direction

The radius of investigation represents how far into the reservoir the transient ef-

fects can propagate. The radius of investigation reflects how much reservoir vol-

ume is investigated for a given duration of a test. rinv = α√

ktφµct

in a homogeneous

reservoir. The choice of α varies by authors (Kuchuk, 2009), but is of order 0.03.

In the pulse testing context, we can define the cyclic radius influence rc inf for

a radial model, which can be defined in terms of frequency instead of time. The

cyclic influence of a test must be greater than or equal to a specific distance so that

the quantity such as pressure reflects the reservoir physics within the distance.


The choice of sourcing frequencies is closely related to the distance over which

each grid block spans. The cyclic influence relationship for a single well test for a

homogeneous reservoir is the following as defined by Rosa (1991).

rDc inf = 1.1√

1/ωD (2.71)

Considering the fact that frequency is reciprocal of time length ( f : T = 1f ), the

definition by Rosa (1991) is a corresponding term to radius of investigation for

time.

The challenge with cyclic influence is that the exact cyclic influence is usually

considered for a single well test. It is hard to define when the following two as-

pects are involved: (1) for multiwell testing, a more complicated structure for cyclic

influence is formed near an observation well and an injection well; (2) for heteroge-

neous reservoir, there is no clear way to explain and define the reservoir dynamics

with the cyclic influence.

Investigation in Vertical Direction

For vertical pulse testing, the depth of investigation is not a well-defined term:

only a few papers discuss the term conceptually (Stewart et al., 1989) in a manner

similar to a radius of investigation.

Uncertainty of Cyclic Influence for the Study

For heterogeneous reservoir models where a permeability distribution is the un-

known to estimate, the study of cyclic influence does provide the understanding

as follows. By sourcing an appropriate set of frequencies (ωj) with which the sig-

nal ideally reaches the jth block distance well enough with a certain physical be-

havior, a permeability distribution can be revealed because the multiple frequen-

cies replace multiple measurements at different observation points over distance.

It is important to note, however, that there exist multiple permeability distribu-

tions which result in the given pressure data, hence the corresponding frequency


attributes. The exact sourcing frequencies are not attainable directly even with

known distance and permeability information.

However, as it was demonstrated with two reservoir models that a specific het-

erogeneity corresponds to attenuation and phase shift pair at multiple frequencies,

it is reasonable to test the hypothesis that by actively using such frequency at-

tributes, a permeability distribution is revealed. Thus it is necessary to investigate

the quality of permeability estimation by using frequency data, which range of fre-

quencies should be selected in revealing such permeability result, how robust the

method is under various conditions, how accurate the permeability distributions

are compared to other methods, how the frequency method can be improved, and

what types of limitations exist for the method. All of the aforementioned would

be investigated as a function of the sourcing frequency which impacts the range of

investigation.

2.8 Discussion of Relationship between Attenuation

and Phase Shift

One might wonder if a relationship exists between a magnitude response and a

phase response, such that one data can be implied from another. Only for a partic-

ular linear time invariant system called a minimum-phase system, the phase shift

of a system is related to attenuation by the following (Smith, 2007):

∠H(ω) = −Hilbert{log |H(ω)|} (2.72)

Hilbert{x(t)} = 1π

∞∫−∞

x(τ)t− τ

dτ (2.73)

In signal processing, an LTI system is said to be minimum-phase if the system

and its inverse are causal and stable in the time domain. It is worthwhile to in-

vestigate whether a multiwell testing environment satisfies the two conditions to

be minimum-phase such that phase shift data can be estimated from attenuation


data. The aim is to see if the phase shift data become redundant information in

characterizing a reservoir which can be estimated by the attenuation data, or vice

versa. The conditions in both time and frequency domain are considered in this

section, starting with a time domain analysis using a homogeneous radial model

as an example.

Consider a time-invariant pressure pulsing environment where h(t) is a reser-

voir’s impulse response when pinj(t) is applied as an input and pobs(t) is an out-

put. The pressure response is the result of the system’s impulse response func-

tion convoluted with an input, which is, pobs(t) = h(t) ∗ pinj(t). Then, an inverse

system hinv(t) enables us to determine the input from the given output, that is

pinj(t) = hinv(t) ∗ pobs(t). This maps the output of the system to the input of the

system such that the following relationship holds between h(t) and hinv(t).

h ∗ hinv(t∗) =∞

∑τ=−∞

h(τ)hinv(t∗ − τ) = δ(t∗) (2.74)

where δ(t) is the Kronecker delta function which becomes an identity at time t∗

only. In frequency domain, a frequency response of an inverse of an LTI system is

the inverse of the frequency response of the original system, which is H(ω)Hinv(ω)

= 1.

The minimum-phase system requires that it meets the following conditions in

the time domain (Hassibi et al., 2000).

- Causality condition:

h(t) = 0, ∀t < 0 (2.75)

hinv(t) = 0, ∀t < 0 (2.76)

- Stability condition:

∞

∑t=0|h(t)| = ‖h‖1 < ∞ (2.77)

∞

∑t=0|hinv(t)| = ‖hinv‖1 < ∞ (2.78)


From periodic pressure pulses with pinj(t) and pobs(t), the impulse responses

h(t) and hinv(t) can be estimated. The system response functions are estimated by

minimizing the difference between the given pressure with an estimate: for esti-

mating a forward system response, the objective function is∥∥pobs(t)− pinj ∗ h(t)

∥∥2,

and for estimating an inverse system response,∥∥pinj(t)− pobs ∗ hinv(t)

∥∥2.

Figure 2.19 (a, c) show the estimated impulse responses h(t) and hinv(t) for

a homogeneous radial ring model with 100 md. The shape of h(t) is similar to

the shape of a kernel function introduced by Oliver (1992); Feitosa et al. (1994).

The shape of hinv(t) shows an oscillation with high amplitude swings in the early

time. The causality condition is met for both impulse responses, because there is

no output unless an input pressure signal is imposed in the first place. The stability

condition for h(t) is satisfied because the sum over entire time range is finite with

limt→∞

h(t) = 0 as indicated. The stability condition is met for hinv(t) because the sum

over entire time range is finite: hinv(t) approaches towards zero asymptotically as

t→ ∞ ( limt→∞

hinv(t) = 0). Figures 2.19 (b, d) show the reconstructed pressure ouput

and input data by h(t) and hinv, respectively.∥∥pinj(t)− pobs ∗ hinv(t)

∥∥2/∥∥pinj(t)

∥∥2

= 0.076 and∥∥pobs(t)− pinj ∗ h(t)

∥∥2/∥∥pinj(t)

∥∥2 = 3.6× 10−4. Here hinv(t) performs

worse than h(t), as the sharp transition between flow and shut-in periods in the

injection pressure is difficult to capture. All in all, the examination of impulse re-

sponse function in time shows that the pressure pulse environment is a minimum-

phase system.

For a frequency analysis in Laplace space, the equivalent condition is that a

transfer function H(s) = B(s)/A(s) with a Laplace variable s must have poles

and zeros in the strict left half of the s-plane. The values of s in the denomina-

tor A(s) which make the denominator zero are the poles, and the values of s in

the numerator B(s) which make the denominator zero are the zeros. A transfer

function can only satisfy causality and stability requirements if, and only if, all the

poles of the Laplace transform of the impulse response function have negative real

part (Hassibi et al., 2000). With a consideration of a transfer function for an inverse

system, Hinv(s) = 1/H(s), in order for Hinv(s) to be stable and causal, the zeros

are required to be in the left half plane. In short, the zeros and poles of a transfer


function have to lie within the left half s-plane to be minimum-phase. To apply this

principle to a transfer function in Fourier domain a variable s = iω is evaluated.

In our case, the frequency response function in pressure pulse tests is estab-

lished when pressure reaches a periodically steady state. This is attained by mod-

eling pressure data to be of the form, p(r, t) = g(r, ω)eiωt, where oscillations are

sustained at a fixed pressure amplitude. The amplitude does not increase or decay

over time; this means that from the perspective in frequency domain, the pressure

pulsing is minimum phase in steady state.

Implementing the relationship in Equation 2.73 with the discrete Hilbert trans-

form, however, an accurate estimate for the phase shift data from the attenuation

data is hard to attain. The multiple frequency data from a homogeneous radial

flow model with 100 md is used as an attempt to evaluate the relation between the

data by Hilbert transform (Figure 2.20). The discrepancies between the true phase

shift and the estimate are visible. This illustrates that there exists a practical limi-

tation to predict phase shift data from attenuation data and one cannot be inferred

accurately from another.

On another note, when the constraints of causality and stability are satisfied,

the inverse system hinv(t) is unique. For pressure pulse testing, hinv(t) is unique.


0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

x 10−3

Time, hr

Impu

lse

resp

onse

, h

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time, hr

Obs

erva

tion

pres

sure

cha

nge

(psi

)

TrueReconstructed

(a) (b)

1 2 3 4 5 6 7 8 9 10

−1

−0.5

0

0.5

1

x 104

Time, hr

Impu

lse

resp

onse

, hin

v

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time, hr

Inje

ctio

n pr

essu

re c

hang

e (p

si)

TrueReconstructed

(c) (d)

Figure 2.19: Characteristic of system’s response for pressure pulse testing environ-ment: (a) impulse response, (b) true and reconstructed observation pressure data,(c) inverse impulse response, (d) true and reconstructed injection pressure data.


0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Attenuation

Pha

se s

hift

10−10

10−8

10−6

10−4

10−2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Dimensionless frequency

Pha

se s

hift

TrueEstimated from attenuation

Figure 2.20: Estimation of phase shift from attenuation using Hilbert transform:attenuation and phase shift of a homogeneous 100 md reservoir (left) and the mis-match between the true phase shift and the estimate from attenuation (right).


2.9 Summary

An overview for the frequency method was provided. The mathematical theories

behind the frequency method were explained in detail: frequency response that

is characterized by attenuation/phase-shift, and the Fourier transform of periodic

pulses. Depending on their pulse shapes, square pulses contain a range of multiple

frequencies, which are integer multiples of the fundamental sourcing frequency.

When excited with periodic pulses, attenuation and phase shift pairs at multiple

frequencies form an indicative feature for a reservoir model with a particular het-

erogeneous permeability distribution. The periodic steady-state space becomes a

basis for the frequency analysis. The steady-state solutions were developed for two

reservoir models: a multicomposite radial model to inspect vertical permeability

distributions, and a multilayered model with a partial perforation to inspect ver-

tical ones. The analytical expressions for the frequency response function, or the

attenuation and phase shift, were formulated with the steady-state assumption for

both models. The appropriate sourcing frequency range should be determined in

consideration of the observation distance and the magnitude of permeability.

Chapter 3

Pressure Data Preprocessing

Preprocessing of time-series pressure data is carried out to enhance steadily peri-

odic characteristics of the data by removing a transient trend. Also, the necessary

Fourier analysis procedure in obtaining attenuation and phase shift can vary ac-

cording to the number of pulses, sampling rate, time window of where the signals

are placed.

The subsequent analysis can be trend-independent by what is called detrend-

ing. Detrending is a necessary preprocessing step to obtain accurate frequency

information which is free of an upward aperiodic transient trend. The purpose

of detrending in this study is to generate the attenuation and phase shift that are

close enough to periodic steady-state solutions; with an upward transient in pres-

sure data the frequency components are not decomposed properly because the

periodicity condition in Equation 2.1 is not met for Fourier analysis to take place.

There are general algorithmic detrending approaches such as the work by Tar-

vainen et al. (2002). In this chapter a simpler detrending approach is introduced

that is proved successful for repetitive pulsing data.

The sensitivity of the method is examined by varying the number of pulses and

the sampling intervals.

66

CHAPTER 3. PRESSURE DATA PREPROCESSING 67

3.1 Quantification of Pressure Transient from Pulses

In order to equate the frequency information in the sinusoidal steady-state frame-

work, it is necessary to quantify how much of a transient is included in the given

pressure data before the removal of the transient trend. For any type of pulses,

a0 component in the nonperiodic term appearing in the signal decomposition part

in Equation 2.2. Thus by calculating Equation 2.3, the pressure transient to be re-

moved is the corresponding fraction of constant rate response pressure that results

from q(t) for pinj(t) and for pobs(t) each.

When the signal shape is square pulses, or to be exact, if the flow rate is sourced

with 50% duty cycle, the detrending process is to deduct a transient pressure

caused by a half of the flow rate value. This is because the periodic square pulse

pressure is a superimposed response from the flow rate train of [q0, 0, q0, 0, · · · ]as follows, which is consistent with the calculated average value of (a0) by us-

ing Equation 2.3:

q(t) =q0

2+

2q0

π

[sin ωt +

13

sin 3ωt + ...]

(3.1)

Of course the flow rate is assumed unknown, the detrending in pressure data is

to eliminate the pressure transient that corresponds to the flow rate of q0/2 because

the magnitude of the flow rate impacts pressure in a linear fashion.

For unequal pulses, for a duration of k out of Tp, a0 = Ak/Tp. For instance, for

a square pulsing, which is 50% duty cycle a0 = A/2; for 25% duty cycle a0 = A/4.

3.2 Heuristic Detrending Method

Detrending essentially requires obtaining the original transient pressure for a con-

stant flow rate. The challenge is that often the flow rate is not known, which

means that the perfect detrending is not possible. In that case, one method is to

reproduce the whole transient pressure sequence directly by inferring all pressure

points backward in time, which falls into a category of deconvolution with two


pressure signals. This is computationally extensive to perform as the data length

becomes large and an efficient way can be devised by focusing on the regularity of

signals, which is periodicity.

Here two different detrending schemes are proposed for two scenarios: one

with square pulsing, and the other with unequal pulses where the duration of flow

is shorter than the duration of shut-in. The term “heuristic” is used because the

selected pivot points where a transient is reconstructed are connected linearly. It

was found that heuristic detrending performs what a perfect detrending does with

a known flow rate as is demonstrated later. At the selected points, the replicated

pressure transient pressure value is exactly the same as the original transient.

3.2.1 Heuristic Detrending for Square Pulses

The following steps are used to reproduce the pressure transient response out of

square pressure pulses when flow rate information is not given. The constant rate

pressure response to be estimated is denoted as p(t). The initial condition p(t0) =

p(t0) = 0 applies. Then, the periodic pulse p(t) with period Tp is a superimposed

response from p(t) as follows.

p(t) = p(t)− p(t−Tp

2) + p(t− Tp) . . . (3.2)

For a set of chosen pivot points at t∗ ∈(Tp/2, tend

], the p(t∗) can be reconstructed

as follows. For the rest of the time segment t∗ ∈(0, Tp/2

], the transient can be

applied directly from first pressure because the superposition has not taken place

yet. The difference between the transient and the pressure pulse equals to the pulse

point a half period ago for any pivot points:

p(t∗) = p(t∗) + p(t∗ −Tp

2) (3.3)

The proof is as follows. Rewriting Equation 3.2 with respect to time t∗ −


Tp2 (t∗ > Tp

2 ), the following holds:

p(t∗ −Tp

2) = p(t∗ −

Tp

2)− p(t∗ − Tp) + p(t∗ −

3Tp

2)... (3.4)

Also apply Equation 3.2 for time t∗ and add these two equations together. Then

the equation is summarized such that the transient response values at t∗ are ob-

tained from the given time series pressure pulse as p(t∗) = p(t∗) + p(t∗ − Tp2 ). A

graphical illustration of the relation is shown at Figures 3.1 (a, b) for a square pulse.

This illustrates that any point in a transient can be reconstructed by combining two

pressure points directly: one at time t which satisfies t > Tp/2 and the other one

which is located Tp/2 ahead.

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time, hr

Pre

ssur

e ch

ange

(ps

i)

ObservationTrue transientEstimated transientMaximum / cycleMinimum / cycle

Figure 3.1: Illustration of transient reconstruction: any transient point for t > Tp/2can be reconstructed using a pressure point which is located Tp/2 ahead for asquare pulse.

Based on this relation, the heuristic detrending process is illustrated in Fig-

ure 3.2 with select pivot points about every Tp/2. Here in this case the maximum


and minimum points each period are selected.

Thus the detrending in this case is to effectively sample the transients heuristi-

cally at some major points such as maximum and minimum points each period on

top of a pressure response during the first half of the period. The rest of the points

can be linearly interpolated in between these pivot points. The linear interpolation

approximates the actual transient because the response follows a linear trend over

time with the superposition of alternating positive and zero flow rates.

The detrending process for square pulses is summarized as below.

1. For t∗ ∈(0, Tp/2

]:

Replicate all data from the pressure data, p(t∗) = p(t∗).

2. Select pivot points. The suggestions for choices of points are (1) every Tp/2 or

Tp points, or (2) some noticeable points such as the maximum and minimum

points.

3. For t∗ ∈(Tp/2, tend

]:

Compute p(t∗) = p(t∗) + p(t∗ − Tp2 ), which is adding points located at Tp/2

ahead in time.

4. Approximate the transient trend by linearly connecting those points.

5. Deduct half of the estimated transient from the given pressure data at injec-

tion and observation point respectively.

3.2.2 Heuristic Detrending for Unequal Pulses

In reality, it might be hard to obtain square pulses, which are generated by 50%

duty cycle flow rate. When a duration of flow event is shorter than a shut-in pe-

riod, the pulses are less than 50% duty cycle; when a duration of flow period is

longer than a shut-in period, the pulses are more than 50% duty cycle. The chal-

lenge becomes how to infer the transient response in an efficient way.

Here we propose a similar detrending method for unequal pulses as introduced

in Section 3.2. Some differences are that a transient is reconstructed in an iterative


manner, which means that the next transient pressure point depends on the pre-

viously calculated pressure point. Also for this iteration to be efficient, the choice

of pivot points should be such that they are evenly spaced. The derivation is as

follows.

For a duty cycle α = k/Tp (0 < α < 1), the rectangular pulse p(t) with period

Tp is a superimposed response from p(t) as follows.

p(t) = p(t)− p(t− αTp) + p(t− Tp)− p(t− (α + 1)Tp) + p(t− 2Tp) . . . (3.5)

Rewriting Equation 3.5 with respect to time t∗ − Tp (t∗ > Tp), the following

holds:

p(t∗ − Tp) = p(t∗ − Tp)− p(t∗ − (α + 1)Tp) + p(t− 2Tp) . . . (3.6)

Also apply Equation 3.5 for time t∗ and deduct Equation 3.6 from the equation

obtained. Then the following relation holds:

p(t∗)− p(t∗ − Tp) = p(t∗)− p(t∗ − αTp) (3.7)

Thus at a set of pivot points t∗ the reconstruction is conducted according to the

following equation:

p(t∗) =

p(t∗) if t∗ ∈

(0, αTp

]p(t∗ − αTp) + p(t∗) if t∗ ∈

(αTp, Tp

]p(t∗ − αTp) + p(t∗)− p(t∗ − Tp) if t∗ ∈

(Tp, tend

] (3.8)

It is important to note that a transient pressure p(t∗) requires a calculation of a

transient which is ahead of time by αTp, which is different from the square pulsing

detrending scheme. Thus, in selecting pivot points it is efficient to space them

evenly by αTp to be used for the calculation of a transient pressure value at the

next pivot. For instance, for a 25% duty cycle pulse, it is wise to put at least four

pivot points per cycle, except during the first pulse.


Figures 3.2 (c, d, e ,f) show the heuristic detrending applied on 25% and 75%

duty cycle pulses. The pivot points are selected every Tp/4 interval for both cases.

For 75% duty cycle pulses, although it is acceptable to sample pivots at every time

interval of 3Tp/4, a denser pivot selection of Tp/4 is used to depict the curvature of

a transient. At the selected points, the reconstructed pressure values exactly match

the original pressure transient.

The detrending process for rectangular pulses is summarized as below.

1. For t∗ ∈(0, αTp

]:

Replicate all data from the pressure data p(t∗) = p(t∗)

2. Select pivot points at least every αTp period.

3. For t∗ ∈(αTp, Tp

]:

Iteratively compute p(t∗ − αTp) + p(t∗) which uses a transient point αTp

ahead in time.

4. For t∗ ∈(Tp, tend

]:

Iteratively compute p(t∗ − αTp) + p(t∗)− p(t∗ − Tp) which uses a transient

point αTp ahead in time.

5. Approximate the transient trend by linearly connecting those points.

6. Deduct α % of the estimated transient from the given pressure data at injec-

tion and observation point respectively.


5 10 15 20

1

2

3

4

5

6

7

8

9

10

11

Time, hr

Pre

ssur

e ch

ange

(ps

i)

InjectionTrue transientEstimated transientMaximum / cycleMinimum / cycle

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time, hr

Pre

ssur

e ch

ange

(ps

i)

ObservationTrue transientEstimated transientMaximum / cycleMinimum / cycle

(a) (b)

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

Time, hr

Pre

ssur

e ch

ange

(ps

i)

InjectionTrue transient, injectionReconstructed pointsSelected points

1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

Time, hr

Pre

ssur

e ch

ange

(ps

i)

ObservationTrue transient, observationReconstructed pointsSelected points

(c) (d)

1 2 3 4 5 6 7 81

2

3

4

5

6

7

8

Time, hr

Pre

ssur

e ch

ange

(ps

i)

InjectionTrue transient, injectionReconstructed pointsSelected points

1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

Time, hr

Pre

ssur

e ch

ange

(ps

i)

ObservationTrue transient, observationReconstructed pointsSelected points

(e) (f)

Figure 3.2: Detrending on various pulse shapes: (a) square pulses at injection, (b)square pulses at observation, (c) 25% duty cycle pulses at injection, (d) 25% dutycycle pulses at observation, (e) 75% duty cycle pulses at injection, (f) 75% dutycycle pulses at observation.


3.3 Detrending on Radial Ring Model

The pressure pair data from two reservoir models are illustrated for the heuristic

detrending technique, first starting with a radial ring model scenario. The mea-

surements are taken at the wellbore and some distance away. The reservoir is

homogeneous with 100 md, and the observation point is 190 ft away from the in-

jection well. The pressure data have 10 periods, 2.18 hour periodicity with a 15.34

sec sampling interval, which is equivalent to 512 points per period. The reservoir

parameters are rw = 0.5 ft, φ= 0.05, µ=1 cp, ct = 5× 10−6 psi−1.

Estimation of Pressure Transient Response

Plot (a) of Figures 3.3, 3.4 and 3.5 shows in the time domain the estimation of

transient pressure for three different duration of active flow period. The sampling

for heuristic detrending is 4 points per period. All of them depict the trend very

well. Note that the heuristic detrending gives almost the exact reconstruction to

the transient pressure response. Table 3.1 shows the discrepancies between the

heuristic transient denoted as ph and the exact one noted as pexact, which is at most

0.15%.

After estimating the pressure transient, from the entire pressure data the pres-

sure transient is removed by the calculated weight according to different pulse

shapes. The weight of 1/2, 1/4, 3/4 are used for square pulses, 25% duty cycle, and

75% duty cycle respectively by Equation 2.3. For instance, pinj− 1/4pinj,h becomes

the detrended injection pressure which will be used for Fourier transform. The re-

sults of detrended pressure are shown on the plot (b) of Figures 3.3, 3.4 and 3.5.

It is important to note that the periodicity condition of f (t + Tp) = f (t) in Equa-

tion 2.1 is applicable on the detrended signals. The pressure signals are aligned in

such a manner where the average becomes zero (a0 = 1T

∫ T/2−T/2 f (t)dt = 0).


Frequency Decomposition by Fourier Transform

Plots (c) and (d) of Figures 3.3, 3.4 and 3.5 show the Fourier magnitude of the

injection and observation pressure for three different pulse shapes. The frequencies

are odd multiples (1ω, 3ω, 5ω, · · · ) are denoted with dots.

As demonstrated, the detrending makes major difference on how the observa-

tion signal is decomposed, especially at the high frequency components. Without

detrending, the signal decomposition into multiple frequency components is af-

fected by the upward trend of the first pressure transient. It is important to note

that the Fourier magnitude at zero frequency, which is the average component a0,

is no longer the largest after detrending. The dominant frequency becomes the

largest component after detrending. For the injection pressure signal, except at the

zero frequency, the performance of frequency decomposition makes no difference;

this is due to the strong periodicity of the injection signal. The shape of frequency

decomposition agrees with the study shown in Figure 2.3.

Frequency Attributes in Comparison with Steady-State Solutions

Plots (e) and (f) of Figures 3.3, 3.4 and 3.5 show the frequency data in compari-

son with steady-state solutions. The attenuation and phase shift for the first 20

odd harmonic frequencies for the radial ring model case are demonstrated. The

detrending is absolutely necessary for obtaining the accurate frequency data from

the pulses by effectively improving the decomposition of signals. The detrended

pressure has closer frequency values to the sinusoidal solution than the raw pres-

sure, which is more apparent as frequency increases. Without detrending unequal

pulses with 25% and 75% duty cycles, the phase shift oscillates except a few low

frequency points; after detrending, the phase shift closely matches the sinusoidal

steady-state data. This suggests that it is more reliable to preprocess and detrend

the pressure in order to perform parameter estimation in the sinusoidal steady-

state space as its base.


The discrepancy of attenuation and phase shift data with periodically steady-

state tends to increase in general as the frequency increases over 20 harmonic fre-

quencies as shown in Figure 3.6. The frequency attributes match well to steady

state until the frequency at which the decomposed observation pressure magni-

tude is too small. The illustrated misfit for attenuation and phase shift at har-

monic frequencies (ωh) are defined as εx(ωh) = |x(ωh)− xss(ωh)| /xss(ωh) and

εθ(ωh) = |θ(ωh)− θss(ωh)| /θss(ωh). The maximum misfits are the lowest for

square pulses, which are 6% for attenuation and 3% for phase shift. This sug-

gests that the frequency attributes from square pulses are the most closest to the

steady state of all pulse shapes. An oscillating behavior is observed for unequal

pulses: about 20% maximum discrepancy for attenuation and about 4% for phase

shift. The discrepancy for attenuation can be reduced further to be within 5% for

the 10 lowest frequencies. The figure does not mean that a higher the sourcing fre-

quency renders discrepancies with the analytical sinusoidal case; but it indicates

that higher the frequency component contained in the signal shows weaker signal

strength. However how high or low, the first several low frequency components

agree closely with the steady-state solutions.

Also, Figure 3.6 shows that no visible difference between the frequency at-

tributes with heuristic detrending and the exact detrending when the flow rate

is known (xh∼= xexact, θh

∼= θexact).

The mismatch in the frequency information for 20 pairs of attenuation and

phase shift (xh, θh) to the steady-state space (xss, θss) are less than 1% for attenuation

and 2% for phase shift each (Table 3.1). The table also shows little discrepancy be-

tween the frequency data obtained after heuristic detrending and after detrending

with known flow rate. Therefore through heuristic detrending, the benefit of not

having to know the flow rate is maintained. The heuristic detrending works fairly

well without knowing the exact value of the flow rate, yet with the limitations of

the number of useful harmonic frequency components.

Figure 5.4 summarizes the effect of detrending on the first 20 odd harmonic

frequency data. The detrending greatly improves the accuracy of the frequency

attributes for all pulse shapes. The oscillation of frequency components no longer


exists for unequal pulses.

Table 3.1: Performance of heuristic detrending on radial ring model(Duty cycle)

50% 25% 75%∥∥ pinj, h − pinj, exact∥∥

2/∥∥ pinj, exact

∥∥2 0.00018 0.00047 0.00010

‖ pobs, h − pobs, exact‖2/‖ pobs, exact‖2 0.00062 0.0015 0.00035‖xh − xexact‖2/‖xexact‖2 0.000028 0.000035 0.00010‖θh − θexact‖2/‖θexact‖2 0.000045 0.000026 0.000075‖xh − xss‖2/‖xss‖2 0.0087 0.0090 0.0096‖θh − θss‖2/‖θss‖2 0.017 0.023 0.022


2 4 6 8 10 12 14 16 18 20

1

2

3

4

5

6

7

8

9

10

11

Time, hr

Pre

ssur

e ch

ange

(ps

i)

InjectionObservationTrue transient, injectionTrue transient, observationReconstructed transients

2 4 6 8 10 12 14 16 18 20

−4

−3

−2

−1

0

1

2

3

4

Time, hr

Pre

ssur

e ch

ange

(ps

i)


(a) (b)

0 20 40 60 80 100

−20

0

20

40

60

80

100

Mag

nitu

de, I

njec

tion

(dB

)

Frequency, rad/hr

No detrendingDetrending

0 20 40 60 80 100−30

−20

−10

0

10

20

30

40

50

60

70

80M

agni

tude

, Obs

erva

tion

(dB

)

Frequency, rad/hr


(c) (d)

0 20 40 60 80 100 1200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency, rad/hr

Atte

nuat

ion

No detrendingDetrending, known flowDetrending, heuristicSinusoidal steady state

0 20 40 60 80 100 1200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency, rad/hr

Pha

se s

hift


(e) (f)

Figure 3.3: Detrending on radial model with square pulses: (a) estimation of thefirst transients, (b) detrended pressure data, (c) frequency components at injection,(d) frequency components at observation, (e) attenuation over 20 harmonics, (f)phase shift over 20 harmonics.


2 4 6 8 10 12 14 16 18 20

1

2

3

4

5

6

7

8

9

10

11

Time, hr

Pre

ssur

e ch

ange

(ps

i)


2 4 6 8 10 12 14 16 18 20

−2

−1

0

1

2

3

4

5

6

Time, hr

Pre

ssur

e ch

ange

(ps

i)


(a) (b)

0 20 40 60 80 100

−20

0

20

40

60

80

100

Mag

nitu

de, I

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100−30

−20

−10

0

10

20

30

40

50

60

70

80M

agni

tude

, Obs

erva

tion

(dB

)

Frequency, rad/hr


(c) (d)

0 20 40 60 80 100 1200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency, rad/hr

Atte

nuat

ion


0 20 40 60 80 100 1200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency, rad/hr

Pha

se s

hift


(e) (f)

Figure 3.4: Detrending on radial model with 25% duty cycle pulses: (a) estima-tion of the first transients, (b) detrended pressure data, (c) frequency componentsat injection, (d) frequency components at observation, (e) attenuation over 20 har-monics, (f) phase shift over 20 harmonics.


2 4 6 8 10 12 14 16 18 20

1

2

3

4

5

6

7

8

9

10

11

Time, hr

Pre

ssur

e ch

ange

(ps

i)


2 4 6 8 10 12 14 16 18 20

−6

−5

−4

−3

−2

−1

0

1

2

Time, hr

Pre

ssur

e ch

ange

(ps

i)


(a) (b)

0 20 40 60 80 100

−20

0

20

40

60

80

100

Mag

nitu

de, I

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100−30

−20

−10

0

10

20

30

40

50

60

70

80M

agni

tude

, Obs

erva

tion

(dB

)

Frequency, rad/hr


(c) (d)

0 20 40 60 80 100 1200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency, rad/hr

Atte

nuat

ion


0 20 40 60 80 100 1200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency, rad/hr

Pha

se s

hift


(e) (f)

Figure 3.5: Detrending on radial model with 75% duty cycle pulses: (a) estima-tion of the first transients, (b) detrended pressure data, (c) frequency componentsat injection, (d) frequency components at observation, (e) attenuation over 20 har-monics, (f) phase shift over 20 harmonics.


0 20 40 60 80 100 1200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Frequency, rad/hr

Atte

nuat

ion

mis

fit

Detrending, known flowDetrending, heuristic

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

Frequency, rad/hr

Pha

se s

hift

mis

fit


(a) (b)

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

Frequency, rad/hr

Atte

nuat

ion

mis

fit


0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Frequency, rad/hr

Pha

se s

hift

mis

fit


(c) (d)

0 20 40 60 80 100 1200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Frequency, rad/hr

Atte

nuat

ion

mis

fit


0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Frequency, rad/hr

Pha

se s

hift

mis

fit


(e) (f)

Figure 3.6: Misfit for frequency attributes on radial model with three pulse shapes:square pulses- (a) attenuation misfit, (b) phase shift misfit; pulses with 25% dutycycle- (a) attenuation misfit, (b) phase shift misfit; pulses with 75% duty cycle- (a)attenuation misfit, (b) phase shift misfit.


0 0.02 0.04 0.06 0.08 0.10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Attenuation

Pha

se s

hift


0 0.02 0.04 0.06 0.08 0.10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Attenuation

Pha

se s

hift


0 0.02 0.04 0.06 0.08 0.10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Attenuation

Pha

se s

hift


Figure 3.7: Attenuation vs. phase shift for three pulse shapes on radial model:square (top), 25% (bottom left) and 75% duty cycle (bottom right).


3.4 Detrending on Multilayered Model

Next, the detrending is illustrated for the partially-penetrating well with crossflow

model. The model is a multilayered reservoir with crossflow where the pressure

measurements are observed at the wellbore, one at a perforated layer and the other

at some depth away from the perforated layer. In the example, the reservoir is

homogeneous with radial permeability of 100 md and vertical permeability of 10

md. The observation layer is 12 f t deep from the perforated layer. The data has

ten periods with 21.09 second sampling interval, and 512 points per period. One

period is 3 hours. The reservoir parameters are rw = 0.5 ft, φ= 0.30, µ=5 cp, ct =

1.2× 10−5 psi−1.

Estimation of Pressure Transient Response

Plots (a) of Figures 3.8, 3.9 and 3.10 show in the time domain the estimation of

transient pressure for three different durations of active flow period. The results

of detrended pressure are shown on plots (b) of Figures 3.8, 3.9 and 3.10. Table 3.2

shows the discrepancies between the heuristic transient denoted as ph and the ex-

act one noted as pexact, which is at most 0.16%.

Frequency Decomposition by Fourier Transform

Plots (c) and (d) of Figures 3.8, 3.9, and 3.10 show the Fourier magnitude of the

injection and observation pressure for three different pulse shapes. The odd har-

monic frequencies are denoted as dots. Detrending shows difference on how the

observation signal is decomposed.

Frequency Attributes in Comparison with Steady-State Solutions

Plots (e) and (f) of Figures 3.8, 3.9, and 3.10 show the frequency data in comparison

with steady-state solutions. The detrending is absolutely necessary for obtaining

the accurate frequency data from the pulses by improving the decomposition of


signals effectively. The discrepancy of attenuation and phase shift data with pe-

riodically steady-state increases as the frequency increases. This is due to the in-

herent limitation of the square pulsing in that the higher the frequency value, the

larger the observed difference among the steady-state answers are due to smaller

magnitudes for the higher frequency components. As the pressure signal passes

through the medium, the shape of the observation is not quite the same shape

as the injection pressure. For three pulse shapes, the phase shift data are slightly

overestimated compared to the steady-state case. The mismatch in the frequency

information for 20 pairs of attenuation and phase shift (xh, θh) to the steady-state

space (xss, θss) are less than 0.7% for attenuation and 5.1% for phase shift each (Ta-

ble 3.2).

The discrepancy of attenuation and phase shift data with periodically steady-

state increases as the frequency increases as shown in Figure 3.11 over 20 harmonic

frequencies. The misfit for attenuation and phase shift at harmonic frequencies

(ωh) are εx(ωh) = |x(ωh)− xss(ωh)| /xss(ωh) and εθ(ωh) = |θ(ωh)− θss(ωh)| /θss(ωh).

This shows a similar result as for the radial ring model. The maximum misfits are

the lowest for square pulses, which are 4% for attenuation and 9% for phase shift.

An oscillating behavior on frequency attributes is observed for unequal pulses:

about 60% maximum discrepancy for attenuation and about 13% for phase shift.

The discrepancies can be reduced further for the ten lowest frequencies, where the

deviations are within 10% for attenuation and 3% for phase shift.

The small discrepancy between the frequency pair by heuristic detrending (xh, θh)

and detrending with flow rate data (xexact, θexact) suggests that heuristic detrend-

ing works well; the frequency information is almost the same as perfect detrending

with known flow rate information.

Figure 3.12 summarizes the effect of detrending on the first 20 odd harmonic

frequency data. The detrending greatly improves the accuracy of the frequency

attributes for all pulse shapes. The oscillation of frequency components no longer

exists for unequal pulses.


Table 3.2: Performance of heuristic detrending on multilayered model(Duty cycle)

50% 25% 75%∥∥ pinj, h − pinj, exact∥∥

2/∥∥ pinj, exact

∥∥2 0.000071 0.00019 0.000040

‖ pobs, h − pobs, exact‖2/‖ pobs, exact‖2 0.00063 0.0016 0.00035‖xh − xexact‖2/‖xexact‖2 0.000024 0.000030 0.000076‖θh − θexact‖2/‖θexact‖2 0.000053 0.000038 0.00012‖xh − xss‖2/‖xss‖2 0.0042 0.0068 0.0065‖θh − θss‖2/‖θss‖2 0.048 0.039 0.051


5 10 15 20 25 30

1

2

3

4

5

6

Time, hr

Pre

ssur

e ch

ange

(ps

i)


5 10 15 20 25 30

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time, hr

Pre

ssur

e ch

ange

(ps

i)


(a) (b)

0 20 40 60 80 100−40

−20

0

20

40

60

80

100

Mag

nitu

de, I

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100−40

−30

−20

−10

0

10

20

30

40

50

60

70M

agni

tude

, Obs

erva

tion

(dB

)

Frequency, rad/hr


(c) (d)

0 10 20 30 40 50 60 70 80 900

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Frequency, rad/hr

Atte

nuat

ion


0 10 20 30 40 50 60 70 80 900.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequency, rad/hr

Pha

se s

hift


(e) (f)

Figure 3.8: Detrending on multilayered model with square pulses: (a) estimation ofthe first transients, (b) detrended pressure data, (c) frequency components at injec-tion, (d) frequency components at observation, (e) attenuation over 20 harmonics,(f) phase shift over 20 harmonics.


5 10 15 20 25 30

1

2

3

4

5

6

Time, hr

Pre

ssur

e ch

ange

(ps

i)


5 10 15 20 25 30

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time, hr

Pre

ssur

e ch

ange

(ps

i)


(a) (b)

0 20 40 60 80 100−40

−20

0

20

40

60

80

100

Mag

nitu

de, I

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100−40

−30

−20

−10

0

10

20

30

40

50

60

70M

agni

tude

, Obs

erva

tion

(dB

)

Frequency, rad/hr


(c) (d)

0 10 20 30 40 50 60 70 80 900

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Frequency, rad/hr

Atte

nuat

ion


0 10 20 30 40 50 60 70 80 900.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequency, rad/hr

Pha

se s

hift


(e) (f)

Figure 3.9: Detrending on multilayered model with 25% duty cycle pulses: (a)estimation of the first transients, (b) detrended pressure data, (c) frequency com-ponents at injection, (d) frequency components at observation, (e) attenuation over20 harmonics, (f) phase shift over 20 harmonics.


5 10 15 20 25 30

1

2

3

4

5

6

Time, hr

Pre

ssur

e ch

ange

(ps

i)


5 10 15 20 25 30−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Time, hr

Pre

ssur

e ch

ange

(ps

i)


(a) (b)

0 20 40 60 80 100−40

−20

0

20

40

60

80

100

Mag

nitu

de, I

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100

−40

−30

−20

−10

0

10

20

30

40

50

60

70M

agni

tude

, Obs

erva

tion

(dB

)

Frequency, rad/hr


(c) (d)

0 10 20 30 40 50 60 70 80 900

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Frequency, rad/hr

Atte

nuat

ion


0 10 20 30 40 50 60 70 80 900.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequency, rad/hr

Pha

se s

hift


(e) (f)

Figure 3.10: Detrending on multilayered model with 75% duty cycle pulses: (a)estimation of the first transients, (b) detrended pressure data, (c) frequency com-ponents at injection, (d) frequency components at observation, (e) attenuation over20 harmonics, (f) phase shift over 20 harmonics.


0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frequency, rad/hr

Atte

nuat

ion

mis

fit


0 10 20 30 40 50 60 70 80 900

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Frequency, rad/hr

Pha

se s

hift

mis

fit


(a) (b)

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency, rad/hr

Atte

nuat

ion

mis

fit


0 10 20 30 40 50 60 70 80 900

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency, rad/hr

Pha

se s

hift

mis

fit


(c) (d)

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency, rad/hr

Atte

nuat

ion

mis

fit


0 10 20 30 40 50 60 70 80 900

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Frequency, rad/hr

Pha

se s

hift

mis

fit


(e) (f)

Figure 3.11: Misfit for frequency attributes on multilayered model with three pulseshapes: square pulses- (a) attenuation misfit, (b) phase shift misfit; pulses with 25%duty cycle- (a) attenuation misfit, (b) phase shift misfit; pulses with 75% duty cycle-(a) attenuation misfit, (b) phase shift misfit.


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Attenuation

Pha

se s

hift


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Attenuation

Pha

se s

hift


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Attenuation

Pha

se s

hift


Figure 3.12: Attenuation vs. phase shift for three pulse shapes on multilayeredmodel: square (top), 25% (bottom left) and 75% duty cycle (bottom right)


3.5 Determinants for Accuracy of Frequency Attributes

Not only are the accuracies of attenuation and phase shift highly improved by

detrending, but also they can be further enhanced by providing more pulses and

selecting pulses at later time.

3.5.1 Effect of Number and Position of Pulses

The larger the number of pressure square pulses and the later the pulses take place

in time, the closer the extracted frequency attributes are to the steady-state solu-

tions. The more pulses we have, the set of pressure data provides clearer infor-

mation about the frequency attributes. In addition, when pulses take place in time

does matter because the later a pulse takes place the more periodic steady-state the

resulting solution resembles. Figure 3.13 shows how the investigation is conducted

by selecting three pulses at a time and feed into Fourier analysis. The number of

time windows of inspection is six for a total of nine pulses as shown in the figure.

Each pulse has 512 points for the study.

Figure 3.13: Number of pulses and location of windows used for sensitivity check.

Figure 3.14 shows the result of attenuation and phase shift error over varying


number of pulses and location of pulses in time. The mismatch in the attenuation

and phase shift to steady-state solution for the first ten harmonics is measured us-

ing a metric ‖x(ωh)− xss(ωh)‖2/‖xss(ωh)‖2 and ‖θ(ωh)− θss(ωh)‖2/‖θss(ωh)‖2,

respectively. The result validates that gathering more pulses, and particularly

pulses at later time increases the accuracy to that of the periodic steady-state solu-

tion. Processing more pulses increases the frequency resolution, and so is helpful.

At later time as pulses are superimposed, the pulses replicate more towards the pe-

riodic steady-state solution, thus the misfit with the periodic steady-state solution

reduces.

1 2 3 4 5 6 7 8 910

−3

10−2

10−1

100

Atte

nuat

ion

mis

fit

Window number from the first pulse

1 pulse3 pulses5 pulses7 pulses

1 2 3 4 5 6 7 8 910

−2

10−1

100

Pha

se s

hift

mis

fit

Window number from the first pulse

1 pulse3 pulses5 pulses7 pulses

Figure 3.14: Accuracy of frequency attributes to steady state with varying num-ber of pulses and window position: summary of attenuation (left) and phase shift(right) at 10 harmonic frequencies.

One might wonder how using only one pulse can indicate periodicity for the

signal. Fourier analysis, using the discrete Fourier transform, assumes that signals

are infinite in time and periodic. In other words, the provided pulse is assumed

periodic over time. The provided pressure signal p(t) is assumed to repeat with

a period equal to the total sample time Np∆t. Figure 3.15 shows the Fourier mag-

nitude spectrum of one pressure pulse which took place at the third pulse. The

shape of the frequency envelope shows visible difference compared to using mul-

tiple pulses, for instance, Figure 3.3 (c) and (d). This is because the number of


data points between the harmonic frequencies is smaller than the one created by

multiple pulses.

0 20 40 60 80 100

−20

0

20

40

60

80

100

Mag

nitu

de, I

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100−30

−20

−10

0

10

20

30

40

50

60

70

80

Mag

nitu

de, O

bser

vatio

n (d

B)

Frequency, rad/hr


Figure 3.15: Frequency components for a pulse located at the third pulse.

Processing the first pulse does not give good estimates of the frequency at-

tributes to the periodic steady-state solution and should be excluded for pulse

tests. The Fourier decomposition on the injection side is desirable as done simi-

larly for any other pulses (Figure 3.16 (a)). On the observation side, however, there

is degeneracy observed at some high frequencies as shown in Figure 3.16 (b) be-

cause there is an initial time period when the observation pressure stays zero. As

a result, the attenuation and phase shift deviates significantly from the periodic

steady-state behavior.


0 20 40 60 80 100

−20

0

20

40

60

80

100

Mag

nitu

de, I

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100−30

−20

−10

0

10

20

30

40

50

60

70

80

Mag

nitu

de, O

bser

vatio

n (d

B)

Frequency, rad/hr


0 0.02 0.04 0.06 0.08 0.10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Attenuation

Pha

se s

hift


Figure 3.16: Fourier analysis when a first pulse is included: Fourier magnitude forinjection pressure (top left), Fourier magnitude for observation pressure (top right)and attenuation vs. phase shift cross-plot (bottom).


3.5.2 Effect of Sampling Frequency

It is expected that the greater number of points per cycle we have, the better would

be the resolution for frequency analysis thus the attenuation and phase shift values

would be more accurate. To verify, the frequency features were gathered at 20 har-

monics with three different sampling frequencies while other conditions remain

the same.

Two exemplary cases were examined to check the accuracy of attenuation and

phase shift to steady-state solutions. The first case uses seven pulses from the

third, and the second case uses one pulse that takes place at the ninth in the pulse

train. The visible discrepancies are observed for 128 points per cycle on the cross-

plot in Figure 3.17 (a, b); the frequency attributes with a sampling frequency of

more than 512 points per cycle match fairly well to the steady-state points. The

misfit of 20 attenuation and phase shift are summarized in a bar plot in Figure 3.17

(c, d) for two different pulse types. For attenuation the misfit is defined as εx =

|x(ωh)− xss(ωh)| /xss(ωh), and for phase shift, εθ = |θ(ωh)− θss(ωh)| /θss(ωh).

The highest sampling frequency brings the least misfit, thus it is recommended to

record the measurements as frequently as possible to obtain attenuation and phase

shift accurately.


0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.2

0.3

0.4

0.5

0.6

0.7

Attenuation

Pha

se s

hift

Sinusoidal steady state128 points/cycle512 points/cycle1024 points/cycle

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.2

0.3

0.4

0.5

0.6

0.7

Attenuation

Pha

se s

hift

Sinusoidal steady state128 points/cycle512 points/cycle1024 points/cycle

(a) (b)

128 512 10240

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Number of points per cycle

Atte

nuat

ion

mis

fit

7 pulses1 pulse

128 512 10240

0.02

0.04

0.06

0.08

0.1

0.12

Number of points per cycle

Pha

se s

hift

mis

fit

7 pulses1 pulse

(c) (d)

Figure 3.17: Accuracy of frequency attributes with varying sampling frequency:the cross-plot using (a) seven pulses from the third, (b) one pulse at the ninth;misfit of frequency attributes with steady-state for (c) attenuation, (d) phase shift.


3.5.3 Effect of Pressure Noise

Normally a signal contains some noise due to inaccurate measurement precision,

other parasitic phenomena such as electronic noise and external events such as

variations in the wellbore environment. It is important to examine if the three

methods used in estimating permeability are robust to additive noise in pressure.

To check the robustness of the frequency method to pressure noise, 1% Gaus-

sian pressure noise was added to the pressure signal: pnoise(t) = p(t)+ η(t), where

η has a distribution of N(0, σ2), where the variance corresponds to 1% of pressure

peak-to-peak magnitude.

The noise corrupts the observation signal more than the injection signal as

shown in the Fourier magnitude plot in Figure 3.18. The amount of perturbation

is distributed equally over the spectrum in the Fourier domain. This characteristic

by additive Gaussian noise is more commonly known as white noise. It is expected

that the high frequency component would be relatively more affected by the noise

due to the weak magnitudes. Thus the frequency analysis would be less reliable at

the higher harmonic frequencies.

Figure 3.18: Fourier magnitude plots with added Gaussian noise in pressure atinjection point (left) and observation point (right).

Ten realizations of 1% Gaussian noise were added to the pressure data. The


attenuation and phase shift were recorded at harmonics. More aberration of phase

shift values is apparent than of attenuation values, as shown on Figure 3.19. The

first several harmonics still retain their original attenuation and phase shift values

from the noise-free case.

Figure 3.19: Attenuation (left) and phase shift (right) of ten realizations of noisypressure with 128 points per cycle.

Noise and Sampling Rate

Another important factor about noise is that the effect is highly correlated with

the sampling frequency in measuring pressure data. The higher the sampling rate,

the less is the effect of noise thus the frequency attributes are more accurate. The

sensitivity of the accuracy of attenuation and phase shift with varying sampling

frequency was checked. To quantify the robustness of pressure noise to attenua-

tion and phase shift, the normalized mean absolute error was calculated: the ten

realizations of Gaussian noise are summarized with respect to mean absolute error

(MAE) in Figure 3.20. The normalized MAE for attenuation is

M∑

i=1

∣∣∣xinoisy−x

∣∣∣/|x|M and

for phase shift,

M∑

i=1

∣∣∣θinoisy−θ

∣∣∣/|θ|M for M number of realizations. The effect of noise is


less significant when the sampling rate is higher. The three MAE values at dif-

ferent frequencies are summarized. Each sampling rate is 128 points/cycle case

(or, 22.6 sec sampling rate); 512 points/cycle case (or, 5.65 sec sampling rate); 2048

points/cycle case (or, 1.4 sec sampling rate). Therefore although expensive, it is

desirable to have higher resolution for the time series measurement in pressure

when utilizing the frequency method.

Figure 3.20: MAE summary for attenuation (left) and phase shift (right).

The implication of Figure 3.20, that higher sampling frequency improves the

accuracy of frequency attributes, can be proved with the analysis of the variance

of noise in Fourier domain and its relationship with the sampling points Ns per cy-

cle. When the Equation 2.4 is discretized, with added noise η(t), the new Fourier

coefficients an have the following relation with the previous noise-free an as fol-

lows.

an =2

Ns

Ns−1

∑j=0

(f (tj) + n(tj)

)cos(

2πntj

T)

= an +2

Ns

Ns−1

∑j=0

n(tj) cos(2πtjn

T) (3.9)


For nonzero nth harmonic, the variance is calculated as follows:

var(an) = (2

Ns)2

Ns−1

∑j=0

σ2 cos2(2πntj

T)

=4σ2

N2s

Ns−1

∑j=0

cos2(2πntj

T)

=4σ2

N2s· Ns

2

=2σ2

Ns(3.10)

This proves that the variance of the complex Fourier coefficients is equal to the

noise variance times 2/Ns. Therefore the way to reduce the variance of the estimate

is to increase the sampling points per period.

3.6 Preprocessing on Field Data

A total of four field data examples were investigated to obtain attenuation and

phase shift data. Field data 1 and 2 are examples for radial ring models, and Field

data 3 and 4 are for multilayered models. The same process was used to retrieve

frequency attributes at harmonic frequencies. Before that, there was a common

type of noise in the field data which appears universally in all the field examples,

which is referred to as quantization noise.

3.6.1 Quantization Noise in Field Data

Discretization is inevitable in recording a continuous pressure signal because there

is a limit to how often the data can be measured (time interval sampling) or to how

precise the amplitude quantization can be. In other words, discretization happens

in time series signal in practice in two directions, horizontally and vertically. Both


discretization errors cause a phenomenon called aliasing, which becomes a con-

cern in the frequency domain because the original frequency contents for a reser-

voir model are distorted.

For practical well tests, the quantization seems to appear in both directions:

horizontally, there is finite precision to record in time; vertically, the measurement

quantizes the pressure magnitude values into a finite-bit representation. The artifi-

cial staircase signal introduces high frequency components which creates a distor-

tion in the frequency domain: for a sampling frequency fs, and an alias frequency

falias, new frequency components are created at fs ± falias, 2 fs ± falias, 3 fs ± falias,

· · · . The accurate retrieval of attenuation and phase shift can be hard with this

type of distortion if harmonic frequency components from square pulses are af-

fected with the additional frequency contents. The aliasing can also be explained

in terms of the difference between the actual continuous pressure values and quan-

tized digital values. This is an error due to rounding or truncation. The error sig-

nal is sometimes considered as an additional random signal called quantization

noise (Valentinuzzi, 2004). The quantization noise is illustrated in Figure 3.21 for a

case of sampling continuous sinusoidal signal for a limited precision. A green line

is the difference between the original signal (in red) and the quantized signal (in

blue).

Figure 3.21: Illustration of quantization error (in green) caused by a limited preci-sion in amplitude.


As it is acknowledged that quantization noise influences the frequency con-

tents, the following concern is raised: does aliasing, or discretization of pressure

data, deteriorate the frequency attributes and if so, is it possible to recover the

original frequency contents? One way of reducing the aliasing problem is to pass

the continuous-time signal through a low-pass filter before sampling to remove

such high frequency edges. The followings discuss the aliasing effect caused by

discretization in time and pressure amplitude.

Aliasing Effect by Quantization of Time

Figure 3.22 illustrates aliasing effect in frequency domain by discretizing synthetic

pressure pulses every five points in time. The discretized pressure is denoted with

a green line in the top panel of Figure 3.23, which has a staircase shape. The time

discretization produces severe aliasing effects in frequency domain: ringing arti-

facts are observed periodically for both injection and observation frequency spec-

tra. Up to the 20th harmonics, the frequency contents remain very close to the

original. This is demonstrated by the green line formed from 20 harmonics on

attenuation and phase shift cross-plot in Figure 3.23, which is fairly close to the

original data. Therefore the aliasing effect is minimal on low harmonic frequency

contents.

Smoothing is performed by averaging over five points in an attempt to elim-

inate the aliasing effect. The red line in the top panel of Figure 3.22 shows the

smoothed injection pressure. Note that by averaging the amplitude of pressure

signal, the pressure amplitude is reduced and damaged around the end of the

flow period. Therefore smoothing in this case is not helpful; one can think of re-

constructing the pressure by knowing particularly how the discretization is con-

ducted in time. In real data, however, it is not simple to find such a rule. It is wise

to put more weight to the low frequency components in the objective function for

the frequency method because the attenuation and phase shift at low frequencies

are relatively more reliable than the ones at high frequencies.


0 50 100 150 200 250 300 350

−30

−20

−10

0

10

20

30

40

50

60

70

Mag

nitu

de, I

njec

tion

(dB

)

Frequency, rad/hr

Original injectionDiscretized injection

0 50 100 150 200 250 300 350

−50

−40

−30

−20

−10

0

10

20

30

40

50M

agni

tude

, obs

erva

tion

(dB

)

Frequency, rad/hr

Original observationDiscretized observation

Figure 3.22: Effect of quantization of time in frequency domain: Fourier magnitudespectra of injection (left) and observation pressure (right).


1560 1580 1600 1620 1640 16603.8

3.9

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

Time, hr

Pre

ssur

e ch

ange

, inj

ectio

n (p

si)

OriginalDiscretizedSmoothed

0 20 40 60 80 100 120 140 160

−30

−20

−10

0

10

20

30

40

50

60

70

Mag

nitu

de, i

njec

tion

(dB

)

Frequency, rad/hr


20 40 60 80 100 120 140 160

−50

−40

−30

−20

−10

0

10

20

30

40

50M

agni

tude

, obs

erva

tion

(dB

)

Frequency, rad/hr


0 0.02 0.04 0.06 0.08 0.10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Attenuation

Pha

se s

hift


Figure 3.23: Effect of smoothing on frequency data when quantized in time: mag-nified view of discretized injection pressure (top), magnitude spectrum by smooth-ing on injection pressure (center left), magnitude spectrum by smoothing on obser-vation pressure (center right), and frequency attributes after smoothing (bottom).


Aliasing Effect by Quantization of Pressure Amplitude

Figure 3.24 illustrates the aliasing effect in the frequency domain by discretizing

the synthetic pressure amplitudes which are obtained by rounding up to the first

decimal place. The discretized pressure is denoted with a green line in the top

panel of Figure 3.25, which has a similar staircase shape as the previous example.

The amplitude discretization produces an evenly spread noise over the entire fre-

quency range for both injection and observation pressure. On the injection side,

the harmonic frequency contents are robust to quantization noise. The observa-

tion pressure, on the other hand, is affected highly by the quantization noise; the

frequency contents up to the first four harmonics are preserved, but frequencies

higher than the fifth harmonic are masked by noise.

0 50 100 150 200 250 300 350

−40

−20

0

20

40

60

Mag

nitu

de, i

njec

tion

(dB

)

Frequency, rad/hr

Original injectionDiscretized injection

0 50 100 150 200 250 300 350

−30

−20

−10

0

10

20

30

40

50

Mag

nitu

de, o

bser

vatio

n (d

B)

Frequency, rad/hr

Original observationDiscretized observation

Figure 3.24: Effect of quantization of pressure amplitude on Fourier magnitudespectra: Fourier magnitude spectra of injection (left) and observation pressure(right).

To demonstrate the elimination of aliasing effect, smoothing is performed by

averaging over five points in time. The red line in the top panel of Figure 3.25

shows the smoothed injection pressure. The smoothed pressure tends to underes-

timate the actual pressure and is damaged around the sharp edges. In the center

of Figure 3.25, the frequency spectra of the injection pressure are shown on the


left, and the observation pressure on the right. It is important to note that the

pressure magnitude envelope is reduced for the injection at high frequencies. The

discretized injection pressure does contain noise at nonharmonic frequencies, but

is very close to the original one. On the observation pressure side on the right, the

noise is effectively removed at high frequencies. The attenuation/phase-shift cross

plots in the bottom of Figure 3.25 suggests that smoothing might help improve the

quality of the frequency data slightly at low frequencies. For frequencies higher

than the several lowest harmonics, both discretized and smoothed pressure shows

noisy behavior.

All in all, smoothing is not recommended because the pressure signal contents

can be deteriorated. It is recommended to keep in mind that the objective function

for the frequency method can be refined in the least squares calculation so that the

low frequency points have more importance than the high frequency points. The

smoothing effect is checked once more for Field data 1 in the subsequent section.


1560 1580 1600 1620 1640 16603.8

3.9

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

Time, hr

Pre

ssur

e ch

ange

, inj

ectio

n (p

si)


0 20 40 60 80 100 120 140 160

−30

−20

−10

0

10

20

30

40

50

60

70

Mag

nitu

de, i

njec

tion

(dB

)

Frequency, rad/hr


20 40 60 80 100 120 140 160

−40

−30

−20

−10

0

10

20

30

40

50M

agni

tude

, obs

erva

tion

(dB

)

Frequency, rad/hr


0 0.02 0.04 0.06 0.08 0.10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Attenuation

Pha

se s

hift


0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Attenuation

Pha

se s

hift


Figure 3.25: Effect of smoothing on frequency data when quantized in pressureamplitude: magnified view of discretized injection pressure (top), magnitude spec-trum by smoothing on injection pressure (center left), magnitude spectrum bysmoothing on observation pressure (center right), frequency attributes of 7 har-monics after smoothing (bottom left), and frequency attributes of 20 harmonicsafter smoothing (bottom right).


3.6.2 Field Data 1

A real field example in Figure 3.26 (a) has 2.7 second sampling interval, 2 hr peri-

odicity, 13330 points in time with a total of five pulses. A magnified view of the

pressure signal is shown in Figure 3.26 (b). Visible quantization noise in pressure

amplitude is observed. The injection pressure has high noise during the shut-in

time. Figure 3.26 (c) and (d) show the result when heuristic detrending is applied.

The pivot points are selected at every half cycle.

Figure 3.27 (a) shows the detrended injection and observation pressure. They

are aligned in a manner where the average is located at zero, which is desirable.

Figure 3.27 (b) illustrates the Fourier magnitude spectrum for a set of pressure data.

Aliasing effect is seen around the low frequency range. The Fourier magnitude for

the injection pressure in Figure 3.27 (c) shows that the harmonic data remain ap-

proximately the same, with or without detrending. Figure 3.27 (d) shows a visible

difference on the frequency decomposition caused by detrending. The nonlinear-

ity in the attenuation data is removed as shown in Figure 3.27 (e). On the phase

shift side, however, more fluctuation is created by detrending the pressure data.

A simple moving average filter was applied as a low-pass filter on the pressure

pair in an effort to remove the aliasing effect. A moving average of five points

was chosen, that is, pnew(t) = 1/5N∑

i=1p(t− i). The top graph in Figure 3.28 shows

the smoothed injection pressure compared to the original one. The staircase trend

is removed by smoothing. On the frequency side, the ringing artifact is reduced

and high frequency components are less noisy as shown in the central panels in

Figure 3.28 for injection and observation pressure; the ringing artifact is removed

at higher frequency range. However, the noise in this example is not particularly

detrimental to affect the attenuation and phase shift pairs significantly as shown

in the bottom plot in Figure 3.28. The quality of the frequency data remain about

the same as the original data. The smoothing does not improve the quality of

frequency values in the lower end of harmonic frequencies.

The attenuation and phase shift at the lowest 30 harmonic frequencies are sum-

marized in Figure 3.29. The sensitivity to a variation in the number of pulses is


checked. As decreasing the conditioning number of pulses, the last n pulses out of

five are checked. The relationship between attenuation and phase shift do not ex-

hibit a typical trend; over frequencies, the attenuation data do not increase mono-

tonically and phase shift data do not decrease. This is due to a nonlinear trend in

the frequency contents in the observation pressure data; there exist visible spikes

in the data in time domain which are not derived from the injection pressure. Pro-

cessing the entire five pulses produces the most nonlinear attenuation and phase

shift. The frequency data from the last two pulses are the most reliable because

the pressure data are likely to reach the steady state after the repetitive pulses,

and processing one pulse is risky by relying on one pulse only. For synthetic data,

extracting frequency data from the last pulse gives the best estimate to the steady-

state, but it is likely that the frequency resolution is not sufficient when noise affects

a single pulse.


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Time (hr)

Pre

ssur

e ch

ange

(ps

i)

Injection pressureObservation pressure

3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (hr)

Pre

ssur

e ch

ange

(ps

i)

Injection pressureObservation pressure

(a) (b)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Time, hr

Pre

ssur

e ch

ange

(ps

i)

InjectionReconstructed transient

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time, hr

Pre

ssur

e ch

ange

(ps

i)

ObservationReconstructed transient

(c) (d)

Figure 3.26: Preprocessing of Field data 1 in time domain: (a) pressure pair, (b)amplitude quantization in a magnified view, (c) reconstruction of constant ratepressure response at injection, (d) reconstruction of constant rate pressure responseat observation.


1 2 3 4 5 6 7 8 9 10

−4

−3

−2

−1

0

1

2

3

4

Time, hr

Pre

ssur

e ch

ange

, psi

Detrended injectionDetrended observation

0 100 200 300 400 500 600

−20

0

20

40

60

80

Mag

nitu

de (

dB)

Frequency, rad/hr


(a) (b)

0 20 40 60 80 100 120 140 160 180

20

30

40

50

60

70

80

90

Mag

nitu

de, i

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100 120 140 160 180

−20

0

20

40

60

80M

agni

tude

, obs

erva

tion

(dB

)

Frequency, rad/hr


(c) (d)

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency, rad/hr

Atte

nuat

ion

No detrendingHeuristic Detrending

100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, rad/hr

Pha

se s

hift


(e) (f)

Figure 3.27: Detrending of Field data 1: (a) detrended pressure pair, (b) Fouriermagnitude of pressure pair, (c) magnitude of injection pressure, (d) magnitude ofobservation pressure, (e) effect of detrending on attenuation, (f) effect of detrend-ing on phase shift.


1.64 1.66 1.68 1.7 1.72

8.38

8.4

8.42

8.44

8.46

8.48

8.5

8.52

8.54

8.56

8.58

Time, hr

Pre

ssur

e ch

ange

(ps

i)

InjectionSmoothed Injection

0 50 100 150 200 250 300 350

10

20

30

40

50

60

70

80

Mag

nitu

de (

dB)

Frequency, rad/hr

Original injectionSmoothed injection

0 50 100 150 200 250 300 350

−30

−20

−10

0

10

20

30

40

50

60M

agni

tude

(dB

)

Frequency, rad/hr

Original observationSmoothed observation

101

102

0.05

0.1

0.15

Atte

nuat

ion

101

102

0.2

0.4

0.6

0.8

Frequency, rad/hr

Pha

se s

hift

OriginalAfter smoothing

Figure 3.28: Effect of smoothing, Field data 1: magnified view of smoothed in-jection pressure, magnitude spectrum by smoothing on injection pressure (centerleft), magnitude spectrum by smoothing on observation pressure (center right),and frequency attributes after smoothing (bottom).


10−2

10−1

0

0.05

0.1

0.15

0.2

Frequency, rad/hr

Atte

nuat

ion

afte

r de

tren

ding

5 pulses4 pulses3 pulses2 pulses1 pulse

10−2

10−1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequency, rad/hr

Pha

se s

hift

afte

r de

tren

ding


Figure 3.29: Frequency attributes by varying number of pulses, Field data 1.


3.6.3 Field Data 2

Figure 3.30 (a) and (b) illustrate the injection pressure and observation pressure for

Field data 2, respectively. The data show a sampling interval of 2.68 seconds, a pe-

riodicity of 2 hrs, and a total of 13520 points in time with a total of five pulses.

There is a small fluctuation in the observation pressure, similar to the dicrotic

notch, every 3/4 of the period. The detrending is performed successfully as shown

by Figure 3.30 (c), where the detrended data are aligned around zero. The ratio of

pressure magnitude in Figure 3.30 (d) determines attenuation over frequencies.

Figure 3.30 (e) and (f) illustrate the effect of detrending; the injection pressure re-

mains almost the same, and the observation pressure have a different frequency

decomposition after detrending.

After detrending, the attenuation is less nonlinear than before (Figure 3.31 (a)).

A decreasing trend over frequencies is pronounced. The phase shift data in Figure

3.31 (b) fluctuate more after detrending. The attenuation and phase shift at 30

harmonic frequencies are summarized in Figure 3.31 (c) and (d) with a varying

number of pulses. The frequency data are gathered from the n last pulses. They

are similar regardless of the position of pulses.


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Time, hr

Pre

ssur

e ch

ange

(ps

i)


1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time, hr

Pre

ssur

e ch

ange

(ps

i)


(a) (b)

1 2 3 4 5 6 7 8 9 10

−3

−2

−1

0

1

2

3

Time, hr

Pre

ssur

e ch

ange

, psi


0 50 100 150 200 250 300−20

−10

0

10

20

30

40

50

60

70

80

Mag

nitu

de (

dB)

Frequency, rad/hr


(c) (d)

0 20 40 60 80 100 120 140 160 180

0

10

20

30

40

50

60

70

80

90

Mag

nitu

de, i

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100 120 140 160 180

−10

0

10

20

30

40

50

60

70

Mag

nitu

de, o

bser

vatio

n (d

B)

Frequency, rad/hr


(e) (f)

Figure 3.30: Field data 2: (a) injection pressure and reconstruction of constant ratepressure response, (b) observation pressure and reconstruction of constant ratepressure response, (c) detrended pressure pair, (d) Fourier magnitude of pressurepair, (e) magnitude of injection pressure, (f) magnitude of observation pressure.


100

101

102

103

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Frequency, rad/hr

Atte

nuat

ion


100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, rad/hr

Pha

se s

hift


(a) (b)

101

102

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Frequency, rad/hr

Atte

nuat

ion

afte

r de

tren

ding


101

102

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequency, rad/hr

Pha

se s

hift

afte

r de

tren

ding


(c) (d)

Figure 3.31: Field data 2: (a) effect of detrending on attenuation, (b) effect of de-trending on phase shift; frequency attributes by varying number of pulses: (c)attenuation, (d) phase shift.


3.6.4 Field Data 3

A real field example with the injection pressure and observation pressure in Fig-

ure 3.32 (a) and (b) has 2.68 second sampling interval, 2 hr periodicity, and 13410

points in time with a total of five pulses. The detrending is performed successfully

as shown by Figure 3.32 (c), where the detrended data are aligned around zero.

The ratio of pressure magnitude in Figure 3.32 (d) determines attenuation over

frequencies. Figure 3.32 (e) and (f) illustrate the effect of detrending; the injection

pressure remains almost the same, and the observation pressure have a different

frequency decomposition after detrending. Severe aliasing artifacts are observed

for both pressure magnitude spectra in the frequency domain.

After detrending, the attenuation is less nonlinear than before (Figure 3.33 (a))

as a decreasing trend over frequencies is pronounced. An illustration of the de-

trending effect on phase shift data is shown in Figure 3.31 (d). The attenuation

and phase shift at 30 harmonic frequencies are summarized in Figure 3.31 (c) and

(d) with a varying number of pulses. The frequency data obtained from the entire

pulse train shows a different trend compared to the rest of the choices, because the

steady-state has not been reached at the first pulse.


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Time, hr

Pre

ssur

e ch

ange

(ps

i)


0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time, hr

Pre

ssur

e ch

ange

(ps

i)


(a) (b)

1 2 3 4 5 6 7 8 9 10

−4

−3

−2

−1

0

1

2

3

4

Time, hr

Pre

ssur

e ch

ange

, psi


0 50 100 150 200 250 300 350

−20

0

20

40

60

80

Mag

nitu

de (

dB)

Frequency, rad/hr


(c) (d)

0 20 40 60 80 100 120 140 160 18010

20

30

40

50

60

70

80

90

Mag

nitu

de, i

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100 120 140 160 180

−30

−20

−10

0

10

20

30

40

50

60

70

Mag

nitu

de, o

bser

vatio

n (d

B)

Frequency, rad/hr


(e) (f)

Figure 3.32: Field data 3 : (a) injection pressure and reconstruction of constantrate pressure response, (b) observation pressure and reconstruction of constant ratepressure response, (c) detrended pressure pair, (d) Fourier magnitude of pressurepair, (e) magnitude of injection pressure, (f) magnitude of observation pressure.


100

101

102

103

0

0.01

0.02

0.03

0.04

0.05

0.06

Frequency, rad/hr

Atte

nuat

ion


100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, rad/hr

Pha

se s

hift


(a) (b)

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

Frequency, rad/hr

atte

nuat

ion

afte

r de

tren

ding


101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, rad/hr

Pha

se s

hift

afte

r de

tren

ding


(c) (d)



3.6.5 Field Data 4

A real field example in Figure 3.30 (a) and (b) has 2.67 second sampling interval, 2

hr periodicity, 8094 points in time with a total of five pulses. The figure (b) shows

many outliers appeared in the observation pressure. The detrending is performed

successfully as shown by Figure 3.30 (c), where the detrended data are aligned

around zero. The ratio of pressure magnitude in Figure 3.30 (d) determines at-

tenuation over frequencies. It is important to note that the frequency envelope

for the observation pressure is highly contaminated with noise. Figure 3.32 (e)

and (f) illustrate the effect of detrending; the injection pressure remains almost the

same, and the observation pressure does not demonstrate a desirable frequency

decomposition. It is worthwhile to examine if removing outliers improve the per-

formance of the frequency analysis.

As demonstrated by Figure 3.35, the frequency components have more well-

defined behavior over frequencies. The plots in the bottom panel of Figure 3.35

show that high frequency noise is removed. The three lowest harmonic frequency

components remain the same.

After detrending, the phase shift has a more pronounced increasing trend over

frequencies (Figure 3.33 (b)). The attenuation and phase shift at 30 harmonic fre-

quencies are summarized in Figure 3.31 (c) and (d) with a varying number of

pulses. The frequency data obtained from the entire pulse train shows a differ-

ent trend compared to the rest of the choices, because the steady-state has not been

reached at the first pulse.


1 2 3 4 5 60

2

4

6

8

10

12

14

Time, hr

Pre

ssur

e ch

ange

(ps

i)


1 2 3 4 5 6

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Time, hr

Pre

ssur

e ch

ange

(ps

i)


(a) (b)

1 2 3 4 5 6

−4

−2

0

2

4

6

8

Time, hr

Pre

ssur

e ch

ange

, psi


0 100 200 300 400 500 600 700

0

10

20

30

40

50

60

70

80M

agni

tude

(dB

)

Frequency, rad/hr


(c) (d)

0 20 40 60 80 100 120 140 160 180

10

20

30

40

50

60

70

80

90

Mag

nitu

de, i

njec

tion

(dB

)

Frequency, rad/hr


0 20 40 60 80 100 120 140 160 180

0

10

20

30

40

50

60

Mag

nitu

de, o

bser

vatio

n (d

B)

Frequency, rad/hr


(e) (f)

Figure 3.34: Field data 4: (a) injection pressure and reconstruction of constant ratepressure response, (b) observation pressure and reconstruction of constant ratepressure response, (c) detrended pressure pair, (d) Fourier magnitude of pressurepair, (e) magnitude of injection pressure, (f) magnitude of observation pressure.


0 20 40 60 80 100 120 140 160 180

−20

−10

0

10

20

30

40

50

Mag

nitu

de, o

bser

vatio

n (d

B)

Frequency, rad/hr

With outliersNo outliers

101

102

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency, rad/hr

Atte

nuat

ion


101

102

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency, rad/hr

Pha

se s

hift


Figure 3.35: Improved signal decomposition by removing outliers from observa-tion pressure: difference on the Fourier magnitude (top), improved quality of at-tenuation without outliers (left), and improved quality of phase shift without out-liers (right).


100

101

102

103

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Frequency, rad/hr

Atte

nuat

ion


100

101

102

103

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency, rad/hr

Pha

se s

hift


(a) (b)

101

102

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Frequency, rad/hr

Atte

nuat

ion

afte

r de

tren

ding

3 pulses2 pulses1 pulse

101

102

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency, rad/hr

Pha

se s

hift

afte

r de

tren

ding

3 pulses2 pulses1 pulse

(c) (d)



3.7 Summary

As demonstrated by the two reservoir models (radial ring and multilayered), the

heuristic detrending technique brings closer match of attenuation and phase shift

values to the sinusoidal steady-state space than the ones obtained without using

detrending. A constant rate response pressure signal is estimated at major time

points by using a periodic trend according to a specific duty cycle. The extrap-

olated constant rate pressure response is as accurate as the exact one that can be

obtained with known flow rate. There are factors which affect the accuracy of the

frequency data to the steady-state solutions. The larger number of pulses, later

occurrence in time, higher sampling frequency, the closer the frequency attributes

are to the steady-state case. The error produced by quantizing a signal, known

as quantization noise, deteriorates the accuracy of the attenuation and phase shift

at high frequencies. Detrending is applied on four field data and the attenuation

and phase shift are gathered at harmonic frequencies. Smoothing on the field data

did not greatly improve the nonlinear trend of attenuation and phase shift over

frequencies. Removing outliers improved the behavior of the frequency attributes.

Chapter 4

Inverse Problem Frameworks

Estimating the permeability distribution that produces the observed pressure data

requires an inverse problem formulation. This involves using the actual observa-

tions to infer the values of the parameters characterizing the system under inves-

tigation, which are the radial and multilayered models for this study. Aside from

the newly developed frequency method two other parameter estimation methods

are introduced to compare the performance of the method. These two comparable

methods are pressure history matching and wavelet thresholding.

The permeability estimation involves inverting the model M which matches

the pressure data p = p(t), that is, M(k) = p, so that k = k(r) is recovered. One

common approach for solving this is to obtain k that minimizes ‖M(k)− p‖2 in

the sense of Euclidean distance, which is least squares. The least squares prob-

lem corresponds to the maximum likelihood criterion if the errors have a normal

distribution.

Nonlinear inverse problems in general have a limitation that multiple solutions

are possible, meaning that different values of the model parameters may be con-

sistent with the data. Although it is of high interest to extract the characteristics

from pressure data that map uniquely into one reservoir behavior, due to its dif-

fusive nature the permeability distribution can permit multiple solutions. This is

the prevalent limitation in many inverse problems for characterization of a reser-

voir with measurement data as has been observed in the smooth estimation of

125

CHAPTER 4. INVERSE PROBLEM FRAMEWORKS 126

permeability distribution results shown by Oliver (1992) and Feitosa et al. (1994)

in general well testing practices.

The solutions obtained by the least-squares approach give no insight into the

uniqueness of the solution. Instead, the robustness of the inverse problem will

be examined by adding noise or perturbation in pressure and frequency domain

in Chapter 5 and 6. This is to answer how stable each method is in estimating

permeability distribution when noise is present in the data.

To impose the nonnegativity condition of the permeability, the logarithm of

the permeability is treated as the unknown of inverse problems for all methods.

This parameter transform also aids in flattening the permeability search space as

examined by Dastan (2010).

4.1 Frequency Method with Attenuation and Phase Shift

The objective is to obtain permeability distribution that minimizes the following

least squares objective function. The input data consists of attenuation and phase

shift at multiple frequencies from square pulses: [xω1 , · · · , xωn ] and [θω1 , · · · , θωn ]

for n selected frequencies.

mink‖xω1 − x(k, ω1)‖2

2 ... + ‖xωn − x(k, ωn)‖22 + ‖θω1 − θ(k, ω1)‖2

2 ... + ‖θωn − θ(k, ωn)‖22

(4.1)

The attenuation and phase shift data (Figure 4.1) should be sufficient to span

the frequency range that is indicative of the spatial heterogeneity. In addition, the

measurements of attenuation and phase shift should be as accurate as possible and

their values should lie between zero and one.

The order of computation is O(2N), with N being the total number of frequen-

cies. The factor 2 is applied because at each frequency, an attenuation and phase

shift pair is provided. It is important to note that the data size is significantly

reduced by extracting average frequency features from pressure pulses. Another

computational effort for the frequency method is in estimating the frequency data

by Fourier transform. A fast Fourier transform, which is an efficient algorithm to


compute the Fourier transform, can compute the same result in only O(n log n) op-

erations with n number of time points for pressure data. As the number of time se-

ries data grows, the frequency method requires much less computing power than a

full pressure matching method, which provides a big advantage for the frequency

method.

Figure 4.1: Input data for frequency method

As to the requirement of number of data points, traditionally, the Fast Fourier

Transform (FFT) algorithm is more efficient when it is dealing with signals that

contain n = 2N data points. However, the number of data points does not strictly

have to be power-of-two length, because the number of pressure data can be in-

creased by zero padding. This is to append a string of zeros to the time domain

sequence to increase the total sample time. When appended with nz zeros to the

original pressure data of np samples (so that np + nz = 2N), the FFT output be-

comes (np + nz)/2 + 1 samples, which spreads over 0 to fs/2. Because append-

ing zeros does not change the input sampling rate, the frequency span of the FFT

output remains the same. Frequency resolution can be improved by zero padding.

The cost is increased data processing. There is no new information added although

the frequency resolution increases.


4.2 Two Other Methods for Comparison

4.2.1 Pressure History Matching

Pressure history matching is the very basic method for estimating permeabilities

when the following data are given: injected and observed pressure data, and flow

rate denoted respectively as pinj(t), pobs(t), and q(t). The objective function for

permeability estimation is to obtain the permeability distribution that best matches

the pressure pair:

mink

(‖pt1 − p(k, t1)‖2

2 + ... + ‖ptm − p(k, tm)‖22

)(4.2)

The computational effort is of the order O(2m× s) with m number of time series

points and s number of Stehfest coefficients. The factor 2 is applied in matching

injection and observation signals. The typical choice for a number of Stehfest coef-

ficients was 8 in this study.

4.2.2 Wavelet Thresholding

Wavelets can be thought of as a generalization of the Fourier transform; wavelets

express any signal as a linear combination of well-defined functions, while for

Fourier analysis only trigonometric polynomials forms the basis. Wavelets also

provide more localized temporal information, which works well also for the non-

stationary signal. A significant reduction in time can be achieved, similar to the

way the frequency method works, and the wavelet-based methodology was com-

pared in this study with the same number of frequency data points to compare the

parameter estimation performance.

The following Haar wavelet function (Mallat, 2008) was chosen as the basis for


compressing the pressure data.

ψ(t) =

1 t ∈ [0, 1/2)

−1 t ∈ [1/2, 1)

0 t /∈ [0, 1)

(4.3)

Figure 4.2 illustrates the general decomposition procedure of wavelet, where

p(t) is the original signal to be decomposed, and LPF and HPF denote low-pass

and high-pass filter respectively. After the ith level decomposition, the detailed

coefficients are generated after LPF and are denoted as cDi; the approximation

coefficients are created after HPF and are denoted as cAi. The bandwidth of the

signal at every level is marked on the figure as frequency, ω, which is normalized

in general to lie between (0, π) in this diagram. Note that the most straightforward

procedure requires that the number of points is power-of-two length (2N form) and

each filter subsamples the signal by 2, meaning that the data size is reduced by half.

The procedure can be explained in detail using a pressure signal of 2048 sample

points which spans a frequency band of zero to π rad/s. At the first decomposi-

tion level, the signal is passed through the highpass and lowpass filters, followed

by subsampling by 2. The output of the highpass filter has 1024 points (hence half

the time resolution), which spans the frequencies π/2 to π rad/s (hence double

the frequency resolution). These 1024 samples constitute the first level of wavelet

transform coefficients. The output of the lowpass filter also has 1024 samples, but it

spans the other first half of the frequency band (0 to π/2 rad/s). This signal is then

passed through the same lowpass and highpass filters for further decomposition.

The output of the second lowpass filter followed by subsampling has 512 samples

spanning a frequency band of 0 to π/4 rad/s, and the output of the second high-

pass filter followed by subsampling has 512 samples spanning a frequency band

of π/4 to π/2 rad/s. The same process repeats for further decompositions, each

having half the number of samples from the previous level. After the sixth decom-

position level, the cA6 is composed of 32 (2048/26) lowest frequency components.

The total wavelet coefficients have the same number of coefficients as the original


signal.

Figure 4.2: Wavelet decomposition process

Figure 4.3 shows an example of the wavelet decomposition of four pressure

pulses by applying Haar wavelet function in Equation 4.3. In particular, we se-

lected the wavelet coefficients at the sixth level of decomposition because we found

that the wavelet coefficients at such level capture the significant differences due to

the varying permeability distributions. Note that the approximation coefficients

cA6 still carries the periodicity. The periodicity can be found in all the detailed

coefficients as well, which show peaks at every abrupt pressure change over time.

The cD1 indeed carries the highest frequency components where the sharp transi-

tions of pressure data are contained. Smoother pressure information is seen as the

decomposition proceeds until cD6.

For a given signal, we can reconstruct the original signal by simply padding


10 20 30

2000

4000

cA6

10 20 30

−2000

200

cD6

20 40 60−200

0

200

cD5

20 40 60 80 100 120

−1000

100

cD4

50 100 150 200 250−100

−500

50

cD3

100 200 300 400 500−50

0

50

cD2

200 400 600 800 1000

−200

20

cD1

Figure 4.3: Wavelet decomposition approximation (cA6) and detail coefficients(cD1,· · · ,cD6)

or approximating nonsignificant wavelet coefficients with zeros and inverse trans-

form the coefficients that are bigger than the threshold to reconstruct the original

signal. A significant data reduction is achieved this way. In this study when using

the wavelet coefficients for the inverse problem, the same number of wavelet coef-

ficients as the number of frequency data were used for comparing the performance

of estimating permeability distribution.

The objective function uses t1, · · · , tl selected wavelet coefficients from injected

and observed pressure which satisfy the conditions |wti | > threshold. By a thresh-

olding criterion, information carried in a signal is effectively compressed by elim-

inating certain wavelet parameters that do not meet a threshold criterion. In gen-

eral, the threshold is defined on the basis of the desired level of accuracy of the


reproduction of a function. Various thresholding such as on the magnitude of the

sensitivity of the pressure data can be used as shown by many authors (Dastan,

2010; Awotunde, 2010; Sahni, 2006), but in this study, the simplest direct transfor-

mation of pressure is considered. The computational effort is similar to pressure

history matching, because the wavelet in this case is an additional function on top

of the calculated pressure series in this case. The input data stream consists of

winj(τ), wobs(τ) for τ ∈ (t1, .., tl) , and q(t).

mink

(‖wt1 − w(k, t1)‖2

2 + ... + ‖wtm − w(k, tl)‖22

)(4.4)

Representation of Pressure Data with Different Number of Wavelets

The comparison of the estimation performance will be conducted with the same

number of wavelet coefficients as is used for the frequency method. One thing to

note beforehand is that the larger the number of significant wavelet coefficients is,

the closer the reconstructed pressure resembles the true pressure. The performance

of the wavelet approach depends largely on the number of selected wavelet points.

By tuning the level of threshold, a different set of wavelets can be used to represent

the reservoir characteristic while replacing the given pressure data.

A total of 8, 30, and 48 largest wavelet coefficients were used as conditioning

data with the threshold criterion of |w(t)| > α. The pressure data consist of 1024

points. Figure 4.4 demonstrates the pressure signals reconstructed by the three

different set of wavelet coefficients. For this example where the wavelets were de-

composed to the sixth level, the eight coefficients are gathered from the two lowest

frequency ranges, cA6 and cD6. For the case of 30 wavelets, additional detailed

coefficients from cD5 were included. For 48 wavelets, additional coefficients from

some of the high frequency ranges, cD3 and cD4 were included. As the high fre-

quency components are included, the pressure resemble the true pressure.

The discrepancies to the true pressure |preconstructed − ptrue| /ptrue are 0.060, 0.021,

0.014 for the chosen 8, 30, and 48 largest wavelet coefficients, respectively. Only

with 0.8 % of the pressure data, the reconstructed pressure resembles close to the

true pressure. This ability of wavelets to compress the information is noteworthy.


1 2 3 4 5 6 7 80

20

40

60

80

100

Time (hr)

Inje

ctio

n pr

essu

re (

psi)

True48 wavelets30 wavelets8 wavelets

Figure 4.4: Reconstruction of pressure by varying number of wavelet coefficients.

For the demonstration in later chapters, the number of chosen wavelet coeffi-

cients was ten so as to be compatible with the number of frequency points in the

frequency method. This allows the estimation performance to be compared with

the frequency method with the same number of attenuation and phase shift data

pairs.

4.3 BFGS Algorithm

All three inverse problems (frequency, pressure history matching, and wavelet

thresholding) are based on least squares. The Broyden-Fletcher-Goldfarb-Shanno

(BFGS) Quasi-Newton optimization with a cubic line search technique was used

for estimating permeability (Gill et al., 1981). BFGS approximates Newton’s method

which seeks a point where a gradient becomes zero. The method does not need to

converge unless the function has a quadratic Taylor expansion near an optimum


permeability. BFGS has proven good performance even for nonsmooth optimiza-

tions. The gradient information should be calculated by finite-difference approx-

imations (∇ f (ki)) for the three objective functions. Also note that the gradient-

based method suffers from the shortcoming of often converging to a local mini-

mum instead of reaching the globally optimum parameter values.

Among many Matlab (MATLAB, 2011) optimization functions, the fminunc func-

tion was used to apply the BFGS technique with cubic line search. The Hessian

matrix is approximated using rank-one updates specified by gradient evaluations.

Let f (k) denote the objective function to be minimized, and k is the unknown

permeability to be estimated. From an initial guess of permeability distribution

k0 and an approximate Hessian matrix B with B0 = I, the following steps are

repeated until ki converges to the solution.

1. Obtain a search direction zi at stage i by solving the Newton equation:

Bizi = −∇ f (ki) (4.5)

2. Perform a line search with an acceptable size αi, and update the permeability:

ki+1 = ki + αizi (4.6)

3. With si = αizi and yi = ∇ f (ki+1)−∇ f (ki)

Bi+1 = Bi +yiyT

iyT

i si−

BisisTi Bi

sTi Bisi

(4.7)

4. Repeat the steps with the inverse of Bi, which is obtained efficiently by ap-

plying the Sherman-Morrison formula to the previous step:

B−1i+1 = B−1

i +(sT

i yi + yTi B−1

i yi)(sisTi )

(sTi yi)2

−B−1

i yisTi + siyT

i B−1i

sTi yi

(4.8)


4.4 Convergence Performance Comparison

The three methods of parameter estimation were carried out using Matlab (MAT-

LAB, 2011). Figure 4.5 describes the three objective function values versus itera-

tion level for the three methods on a radial ring model example with a uniform

permeability of 100 md. The initial guess for permeability was 200 md for all runs.

The objective function in each case has been calculated with a radial permeability

estimate at each iteration. It is seen that the scale of the objective function for fre-

quency method starts with a value as low as 1× 10−2 and terminates with a value

around 1× 10−5, meaning that the frequency method requires a high precision to

have a successful estimate. The plot also shows the comparison of convergence

speeds for different methods. The history matching method usually requires more

iterations, and converges the slowest of all methods. It is important to observe,

however, that convergence rate is strongly dependent on the nature of each case.

Furthermore, each method involves different computational effort at each itera-

tion step, with history matching taking the largest effort depending on the size of

the given pressure data. Note that wavelet method was used for this study in a

manner such that the same computational effort as history matching is required

to calculate the pressure signal at each iteration. All in all, the frequency method

saves computational time drastically, although history matching gives the most ac-

curate estimate of permeability distribution almost surely by conditioning on the

whole set of pressure data.

The elapsed CPU time for the three inverse problem frameworks was mea-

sured using an example of pressure pulses with 5000 points. When pressure his-

tory matching or wavelet method was applied, the CPU time was approximately

30 mins. In contrast, using ten pairs of attenuation and phase shift data from the

given pressure data, the frequency method took approximately 30 seconds. Mat-

lab (MATLAB, 2011) was run on a computer with the following specifications:

Windows 32-bit Operating System, 2.26GHz processor, and 2GB RAM memory.

Due to the fact that MATLAB is an interpreted language, it is important to note

that the Matlab scripts execute slower than compiled programs written in other


languages, such as C, C++, and Fortran.

0 10 20 30 40 50 6010

−6

10−4

10−2

100

102

104

106

108

Number of iterations

Obj

ectiv

e fu

nctio

n va

lue

History matchingWaveletFrequency method

Figure 4.5: Comparison of convergence curves (objective function versus itera-tions)

4.5 Reconstructed Pressure by Three Methods

Figure 4.6 illustrates pressure match results at injection and observation point ac-

cording to the three methods. The two vertical layered models, Model 4 and

Model 5, are used as examples.

There is a visible difference in the injection pressure value when the frequency

method is used, with a discrepancy of 2.4 psi at the injection pressure peak of 110.8

psi. The overall performance using various methods is summarized later in Ta-

ble 6.2. The history matching shows the least mismatch error in terms of pressure

norm difference which is attained at a higher computational cost than other meth-

ods. The wavelet method with ten coefficients works as well as the pressure his-

tory matching. For other models, similar performance is observed with pressure

history matching showing the best fit to the pressure by directly relying on pres-

sure data; and frequency method and wavelet thresholding showing the similar

mismatch in pressure. This is demonstrated later in Tables 5.3, 5.4, 5.5, 6.2, 6.3, 6.4


for the results on Model 1 - 6.

0 1 2 3 4 5 6 7 80

20

40

60

80

100

120

Time, hr

Inje

ctio

n pr

essu

re c

hang

e (p

si)

TrueHistory matchingWavelet, 10 ptsFrequency, 10pts

0 1 2 3 4 5 6 7 8−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time, hrO

bser

vatio

n pr

essu

re c

hang

e (p

si)


(a) (b)

0 2 4 6 8 100

20

40

60

80

100

120

Time, hr

Inje

ctio

n pr

essu

re c

hang

e (p

si)


0 2 4 6 8 10−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time, hr

Obs

erva

tion

pres

sure

cha

nge

(psi

)


(c) (d)

Figure 4.6: Reconstructed injection (left) and observation pressure (right) for mul-tilayered model: Model 4- (a) injection pressure, (b) observation pressure; Model5- (c) injection pressure, (d) observation pressure.


4.6 Summary

The formulation of inverse problems with the three methods of parameter esti-

mation was discussed. The frequency method aims at matching the frequency

attributes at multiple harmonic frequencies. The two other methods, the pres-

sure history matching and the wavelet method, are introduced to assess the per-

formance of permeability estimation by comparison. All three inverse problems

are based on least squares with BFGS Quasi-Newton optimization technique. The

computational effort is saved by using the frequency method.

Chapter 5

Permeability Estimation on Radial

Ring Model

Chapter 2 brought together the analysis on how frequency attributes help distin-

guish permeability distributions from one another. In this chapter, the frequency

method is applied to estimate permeability distributions on radial ring model ex-

amples. One homogeneous and two heterogeneous radial ring models are exam-

ined as synthetic examples. The quality of the permeability estimation is verified

with attenuation and phase shift information from ideal sinusoidal frequencies to

evaluate the ability of frequency data to characterize the reservoir. After this evalu-

ation, square pulses mimicking an actual field test are used to evaluate the method.

Square pulses containing many harmonic frequencies and frequency data can be

collected from these. The performance of the method in revealing the heterogene-

ity over distance is then compared with two other methods, full pressure history

matching and the wavelet thresholding method. The effects of storage, skin and

boundary conditions on frequency data are also demonstrated, and field data ex-

amples are presented as well. While an inverse problem in estimating permeability

from pressure history is an ill-posed problem with many possible permeability re-

alizations, one set of permeability estimates is obtained and presented subject to

specific initial and terminating conditions.

In these examples, the placement of the observation well is at the furthest ring

139

CHAPTER 5. PERMEABILITY ESTIMATION ON RADIAL RING MODEL 140

in the radial ring model as explained in Section 2.3. For the purpose of this analy-

sis, the outer region is an infinite reservoir system unless otherwise defined. The

primary focus of the study was on obtaining the permeability distribution in the

interwell region, so no permeability variation beyond the observation well was

considered. In fact, the pulse-test values strongly reflect the formation properties

between two points and are affected little by the heterogeneities beyond the tested

wells (Ogbe and Brigham, 1987). Although there is no direct relationship between

the scale of permeability and distance in determining the sourcing frequency, the

permeability estimation is expected to be successful if the cross plots of attenua-

tion vs. phase shift from multiple frequencies constitute sufficient data points to

characterize the reservoir’s permeability distribution. Thus, inspections upon the

appropriate frequency range and on varying the number of such frequency points

were conducted.

5.1 Using Multiple Sinusoidal Frequencies

One homogeneous model denoted as Model 1 and two heterogeneous models de-

noted as Model 2, 3 are examined here. Those three models were introduced in

Section 2.3.2, where the frequency data are used to characterize the permeability

distributions. To determine which frequency ranges best characterize the radial

permeability distribution, multiple attenuation and phase shift pairs at different

frequencies were compared. For these tests, the observation well is 29 ft away

from the active well for all three synthetic models. Application of these methods

for a larger interwell distance for practical pulse testing was described in Section

2.3.2.

Parameter estimation was performed to obtain the permeability distribution

based on the attenuation/phase-shift objective function introduced in Section 4.1.

The initial guesses used for the permeability estimation were 200 md for Model

1, 300 md for Model 2 and Model 3. There are ten degrees of freedom in deter-

mining permeability values for ten rings. For all synthetic examples, the reservoir

parameters are: rw = 0.5 ft, φ= 0.25, µ=5 cp, ct = 1.2× 10−5 psi−1.


For each test, the attenuation and phase shift pairs at ten frequencies were used.

First, for the three models, 30 logarithmically spaced frequencies were selected and

they cover a wide range of values in the attenuation and phase shift cross-plots.

Then the range was divided further into four different segments in an effort to

check which frequency segment is the most effective in obtaining the best estimate

of the permeability distribution. Frequencies 1 has the lowest ten frequencies and

Frequencies 3 has the highest ten frequencies. Frequencies 4 has ten frequencies

that span evenly among the whole range. Frequencies 5 contains additional fre-

quency components that reflect the cyclic influence. The cyclic influence is a rule

of thumb used as a reference to check if this relationship is effective in accurately

estimating the permeability distribution.

5.1.1 Homogeneous Radial Ring - Model 1

Figure 5.1 (a) shows logarithmically spaced sourcing periodicities from 2.6 min

to 6.9 hr (Frequencies 1 - 4). Frequencies 5 is calculated from the cyclic radius

of investigation formula (Rosa, 1991), rD = 1.1√

1/ωD. The lowest frequency to

reach the furthest radial ring (rD = 58) has a period of 2.53 hr; while the highest

frequency has a period of 42.5 sec to reach the nearest radial ring (rD = 4). The

high frequency generates values that are too low for attenuation and too high for

phase shift, and so Frequencies 5 are chosen in the range from 2.6 min to 2.9 hr.

Figure 5.1 (b) shows the estimation result. The high frequency range (Frequencies

3) shows the best performance in estimating the permeability, followed by the

evenly spaced range (Frequencies 4). Table 5.3 shows the normalized permeabil-

ity mismatch error, ‖kr,estimate − kr,true‖2/‖kr,true‖2. As shown by the permeability

estimation result in different frequency ranges, there is no particular benefit in fine

tuning the sourcing frequencies for the corresponding radii of cyclic influence.


5.1.2 Heterogeneous Radial Ring - Model 2 and 3

For Model 2, Figure 5.1 and Table 5.4 show that although most frequency ranges

successfully depict the model’s heterogeneous trend over distance, using Frequen-

cies 4 results in the best estimation of the true distribution. The performance is

similar to the studies by Oliver (1992) and Feitosa et al. (1994) in that the perme-

ability estimate depicts a smooth transition rather than the actual sharp transition.

Model 3 investigates another case of the radial ring model with a different per-

meability distribution. Figure 5.1 (e, f) and Table 5.5 show that Frequencies 2 -

5 successfully depict the model’s heterogeneous trend over distance. Frequen-

cies 1 does not accurately represent the permeability distribution and this is due

to the low frequencies not providing sufficient information for the interwell het-

erogeneity; this estimation satisfies the termination criteria, but the permeability

distribution did not reflect the true distribution. Frequencies 4 resulted in the best

estimate of the true distribution.

The average permeability for the two ring models can be calculated by using

the formula log(re/rw)n∑

j=1log(rj/rj−1)/kj

. For Model 2 the average permeability is 315.03 md,

and for Model 3 it is 306.54 md. Applying the average permeability as a refer-

ence, the appropriate sourcing frequency range that covers the entire radial ring

distance translates to a periodicity of 13.50 sec - 47.31 min for Model 2, and 13.88

sec - 48.71 min for Model 3. Fine tuning to the cyclic influence does not show a

better estimation of permeability, although the heterogeneous trend is estimated

successfully for all three models.


2 5 8 11 14 17 20 23 26 290

20

40

60

80

100

120

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d

TrueFrequencies 1Frequencies 2Frequencies 3Frequencies 4Frequencies 5

(a) (b)

0.05 0.1 0.15 0.2 0.25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Attenuation

Pha

se s

hift

Frequencies 1Frequencies 2Frequencies 3Frequencies 4Frequencies 5

2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

700

800

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


(c) (d)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Attenuation

Pha

se s

hift

Frequencies 1Frequencies 2Frequencies 3Frequencies 4Frequencies 5

2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

700

800

900

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


(e) (f)

Figure 5.1: Radial permeability estimation by sinusoidal frequencies: Model 1- (a)Frequency data, (b) radial permeability estimation; Model 2- (c) frequency data, (d)radial permeability estimation; Model 3- (e) frequency data, (f) radial permeabilityestimation.


Extreme Frequency Range

When frequencies are chosen such that the cyclic influence goes far beyond the

distance of the observation well, the permeability distribution cannot be estimated

accurately. The long period listed as Frequencies 6, 7 in Table 5.3 were used to

obtain permeability estimation result shown in Figure 5.2. When the cyclic influ-

ence by the selected ten frequencies goes beyond the observation distance, that is,

rND << rDc inf(ωD), the frequency data do not have the capability to indicate the

permeability distribution of the radial model.

Even with sufficient frequency points, the distance of investigation might not

be covered by the sourcing frequencies. This is the inherent limitation of square

pulse signals in that not all the high frequency components are useful. With insuf-

ficient high frequency components, the near-well region cannot be investigated;

with insufficient low frequency components, longer distances from the well can-

not be investigated.

0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Attenuation

Pha

se s

hift

Frequencies 1Frequencies 2Frequencies 3Frequencies 4Frequencies 5Frequencies 6Frequencies 7

2 5 8 11 14 17 20 23 26 290

20

40

60

80

100

120

140

160

180

200

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d

TrueFrequencies 6Frequencies 7

Figure 5.2: Permeability estimation result (right) with frequencies (left) that reachfar beyond the observation point for Model 1.


Table 5.1: Radial permeability estimation error with varying number of frequenciesModel 1 Model 2 Model 3

Number of frequencies ε(kr) ε(kr) ε(kr)

10 0.013 0.20 0.185 0.044 0.20 0.293 0.088 0.20 0.232 0.37 0.44 0.511 0.34 0.43 0.47

Permeability Estimation with Varying Number of Sinusoidal Frequencies

A varying number of frequencies (10, 5, 3, 2, 1) in the range of Frequencies 4 were

evaluated for all three models (Figure 5.3). Each choice of frequencies is spread

evenly among the entire range. Table 5.1 shows the permeability estimation per-

formance by the metric ‖kr,estimate − kr,true‖2/‖kr,true‖2.

Using three, five, or ten frequencies resulted in a good estimation for the per-

meability distribution. In general, more frequency points resulted in a better es-

timation. However, using only one or two frequency points was not sufficient to

resolve a reservoir with ten degrees of freedom. One or two points are not suffi-

cient because the cross-plot they form do not uniquely describe the reservoir. The

estimation performance also depends on the resolution required for grid blocks;

here in the examples three levels of heterogeneous permeability distributions are

unknown for each model with ten degrees of freedom. Therefore three frequency

points are sufficient to describe the permeability trend over distance.


2 5 8 11 14 17 20 23 26 290

20

40

60

80

100

120

140

160

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d

True10 frequencies5 frequencies3 frequencies2 frequencies1 frequency

2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

700

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

700

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


Figure 5.3: Radial permeability estimate with varying number of sinusoidal fre-quencies for Model 1 (top), Model 2 (left) and Model 3 (right).


5.2 Using Harmonic Frequencies from Square Pulses

For the three models, ten frequency points were applied using the frequency method

for estimating permeability between wells using square pulse sourcing. Since the

odd harmonics are obtained (1ω, 3ω, 5ω, · · · , 19ω), the highest frequency is 19

times the sourcing frequency.

Then the estimation results from the frequency method were compared with

history matching and wavelet thresholding techniques. A total of 2048 points of

pressure data with four pulses were used for history matching with no data re-

duction; the same number of wavelet coefficients as the frequency data were gath-

ered by obtaining the largest wavelet coefficients to compare with the frequency

method. For the frequency method, attenuation and phase shift information from

the first ten harmonic frequency components were used. To make the compari-

son with wavelet method, thresholding is conducted such that ten largest wavelet

coefficients from the sixth decomposition level were used to check the estimation

performance.

5.2.1 Homogeneous Radial Ring - Model 1

Figure 5.5 (a) and Table 5.3 summarize the performance of the three different meth-

ods. A set of square pulses was sourced with a fundamental periodicity of 51.9 min

with 6.1 sec sampling frequency.

The history matching shows the best proximity to the true permeability value

because all the of 2048 pressure time series points are utilized without reducing the

data size. The method sets the upper bound for how accurately permeability dis-

tribution can be estimated with given pressure pulses. In comparing the frequency

method and the wavelet method with the same number of points, the frequency

method results in a closer estimation to the true permeability distribution. Using

all wavelets would give the same performance as the pressure history matching.

For Table 5.3 - 5.5, the error metrics used to summarize the performance are

the following: for pressure, the average pressure misfit is used, which is defined as

‖pestimate − pmeasured‖2/

Np; for permeability, ‖kr,estimate − kr,true‖2/‖kr,true‖2 is used.


Sensitivity of Performance to Termination Criteria

Unlike using sinusoidal frequencies, when harmonic frequency attributes are gath-

ered from square pulses, discrepancies already exist with the steady-state solutions

especially at high frequencies because the pressure magnitude at high frequencies

is small when pressure data are decomposed. The higher the frequency, the big-

ger is the difference because the frequencies are sourced at weaker magnitude as

the harmonic number increases. The harmonic components obtained show that

‖x(ωh)− xss(ωh)‖2 = 9.2× 10−4 and ‖θ(ωh)− θss(ωh)‖2 = 0.034 for Model 1.

As the frequency attributes already have error, depending on how strict the

termination criterion is, the answer can be quite different. Constraining too much

would result in a search path that goes beyond the desirable optimum and runs an

exhaustive number of iterations. Constraining too little would result in a point that

is not near the vicinity of a solution. For a termination criteria α, when | f (ki) −f (ki+1)| < α the iterations end. α is a lower bound on the change in the value of

the objective function during a step.

See Figure 5.4 and Table 5.2 which summarize the trade-off between the tight-

ness of the termination criterion. For selected n number of frequencies, the atten-

uation and phase shift data from the n lowest harmonic frequencies are used. The

permeability estimate by using ten frequency points with a tight termination con-

straint of α = 1× 10−4 deviated significantly from the true distribution, because

the mismatch between the frequency data with the periodic steady-state solutions

at ten frequency points is more than the termination criteria.

Also, using fewer frequency points can result in a better permeability estimate.

But the number of frequency data needs to be sufficient to indicate the reservoir

characteristic. For both termination criteria, the frequency method with five fre-

quency points performed the best in terms of matching the true permeability dis-

tribution. Adding five more frequency points performed not as well as using five,

because the additive five points have errors to the periodic steady-state solutions.


2 5 8 11 14 17 20 23 26 290

20

40

60

80

100

120

140

160

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d

True10 frequencies5 frequencies3 frequencies2 frequencies

2 5 8 11 14 17 20 23 26 290

20

40

60

80

100

120

140

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d

True10 frequencies5 frequencies3 frequencies2 frequencies

Figure 5.4: Effect of termination criteria on permeability estimation performancewith α = 1× 10−4(left) and α = 1× 10−3 (right) for Model 1.

Table 5.2: Radial permeability estimation error with different termination criteriaand number of frequencies for Model 1

α = 1× 10−4 α = 1× 10−3

Number of frequencies ε(kr) ε(kr)

10 0.24 0.0905 0.055 0.0763 0.084 0.102 0.11 0.12

5.2.2 Heterogeneous Radial Ring - Model 2 and 3

For Model 2, four pulses with periodicity of 3.36 hr and 2048 pressure data points

were used. Figure 5.5 (c) along with Table 5.4 reveal that the history matching

method gives the best result in terms of matching true permeability distribution

and resulting in the least misfit for pressure data. The frequency method shows

the next best estimation performance. Wavelet thresholding with ten points failed

in depicting the overall heterogeneous distribution. The frequency method out-

performed with the same number of input data for the wavelet method in this

case.

For Model 3, four pulses with periodicity of 5.04 hr and 2048 points pressure


were applied. Figure 5.5 (e) along with Table 5.5 reveal that next to the history

matching method, the frequency method shows the second best estimation perfor-

mance. However, in terms of reconstructing pressure data, wavelet compression

had a better match in pressure data with the same number of input parameters.

5.3 Permeability Estimation with Added Pressure Noise

Finally, Figure 5.5 illustrates the performance of three methods with one example

with 1% noise in pressure pair. Tables 5.3, 5.4, 5.5 summarize the permeability and

pressure (when reconstructed with permeability estimate) mismatch for Model 1,

2 and 3 respectively. In general the added noise deteriorates the estimation per-

formance, however, the wavelet method in Model 1 outperforms the frequency

method in matching the true permeability. The effect on wavelet coefficients by

the added noise is insignificant because the magnitude of high frequency noise

is filtered as detailed coefficients (cD6 in this case). ‖wnoise − w‖2/‖w‖2 for three

coefficients from injection and observation is 1.3× 10−4 and 4.1× 10−4.


2 5 8 11 14 17 20 23 26 290

20

40

60

80

100

120

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d

TrueHistory matchingWavelet, 10 ptsFrequency, 10 pts

2 5 8 11 14 17 20 23 26 290

20

40

60

80

100

120

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


(a) (b)

2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

700

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

700

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


(c) (d)

2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

700

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


(e) (f)

Figure 5.5: Radial permeability estimation from square pulses: Model 1- (a) with-out noise, (b) with noise; Model 2- (c) without noise, (d) with noise; Model 3- (e)without noise, (f) with noise.


Table 5.3: Model 1 - Radial permeability estimation errorε(pinj(t)) ε(pobs(t)) ε(kr) ε(kr)

with noiseMultiple sinusoidsFrequencies 1 (1.4 - 6.9 hr) - - 0.066 -Frequencies 2 (15.0 min - 1.2 hr) - - 0.052 -Frequencies 3 (2.6 - 12.6 min) - - 0.013 -Frequencies 4 (2.6 min - 6.9 hr) - - 0.013 -Frequencies 5 (2.6 min - 2.9 hr) - - 0.041 -Frequencies 6 (9.9 year - 41.4 day) - - 0.59 -Frequencies 7 (6.9 hr - 25.2 day) - - 0.30 -Square pulsesHistory matching 0.0016 0.001 0.0048 0.0057Wavelet thresholding 0.079 0.037 0.068 0.034Frequency method 0.054 0.041 0.052 0.054


with noiseMultiple sinusoidsFrequencies 1 (4.7 - 61.1 hr) - - 0.37 -Frequencies 2 (16.0 min - 3.5 hr) - - 0.24 -Frequencies 3 (55.2 sec - 12.1 min) - - 0.18 -Frequencies 4 (55.2 sec - 61.1 hr) - - 0.20 -Frequencies 5 (55.2 sec - 50.3 min) - - 0.19 -Square pulsesHistory matching 0.013 0.010 0.21 0.24Wavelet thresholding 0.10 0.050 0.44 0.46Frequency method 0.072 0.042 0.19 0.37



with noiseMultiple sinusoidsFrequencies 1 (7.0 hr - 3.8 day) - - 0.47 -Frequencies 2 (24.1 min - 5.3 hr) - - 0.27 -Frequencies 3 (1.4 - 18.1 min) - - 0.24 -Frequencies 4 (1.4 min - 3.8 day) - - 0.18 -Frequencies 5 (1.4 min - 50.7 min) - - 0.20 -Square pulsesHistory matching 0.077 0.075 0.20 0.21Wavelet thresholding 0.086 0.074 0.40 0.46Frequency method 0.18 0.13 0.30 0.41


5.4 Perturbation in Frequency Space

Figure 5.6 shows the permeability estimate result when frequency information is

10% more and 10% less than the original values for the three different models.

When attenuation data are modified, the phase shift data remain the same, and

vice versa. Table 5.6 summarizes the norm difference of the permeability estimate

between conditioning on perturbed attenuation and phase shift data with the de-

fault case. The performance of estimation is not as good as the default case, as

expected. By changing the phase shift data for Model 1, the permeability distribu-

tion showed a big difference compared to the original permeability distribution.

The heterogeneous models, however, depicted the trend of permeability distribu-

tion fairly well.

Table 5.6: Radial permeability error with perturbation to frequency dataModel 1 Model 2 Model 3

ε(kr) ε(kr) ε(kr)

Default 0.016 0.18 0.24+10% attenuation 0.056 0.30 0.26−10% attenuation 0.065 0.31 0.25+10% phase shift 0.23 0.27 0.25−10% phase shift 0.34 0.27 0.25

5.5 Effects of Storage and Skin

The study considered the addition of wellbore storage and skin effects at the puls-

ing well. Ogbe and Brigham (1987) conducted an intensive study on how the stor-

age and the skin effect influence pulse tests. They concluded that the effects reduce

the amplitude and increase the time lag. The same phenomenon is indeed demon-

strated in the time series pressure in both the injection well and the observation

well. The homogeneous radial ring model with 100 md is examined as an exam-

ple.


2 5 8 11 14 17 20 23 26 290

20

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120

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180

Distance, r, ft

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TrueDefault+10% attenuation−10 % attenuation+10% phase shift−10% phase shift

2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

700

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

700

800

900

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


Figure 5.6: Radial permeability estimate from 10% change in attenuation and phaseshift for Model 1 (top), Model 2 (left) and Model 3 (right).

Effect of Storage on Pressure Pulses

As shown in Figure 5.7 (a, b), for the storage effect, the larger the dimensionless

CD value, the more attenuation is present. When the storage value is too high,

the observation signal is taken over by the transient upward trend and becomes

minimally periodic.

The accuracy of frequency information was evaluated compared with the steady-

state model and it is illustrated over harmonic frequencies in Figure 5.7 (c, d). The

summary of the estimation performance is shown in Table 5.8. The rule of thumb


for the time frame of tD > CD(60 + 3.5s) is suitable in theory to examine the reser-

voir characteristics for a single-phase injection test. Before this time, the pressure

response is affected by wellbore storage and skin effects, making it difficult to diag-

nose reservoir properties such as permeability. The examination with a dimension-

less storage of 100 satisfies tD > CD(60 + 3.5s) for all time and using the last two

pulses best estimates its steady-state solution at the ten lowest harmonic frequen-

cies, as pulses reach the periodically steady state as more pulses are superimposed.

A similar observation is shown in Figure 3.14. For higher CD of 1000 and 10000,

the last two pulses are used to estimate the frequency attributes. For CD of 1000,

the condition (tD > CD(60 + 3.5s)) is satisfied for only 6% at a later time. Except

at the three lowest harmonic frequencies, the attenuation and phase shift deviate

from the steady-state solution. For CD of 10000, the storage effect dominates the

reservoir and tD < CD(60 + 3.5s) holds. As the time frame is masked by a strong

storage effect, a substantial deviation of the harmonic frequencies exists.


1 2 3 4 5 6 7 8

2

4

6

8

10

12

14

16

18

20

22

Time (hr)

Inje

ctio

n pr

essu

re c

hang

e (p

si)

CD = 0

CD = 100

CD = 1000

CD = 10000

1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (hr)

Obs

erva

tion

pres

sure

cha

nge

(psi

)

CD = 0

CD = 100

CD = 1000

CD = 10000

(a) (b)

0 20 40 60 80 100 120 1400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Frequency, rad/hr

Atte

nuat

ion

Sinusoidal steady stateLast 8 pulses (CD=100)Last 4 pulses (CD=100)Last 2 pulses (CD=100)Last 2 pulses (CD=1000)Last 2 pulses (CD=10000)

0 20 40 60 80 100 120 1400.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, rad/hr

Pha

se s

hift


(c) (d)

Figure 5.7: Storage effect on radial ring model: (a) injection pressure, (b) observa-tion pressure; ten harmonic frequency attributes with steady-state: (c) attenuation,(d) phase shift.


Effect of Skin Factor on Pressure Pulses

Figure 5.8 (a) represents the effect of skin on the pressure data when there is no

wellbore storage (CD = 0). A poignant difference is observed in the magnitude for

the injection pressure; meanwhile the observation pressure data remain the same.

The harmonic frequency components are gathered from the last two pulse pairs

with varying skin values (Figure 5.8 (b)). To compare the high frequency compo-

nents in detail, the cross-plot is represented with a semilog axis. As the skin factor

increases the pressure amplitudes at the wellbore, a visible reduction in amplitudes

and a slight increase in phase shift are observed. Contrary to the observation that

storage does not influence the steady-state solutions, the skin factor shifts atten-

uation and phase shift attributes. A good match with the steady-state solution is

attained especially for the first several harmonic frequency components, as sum-

marized in Table 5.8.

1 2 3 4 5 6 7 8

5

10

15

20

25

30

35

40

45

Time (hr)

Inje

ctio

n pr

essu

re c

hang

e (p

si)

s = 0s = 1s = 5

10−4

10−3

10−2

10−1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Attenuation

Pha

se s

hift

Steady state (s=0)Steady state (s=1)Steady state (s=5)Last 2 pulses (s=0)Last 2 pulses (s=1)Last 2 pulses (s=5)

Figure 5.8: Skin effect on radial ring model with CD = 0: (a) injection pressure, (b)cross-plot of attenuation and phase shift with varying skin factors.

The pressure data in Figure 5.9 shows the influence of skin effect on pressure

with a dimensionless storage of CD = 100. The last two pulses are examined in

which the condition tD > CD(60 + 3.5s) is satisfied. Figure 5.10 shows the fre-

quency attributes when combined with a storage effect. The frequency attributes


from square pulses deteriorate from the steady-state solutions. Table 5.8 shows

that the deviation is increased as the wellbore storage coefficient increases.

Interpretation of Reservoir by Pressure Pulses with Storage and Skin Effect

As demonstrated in Figure 5.11, the dominant frequency has almost the same value

for all cases when storage and skin are present. The higher the frequency the more

the frequency information deteriorates. The larger the storage and skin, the more

discrepancy with the steady-state model is observed at the high harmonic frequen-

cies. The sinusoidal steady-state model is unable to capture such change; the at-

tenuation and phase shift remain unchanged with varying storage. This is due

to the fact that frequency information is, by nature, relative information between

sourcing and observed pressure. When both pressure signals are attenuated and

delayed together, the level of storage is not captured in frequency information.

The attenuation and phase shift are used to estimate the same permeability distri-

bution regardless of storage and skin effect, but only a few low frequency points

are reliable in the steady-state space.

With skin factor, however, the magnitude of the injection pressure is changed

and thus the steady-state model changes. The combined effect of storage and skin

deteriorate the frequency attributes at high frequencies. The deviation is increased

as the wellbore storage and skin values increase.

The frequency attributes with varying skin values make the frequency method

more difficult to discern permeability distributions. Figure 5.12 shows the sinu-

soidal steady-state space for the three radial models with two skin values. The

same harmonic frequencies are used for the plot. The storage effect does not re-

flect on the steady-state models and thus these are omitted on the plot.

With these effects, the permeability estimation in sinusoidal space no longer

works with higher frequency terms. With skin effect, estimating both permeability

distribution and skin factor becomes a challenge for the frequency method.


Estimation of Wellbore Storage

The dimensionless pressure in the wellbore is directly proportional to the dimen-

sionless time at early time when the wellbore storage dominates the pressure re-

sponse. The relationship is stated as pD = tD/CD for the constant rate pressure

response shown in Figure 5.13, which can be obtained by the approach introduced

in Chapter 3. As dimensionless time and pressure are only determined with a per-

meability estimate, a storage value can be estimated after applying the frequency

method. The pressure data from the first pulse is a good candidate to calculate

dimensionless storage coefficient, CD = t∗D/p∗D.

Estimation of Skin

As the skin factor changes the steady-state space as shown in Figure 5.8 (b), it

is hard to estimate the true permeability distribution when the skin factor is un-

known. The skin effect does change the frequency values in the periodic steady

space and adds many degrees of freedom such that H(s1, k1, ω) = H(s2, k2, ω)

holds. There are many possible combinations of permeabilities and skin factor

which lie on the same plots in the attenuation and phase shift cross-plot.

When a skin factor is small, the frequency method can be used to estimate the

permeability distribution assuming zero skin effect. Then the skin factor can be

estimated by the conventional well testing method. The following equations do

not generally hold for a heterogeneous radial ring reservoir, but can be thought of

as a rule of thumb for calculating the skin factor.

Let kest be either an permeability estimate at the first ring, or an average of over

the radial distance.

m =162.6q0µB

kesth(5.1)

s = 1.1513(

p1hr − pinit

m− log

(kest

µφctrw2

)+ 3.2275

)(5.2)

A slightly different approach is to update skin factor iteratively. This process

of iteratively calculating and updating reservoir parameter with skin factor has


been introduced in earlier well testing literature; Huang et al. (1998) estimated

an average permeability and skin factor from a buildup test by history matching.

With the obtained permeability estimation at each iteration step, the skin factor can

be estimated. The two pressure points p1hr(t), pinit(t) can be read from a constant

rate pressure response at the injection as shown in Figure 5.13 and Equation 5.2 is

applied to estimate a skin factor. With the obtained skin factor, a new permeability

distribution is recalculated using the frequency method. The skin factor can be

updated accordingly. The process repeats until a skin factor no longer changes

from the previous step.

However, it is important to note that the skin factor is not guaranteed to con-

verge over iterations. When a skin factor is calculated, it is highly sensitive to the

first value or an average permeability value.

Another possible way to interpret the frequency method with a skin factor is

using the reduced permeability ks with a damaged zone of radius rs (Horne, 1995):

s =(

kks− 1)

lnrs

rw(5.3)

The frequency method in this case can be applied to estimate the reduced perme-

ability distribution. The challenge in this case is that the value of rs is typically

unknown.

Result of Permeability Estimate with Storage

Figure 5.14 shows the permeability estimate in the presence of wellbore storage of

CD = 100. Using the frequency method, the wellbore storage can be estimated

with permeability estimation by frequency method. Note that the storage value is

estimated separately from the frequency method once a constant pressure response

is obtained. Note that because storage does not affect the steady-state space, the

wellbore storage is calculated by applying the permeability estimate on the con-

stant rate pressure response, with the relation CD = t∗D/p∗D at early time. Table 5.9

summarizes the estimation performance. The estimation is not as good as the case

without wellbore storage, but a good estimate can be attained for both unknowns.


Table 5.7: Storage effect on radial ring model: mismatch of frequency attributes‖x− xss‖2/‖xss‖2 ‖θ − θss‖2/‖θss‖2

Last 8 pulses (CD = 100) 0.093 0.060Last 4 pulses (CD = 100) 0.041 0.035Last 2 pulses (CD = 100) 0.035 0.031Last 2 pulses (CD = 0) 0.0096 0.0069Last 2 pulses (CD = 1000) 0.26 0.096Last 2 pulses (CD = 10000) 1.62 0.15

Table 5.8: Skin effect: mismatch of frequency attributes‖x− xss‖2/‖xss‖2 ‖θ − θss‖2/‖θss‖2

s=0, CD = 0 0.0096 0.0069s=1, CD = 0 0.0088 0.0051s=5, CD = 0 0.0077 0.0051s=0, CD = 100 0.035 0.031s=1, CD = 100 0.037 0.038s=5, CD = 100 0.046 0.071

Table 5.9: Radial permeability and dimensionless storage estimation errorModel 1 Model 2 Model 3

ε(kr) 0.086 0.40 0.33ε(CD) 0.059 0.45 0.11


1 2 3 4 5 6 7 8

5

10

15

20

25

30

35

40

Time (hr)

Inje

ctio

n pr

essu

re c

hang

e (p

si)

s = 0s = 1s = 5

1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (hr)

Obs

erva

tion

pres

sure

cha

nge

(psi

)

s = 0s = 1s = 5

Figure 5.9: Skin effect on radial ring model with CD = 100: (a) injection pressure,(b) observation pressure

10−4

10−3

10−2

10−1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Attenuation

Pha

se s

hift


Figure 5.10: Effect of skin factors on attenuation and phase shift with CD = 100


10−4

10−3

10−2

10−1

100

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Attenuation

Pha

se s

hift

Steady state (s=0)Steady state (s=5) Square pulses (C

D =0, s=0)

Square pulses (CD =0, s=5)



Figure 5.11: Cross-plot of attenuation and phase shift with storage and skin (sum-mary).

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Attenuation

Pha

se s

hift

Model 1, s=0Model 2, s=0Model 3, s=0Model 1, s=5Model 2, s=5Model 3, s=5

Figure 5.12: Attenuation and phase shift of three models with skin factors.


10−3

10−2

10−1

100

101

0

5

10

15

20

25

Time, hr

Pre

ssur

e ch

ange

(ps

i)

CD=0, s=0

CD=0, s=5

CD=100, s=0

CD=100, s=5

Figure 5.13: Constant rate pressure response with storage and skin effects


2 5 8 11 14 17 20 23 26 290

20

40

60

80

100

120

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d

TrueFrequency method, C

D=100

2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

700

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


D=100

2 5 8 11 14 17 20 23 26 290

100

200

300

400

500

600

Distance, r, ft

Rad

ial p

erm

eabi

lity,

kr, m

d


D=100

Figure 5.14: Radial permeability estimate when CD = 100 for Model 1 (top), Model2 (left) and Model 3 (right).


5.6 Effect of Boundary Conditions

The periodically steady-state solutions formed by attenuation and phase shift at

multiple frequencies are defined differently for different extended boundary con-

ditions. Thus the parameter estimation is done specifically for each boundary con-

dition. The cross-plot in Figure 5.15 illustrates attenuation and phase shift data at

multiple frequencies when the observation well is located at the second to the last

ring at rD = 120 when the outer boundary is located at reD = 141. Among the three

boundary conditions, the no-flow boundary case demonstrated the strongest ob-

servation signal, whereas the constant pressure boundary demonstrated the weak-

est. In addition, the high frequency range is not useful in discriminating the fre-

quency attributes for the constant pressure boundary case. The three conditions

are as follows:

- Infinite reservoir: limrD→∞

pjD(rD, tD) = 0

- No flow boundary:∂pjD∂rD

∣∣∣rD=reD

= 0

- Constant pressure boundary: pjD(reD, tD) = 0

For a detailed formulation, refer to Appendix C. The illustration of how the

pressure transmits over distance is represented with attenuation and phase shift at

each ring.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Attenuation

Pha

se s

hift

InfiniteNo flowConstant pressure

Figure 5.15: Sensitivity of attenuation and phase shift with different boundary con-ditions.


5.7 Application to Field Data

The frequency method was applied on two field data examples to estimate the

radial permeability distributions. Two cases, Field data 1 and 2, which were in-

troduced earlier in Section 3.6 are demonstrated. The attenuation and phase shift

pair at multiple harmonic frequencies that are gathered from the last two pulses

are used. For field data, rw = 0.5 ft, φ= 0.05, µ=1 cp, ct = 1× 10−6 psi−1.

5.7.1 Field Data 1

Frequency Method

The frequency method was used to estimate the unknown permeability distribu-

tion on the Field data 1. The pressure data consist of 13330 points, and have five

square pulses which are sourced every 2 hr. The observation well is 405.5 ft away

from the active well. Denoted with dots in Figure 5.16 (a) were the retrieved atten-

uation and phase shift data pairs at the 20 harmonic frequencies. The data were

gathered from the last two square pulses, as demonstrated in Section 3.6.2. The

harmonic frequencies correspond to the periodicity of 2 hr, 2/3 hr, 2/5 hr, and so

on. Out of those harmonics, only a few data points conform, in physical behav-

ior, to a desirable relationship that the attenuation decreases and the phase shift

increases over the harmonic frequencies. The choice of 5, 10, and 20 attenuation

and phase shift pairs was used to estimate permeabilities for the five radial rings.

Figure 5.16 (b) shows the result of the permeability estimates by the frequency

method from the periodic steady-state space with no skin factor. The initial guess

for permeabilities for all rings was 800 md. The permeability estimates over five

radial rings fits the overall attenuation and phase shift data at multiple frequencies

in the least-squares sense as shown in Figure 5.16 (a). Using five frequencies yields

the permeability distribution that most fits the measured attenuation and phase

shift points especially at the low harmonic frequencies. Figure 5.16 (c) and (d)

are the resulting attenuation and phase shift over frequencies by the permeability


estimate by the choice of five frequencies. Despite the fluctuating trend in attenu-

ation and phase shift over frequencies, the multiple frequency contents generated

a permeability distribution that matches the measured pressure data, which are

plotted in Figure 5.16 (e) and (f) for the detrended injection and observation pres-

sure, respectively. The average pressure misfit, ‖pmeasured − pest‖ /Np, is 0.013 for

the injection pressure, and 0.00048 for the observation pressure. There are discrep-

ancies between the measured pressure data and the reconstructed one at shut-in

time.

When reconstructing the pressure, storage and skin effects were evaluated from

the constant rate pressure response by applying the findings in Section 3.6.2. Fig-

ure 5.17 (b) shows the extracted constant rate pressure response that was obtained

by the method from Section 3.6.2. The flow rate of 0.43 STB/day was used. The

dimensionless storage is calculated to be CD = 10000 at early time. When recon-

structing pressure data, the skin factor is estimated to be s = 0.2 using the relation

in Equation 5.2. It is assumed that the skin factor is minimal such that it does not

affect periodically steady state space formed by the attenuation and phase shift.

Therefore the permeability estimate by the frequency method was not adjusted.

With the obtained flow rate, the history matching was performed to compare

the performance of reconstructing pressure data. Figure 5.17 (a) shows the per-

meability estimates by the frequency method and the pressure history matching.

Figures 5.17 (b, c, d) show that a close match to the true pressure data was at-

tained with the constant rate pressure response, the injection pressure, and the

observation pressure, respectively, by both methods. It was checked that the last

two pulses where the attenuation and phase shift are gathered take place when the

wellbore and storage effect become negligible. Table 5.10 summarizes the perfor-

mance of the pressure match by two methods.


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Attenuation

Pha

se s

hift

Field data 15 harmonics10 harmonics20 harmonics

0 50 100 150 200 250 300 350 400 4500

50

100

150

200

250

300

350

400

450


Rad

ial p

erm

eabi

lity,

kr, m

d

5 harmonics10 harmonics20 harmonics

(a) (b)

0 20 40 60 80 100 120 1400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Frequency, rad/hr

Atte

nuat

ion

MeasuredEstimate

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, rad/hr

Pha

se s

hift

MeasuredEstimate

(c) (d)

Figure 5.16: Field data 1- Radial permeability estimates by the frequency method:(a) attenuation and phase shift data at 20 harmonics from the last two squarepulses, (b) radial permeability estimate results; comparison of measured and es-timated frequency data over 20 frequencies- (c) attenuation, (d) phase shift.


0 50 100 150 200 250 300 350 400 4500

50

100

150

200

250

300

350

400

450


Rad

ial p

erm

eabi

lity,

kr, m

d

History matchingFrequency method

104

105

106

107

0

1

2

3

4

5

6

7

8

9

Dimensionless time

Con

stan

t rat

e pr

essu

re (

psi)

MeasuredHistory matchingFrequency method

(a) (b)

1 2 3 4 5 6 7 8 9 10

−4

−3

−2

−1

0

1

2

3

4

Time (hr)

Inje

ctio

n pr

essu

re c

hang

e (p

si)


1 2 3 4 5 6 7 8 9 10−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time (hr)

Obs

erva

tion

pres

sure

cha

nge

(psi

)


(c) (d)

Figure 5.17: Field data 1- Reconstruction of pressure data in comparison with his-tory matching: (a) radial permeability estimate result over five rings, (b) compar-ison of reconstructed constant rate pressure responses, (c) comparison of injectionpressure, (f) comparison of observation pressure.


5.7.2 Field Data 2

The frequency method was used for the Field data 2. The pressure data consist

of 13520 points, and have five square pulses which are sourced every 2 hr. The

observation well is 405.5 ft away from the active well. The starting guess for per-

meabilities is 200 md. Figure 5.18 (a) shows the retrieved attenuation and phase

shift data pairs at 20 harmonic frequencies. The frequency data points deviate sig-

nificantly from the ideal behavior as they are scattered in the cross plot. Only a few

frequency data conform to a desirable relationship according to frequencies. The

choice of 5, 10, and 20 attenuation and phase shift pair was used to estimate perme-

abilities for the five radial rings. Figure 5.18 (b) shows the result of the frequency

method that was obtained from the periodic steady state space with no skin factor.

The permeability estimate by the set of 20 pairs of frequency data best reconstructs

the measured attenuation and phase shift at low frequencies. Figure 5.18 (c) and

(d) show that, despite the fluctuating trend in attenuation and phase shift over fre-

quencies, the permeability estimates match the measured frequency attributes in

the least-square sense. The initial guess for permeabilities for all rings was 800 md.

The multiple frequency contents generate a permeability distribution that matches

the detrended pressure data, which are plotted in Figure 5.19 (c) and (d) for the

injection pressure and the observation pressure, respectively (Table 5.10).

When reconstructing the pressure data pair, storage and skin effects were eval-

uated from the constant rate pressure response by applying the findings in Section

3.6.2. The flow rate is estimated to be 0.23 STB/day. With this flow information,

the history matching was performed to compare the performance of reconstructing

pressure data. Figure 5.19 (a) shows the permeability estimates by the frequency

method and the pressure history matching. Figures 5.19 (b, c, d) show that a close

match to the true pressure data was attained with the constant rate pressure re-

sponse, the injection pressure, and the observation pressure, respectively, by both

methods. The dimensionless storage is calculated to be CD = 7000 at early time.

The skin factor is estimated to be s = −1.5 using the relation in Equation 5.2. Be-

cause the value of the skin factor is small, it is assumed that permeability estimate


by the frequency method with no skin factor still holds true. It was checked that

the last two pulses where the attenuation and phase shift are gathered take place

when the wellbore and storage effect become negligible. Table 5.10 summarizes

the performance of permeability estimates.


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Attenuation

Pha

se s

hift


0 50 100 150 200 250 300 350 400 4500

20

40

60

80

100

120

140

160


Rad

ial p

erm

eabi

lity,

kr, m

d


(a) (b)

0 20 40 60 80 100 120 1400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Frequency, rad/hr

Atte

nuat

ion

MeasuredEstimate

0 20 40 60 80 100 120 1400.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequency, rad/hr

Pha

se s

hift

MeasuredEstimate

(c) (d)

Figure 5.18: Field data 2- Radial permeability estimates by the frequency method:(a) attenuation and phase shift data at 20 harmonics from the last two squarepulses, (b) radial permeability estimate result over five rings; comparison of mea-sured and estimated frequency data over 20 frequencies- (c) attenuation, (d) phaseshift.


0 50 100 150 200 250 300 350 400 4500

100

200

300

400

500

600

700

800


Rad

ial p

erm

eabi

lity,

kr, m

d


103

104

105

106

0

1

2

3

4

5

6

7

Dimensionless time

Con

stan

t rat

e pr

essu

re (

psi)


(a) (b)

1 2 3 4 5 6 7 8 9 10

−3

−2

−1

0

1

2

3

Time (hr)

Inje

ctio

n pr

essu

re c

hang

e (p

si)


0 2 4 6 8 10−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time (hr)

Obs

erva

tion

pres

sure

cha

nge

(psi

)


(c) (d)

Figure 5.19: Field data 2- Reconstruction of pressure data in comparison with his-tory matching: (a) radial permeability estimate result over five rings, (b) compar-ison of reconstructed constant rate pressure responses, (c) comparison of injectionpressure, (f) comparison of observation pressure.


Table 5.10: Performance of frequency method in comparison with history matchingfor Field data 1 and 2

Frequency method History matchingField 1

∥∥ pinj,measured − pinj,est∥∥

2 /Np 0.0012 0.0038∥∥pinj,measured − pinj,est∥∥

2 /Np 0.013 0.013‖pobs,measured − pobs,est‖2 /Np 0.00048 0.00068

Field 2∥∥ pinj,measured − pinj,est

∥∥2 /Np 0.0041 0.0024∥∥pinj,measured − pinj,est∥∥


5.8 Summary

The performance of the permeability estimation by frequency method was demon-

strated with three synthetic radial models and two field data sets. The robustness

of the frequency method to noise is examined both from the perspective of addi-

tive pressure noise and direct perturbation in the frequency space. The accuracy

of the permeability distribution depends highly on the number of frequency data

points, the sourcing frequency ranges, the tightness of the termination criteria. The

proximity to the true distribution is not as good as the history matching technique.

The wavelet performs as well as the frequency method, but the computational cost

is similar to the pressure history matching in this study. In terms of depicting

the permeability distribution trend, the frequency method wins over the wavelet

thresholding. Storage effect does not alter the periodic steady-state solutions, but

the discrepancy at high frequencies increases with a large storage coefficient. The

skin factor shifts the periodic steady state space, therefore it is hard to obtain skin

and permeability distribution at the same time. The periodic steady-state space

should be adjusted with different boundary conditions.

Chapter 6

Permeability Estimation on

Multilayered Model

In this chapter, the newly developed frequency method is applied to estimating

vertical permeabilities. For a partially-penetrating well with cross flow in a mul-

tilayered system, a flow is injected at a top layer and passes down to the layers at

the bottom. One homogeneous and two heterogeneous vertical permeability lay-

ers are examined as synthetic examples. Demonstration on a couple of field data

examples follow. The similar verification approach applies as in Chapter 5.

There is no definite cyclic influence defined for the multilayered model with

partial flow. Assuming that attenuation and phase shift at multiple frequencies

characterize and differentiate a reservoir with a specific permeability distribution,

the permeability estimation is expected to be successful if the cross plots of attenu-

ation vs. phase shift from multiple frequencies constitute sufficient data points to

characterize the reservoir’s permeability distribution.

6.1 Using Multiple Sinusoidal Frequencies

To determine if frequency information suffices to characterize the vertical perme-

ability distribution, multiple attenuation and phase shift pairs at different frequen-

cies were used to estimate the permeability distribution. One homogeneous model

178

CHAPTER 6. PERMEABILITY ESTIMATION ON MULTILAYERED MODEL 179

denoted as Model 4 and two heterogeneous models denoted as Model 5, 6 were

examined. Those three models were introduced previously in Section 2.4.2 where

the frequency data form an indicator characteristic for each permeability distribu-

tion. The radial permeability is constant at 100 md for all layers.

The well is perforated at the top layer where a series of periodic flow rates

transmits vertically to the bottom layer which is 18 ft down. For sensitivity analysis

with depth, refer to Section 2.4.2.

The parameter estimation is performed in obtaining the permeability distri-

bution based on the attenuation/phase-shift objective function introduced in Sec-

tion 4.1. The initial guess used for the permeability estimation was 20 md for

Model 4 and 30 md for Model 5 and Model 6. There are ten degrees of freedom

in determining permeability values for ten rings. For all synthetic examples, the

reservoir parameters are: rw = 0.5 ft, φ= 0.25, µ=5 cp, ct = 1.2× 10−5 psi−1.

The vertically homogeneous reservoir model consists of ten layers with nine

unknown vertical interlayer permeabilities to estimate, with fixed radial perme-

ability of 100 md for all layers. The pressure data are recorded at the wellbore for

both layers. The multiple attenuation and phase shift pairs at different frequen-

cies are used to estimate the permeability distribution. There are nine degrees of

freedom in determining interlayer vertical permeability values for ten layers.

For each test, the attenuation and phase shift pairs at ten frequencies were used.

Similar to the study conducted on the radial model, four different frequency ranges

from a total of 30 logarithmically spaced frequencies were selected and used for

estimation. Frequencies 1 has the lowest ten frequencies and Frequencies 3 has

the highest ten frequencies. Frequencies 4 has ten frequencies that span evenly

among the whole range.

6.1.1 Homogeneous Multilayered System - Model 4

The attenuation and phase shift values used for these four frequency ranges are

demonstrated in Figure 6.1 (a) with specific ranges. The frequency range of 9.5 min

to 26.2 days is considered. The high frequency range (Frequencies 3) shows the


best performance in obtaining estimates close to the true permeability, followed by

the evenly spaced range (Frequencies 4). This suggests that the curvature formed

by this range of frequency is the most indicative of the character of the permeability

distribution. The other ranges of frequency are less discriminating of permeabil-

ity distributions. All the frequency ranges meet the termination criteria with the

frequency misfit objective on order of 10−7 to 10−13. Figure 6.1 (b) shows the esti-

mation result. The proximity to true permeability represented by the normalized

permeability mismatch error, ‖kr,estimate − kr,true‖2/‖kr,true‖2 is shown in Table 6.2.

6.1.2 Heterogeneous Multilayered System - Model 5 and 6

The attenuation and phase shift values used for these four frequency ranges are

demonstrated in Figure 6.1 (c, e) for Model 5 and Model 6. For both models, all

ranges successfully depict the variation of permeability over the depth of investi-

gation as shown in Figure 6.1 (d, f). For Model 5, the middle range (Frequencies

2) shows the best estimation performance (Table 6.3); for Model 6, the overall

range (Frequencies 4) shows the best estimation performance, next to the middle

range (Frequencies 2) (Table 6.4). In other words, these ranges are most indicative

of the character of the permeability distribution. The performance shows rather

smoother permeability distribution than the true one, which is similar to the radial

case.


0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Attenuation

Pha

se s

hift

Frequencies 1Frequencies 2Frequencies 3Frequencies 4

2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md

TrueFrequencies 1Frequencies 2Frequencies 3Frequencies 4

(a) (b)

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Attenuation

Pha

se s

hift


2 4 6 8 10 12 14 16 180

5

10

15

20

25

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


(c) (d)

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Attenuation

Pha

se s

hift


2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


(e) (f)

Figure 6.1: Vertical permeability estimation by sinusoidal frequencies: Model 4- (a)Frequency data, (b) radial permeability estimation; Model 5- (c) frequency data, (d)radial permeability estimation; Model 6- (e) frequency data, (f) radial permeabilityestimation.


Permeability Estimation with Varying Number of Frequencies

The usage of 10, 5, 3, 2, 1 different frequencies was checked for all three models

(Figure 6.2). A frequency range which has the best estimation among the four fre-

quency ranges is selected for each model: Frequencies 2 is used for Model 4, and

Frequencies 3 is used for Model 5 and Model 6. Each choice of number of fre-

quencies is evenly spread among the range of ten frequencies. Table 6.1 shows the

permeability estimation performance by the metric ‖kr,estimate − kr,true‖2/‖kr,true‖2.

Using two or more frequencies resulted in a good estimation for the perme-

ability distribution. In general, more frequency points bring a better proximity to

the true permeability distribution which is similar to the previous radial case anal-

ysis. Using only one condition is not sufficient to resolve the reservoir with ten

degrees of freedom with the exception of Model 4 in this case. The performance of

estimation depends largely on the number of frequency data that are sufficient to

characterize the reservoir with a specific permeability distribution different from

the alternative distributions.

Table 6.1: Vertical permeability estimation error with varying number of frequen-cies

Model 4 Model 5 Model 6Number of frequencies ε(kv) ε(kv) ε(kv)

10 0.0034 0.14 0.165 0.0086 0.14 0.183 0.0072 0.16 0.182 0.032 0.19 0.181 0.091 0.21 0.41


2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


2 4 6 8 10 12 14 16 180

5

10

15

20

25

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


Figure 6.2: Vertical permeability estimate with varying number of sinusoidal fre-quencies for Model 4 (top), Model 5 (left) and Model 6 (right).


6.2 Using Harmonic Frequencies from Square Pulses

For the three models, ten frequency points were gathered to apply the frequency

method for estimating permeability between wells using square pulse sourcing.

As the odd harmonics are obtained (1ω, 3ω, 5ω, · · · , 19ω), the highest frequency

is 19 times the sourcing frequency.

Then the estimation results from frequency method were compared with his-

tory matching and wavelet thresholding technique. The total of 1024 points of

pressure data with four pulses were used for history matching with no data re-

duction; the same number of wavelet coefficients as the frequency data were gath-

ered by obtaining the largest wavelet coefficients to compare with the frequency

method. For the frequency method, attenuation and phase shift information from

the first ten harmonic frequency components are used. To make the comparison

with wavelet method, thresholding was conducted such that ten wavelet coeffi-

cients from the sixth decomposition level were used to check the estimation per-

formance.

6.2.1 Homogeneous Multilayered System - Model 4

Figure 6.3 (a) and Table 6.2 summarize the performance of three different methods.

The periodic flow was generated with a fundamental periodicity of 2 hr and a 28.1

sec sampling rate. The attenuation and phase shift at the lowest ten harmonics

were used for the estimation of interlayer vertical permeabilities. When accounting

for ten measured frequency data points, the deviation from the steady state was

on order O(10−4) for attenuation and O(10−2) for phase shift. The same inherent

limitation on the radial ring model applies to the multilayer model: the higher

the frequency value, the more discrepancy there is for the steady-state attenuation

and phase shift. Thresholding was used for wavelet coefficients decomposed at

the sixth level. For wavelet, the threshold value is determined such that the same

number of data points was used to compare the performance with the frequency

method. For this case, wavelet and pressure history matching does a better job in

estimating the true permeability distribution.


For Table 6.2 - 6.4, the error metrics used to summarize the performance are

the following: for pressure, the average pressure misfit was used, which is defined

as ‖pestimate − pmeasured‖2/

Np; for permeability, ‖kr,estimate − kr,true‖2/‖kr,true‖2 was

used.

6.2.2 Heterogeneous Multilayered System - Model 5 and 6

For both heterogeneous models, four square pulses with fundamental frequencies

of 2.5 hr and 2 hr are sourced, respectively. The lowest ten harmonics of attenua-

tion and phase shift were used for the estimation of interlayer vertical permeabili-

ties. Figure 6.3 show the comparison of three methods for each model.

Table 6.3 and Table 6.4 summarize the performance of the method by looking at

pressure and permeability misfit. The frequency method performed fairly well in

terms of revealing the vertical permeability trend. However, as seen in Figure 4.6

where pressure is reconstructed for Model 5, there is a visible difference of 2.9 psi

at the peak around 110.5 psi for the injection pressure when reconstructed with

the frequency method. The frequency method behaves similar to Model 6, where

a difference of 3.0 psi is recorded at the 102.3 psi peak value for injection. The

wavelet method produced more difference in terms of matching the true perme-

ability trend, especially for Model 6; the permeability estimation by wavelet is not

too indicative of the trend in this case, where it requires more wavelet coefficients.

6.3 Permeability Estimation with Added Pressure Noise

Robustness of Permeability Estimation to Pressure Noise

Finally, Figure 6.3 illustrates the performance of three methods with one example

of 1% noise in pressure pair. Table 5.3, 6.3, 6.4 summarize the permeability and

pressure (when reconstructed with permeability estimate) mismatch for Model 1,

2 and 3 respectively. In general the added noise deteriorate the estimation perfor-

mance, however, the wavelet method in Model 1 happened to better perform in

terms of matching the true permeability. The difference on wavelet coefficients by


the added noise is insignificant because the magnitude of the high frequency noise

is filtered as detailed coefficients (cD6 in this case). ‖wnoise − w‖2/‖w‖2 for three

coefficients from injection and observation is 1.3× 10−4 and 4.1× 10−4.


2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


(a) (b)

2 4 6 8 10 12 14 16 180

5

10

15

20

25

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


2 4 6 8 10 12 14 16 180

5

10

15

20

25

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


(c) (d)

2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

20

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


(e) (f)

Figure 6.3: Vertical permeability estimation from square pulses: Model 4- (a) with-out noise, (b) with noise; Model 5- (c) without noise, (d) with noise; Model 6- (e)without noise, (f) with noise.


Table 6.2: Model 4 - Vertical permeability estimation errorε(pinj(t)) ε(pobs(t)) ε(kv) ε(kv)

with noiseMultiple sinusoidsFrequencies 1 (2.0 - 26.2 days) - - 0.085 -Frequencies 2 (2.8 hr - 1.5 days) - - 0.084 -Frequencies 3 (9.5 min - 2.1 hr) - - 0.0034 -Frequencies 4 (9.5 min - 26.2 days) - - 0.025 -Square pulsesHistory matching 3.7× 10−4 1.6× 10−4 0.098 0.069Wavelet thresholding 1.4× 10−3 7.0× 10−4 0.12 0.084Frequency method 5.3× 10−2 1.5× 10−3 0.16 0.18


with noiseMultiple sinusoidsFrequencies 1 (2.0 - 26.2 days) - - 0.23 -Frequencies 2 (2.8 hr - 1.5 days) - - 0.13 -Frequencies 3 (9.5 min - 2.1 hr) - - 0.14 -Frequencies 4 (9.5 min -26.2 days) - - 0.16 -Square pulsesHistory matching 1.1× 10−4 1.1× 10−5 0.039 0.063Wavelet thresholding 0.012 2.8× 10−4 0.26 0.25Frequency method 0.067 9.2× 10−4 0.22 0.47



with noiseMultiple sinusoidsFrequencies 1 (2.1 - 26.2 days) - - 0.067 -Frequencies 2 (3.2 hr - 1.6 days) - - 0.029 -Frequencies 3 (11.9 min - 2.4 hr) - - 0.14 -Frequencies 4 (11.9 min - 26.2 days) - - 0.021 -Square pulsesHistory matching 7.5× 10−6 2.5× 10−6 0.10 0.090Wavelet thresholding 1.3× 10−3 2.8× 10−4 0.36 0.27Frequency method 6.3× 10−2 1.3× 10−4 0.22 0.32


6.4 Perturbation in Frequency Space

Figure 6.4 shows the permeability estimate result when frequency information is

10% greater and 10% less than the original values. Overall, the partially penetrat-

ing case is more sensitive to the change in attenuation and phase shift informa-

tion. However, this can be due to the fact that the scale of frequency information

for the partially penetrating case is smaller than for the radial permeability case.

Table 6.5 summarizes the norm difference of the permeability estimate between

conditioning on perturbed attenuation and phase shift data with the default case.

The performance of estimation is not as good as the default case.

2 4 6 8 10 12 14 16 180

5

10

15

20

25

30

35

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


2 4 6 8 10 12 14 16 180

10

20

30

40

50

60

70

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


2 4 6 8 10 12 14 16 180

10

20

30

40

50

60

70

80

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


Figure 6.4: Vertical permeabilty estimate from 10% change in attenuation andphase shift for Model 4 (top), Model 5 (left) and Model 6 (right)


Table 6.5: Vertical permeability error with perturbation to frequency dataModel 4 Model 5 Model 6

ε(kv) ε(kv) ε(kv)

Default 0.0034 0.12 0.029+10% attenuation 0.17 0.97 0.61−10% attenuation 0.21 0.61 0.61+10% phase shift 0.16 1.16 1.80−10% phase shift 0.84 1.36 1.26


Table 6.6: Storage effect on multilayered model: mismatch of frequency attributes‖x− xss‖2/‖xss‖2 ‖θ − θss‖2/‖θss‖2

Last 8 pulses (CD = 100) 0.020 0.12Last 4 pulses (CD = 100) 0.0093 0.064Last 2 pulses (CD = 100) 0.0095 0.056Last 2 pulses (CD = 0) 0.0095 0.056Last 2 pulses (CD = 1000) 0.11 0.22Last 2 pulses (CD = 10000) 0.88 0.48

6.5 Effects of Storage and Skin

The following study considers wellbore storage and skin effects at the perforated

layer. The homogeneous reservoir model with a radial permeability of 100 md and

a vertical permeability of 10 md is examined as an example. The effects and the

estimation of storage and skin are similar to the radial reservoir case, therefore the

multilayered case is demonstrated with graphs in a simple manner.

Effect of Storage on Pressure Pulses

As shown in Figure 6.5 (a, b), for the storage effect, the larger the dimensionless

CD value, the more attenuation is present. When the storage value is too high,

the observation signal is taken over by the transient upward trend and minimally

periodic. The accuracy of frequency information is evaluated compared with the

steady-state model and it is illustrated over harmonic frequencies in Figure 6.5 (c,

d). The similar interpretation applies for the storage effect in a multilayered case

as it applies for a radial case in Secion 5.5. For low CD, the last two pulses show

the best estimates for the steady-state solutions. For high CD, the time frame is

masked by a strong storage effect, a substantial deviation at harmonic frequencies

exists. The summary of the estimation performance is shown in Table 6.6.

Effect of Skin Factor on Pressure Pulses

Figure 6.6 (a) represents the skin effect on the pressure data when there is no well-

bore storage (CD = 0). Poignant difference is observed in the magnitude for the


injection pressure; meanwhile the observation pressure data are kept the same.

The harmonic frequency components are gathered from the last two pulse pairs

with varying skin values ( 6.6 (b)). The skin factor shifts attenuation and phase

shift attributes. A good match with the steady-state solution is attained especially

for the first several harmonic frequency components.

The pressure data in 6.8 shows skin effect on pressure with a dimensionless

storage of CD = 100. The last two pulses are examined in which the condition

tD > CD(60 + 3.5s) satisfies. Figure 5.10 shows the frequency attributes when

combined with a storage effect. The deviation is increased as the wellbore storage

coefficient increases.

Interpretation of Reservoir by Pressure Pulses with Storage and Skin Effect

As demonstrated in Figure 6.9, the dominant frequency almost has the same value

for all cases when storage and skin are present. The higher the frequency the more

deteriorated the frequency information becomes. The larger the storage, the more

discrepancy with the steady-state model is observed. The sinusoidal steady-state

model is unable to capture such change: the attenuation and phase shift remain

unchanged with varying storage. This is due to the fact that frequency informa-

tion is, by nature, relative information between sourcing and observed pressure.

The attenuation and phase shift are used to estimate the same permeability distri-

bution regardless of storage and skin effect, but only a few low frequency points

are reliable in the steady-state space.

With skin factor, however, the magnitude of the injection pressure is changed

and thus its steady-state model changes. The combined effect of storage and skin

deteriorate the frequency attributes at high frequencies. The deviation is increased

as the wellbore storage and skin values increase.

The frequency attributes with varying skin values make the frequency method

more difficult to discern permeability distributions. Figure 6.10 shows the sinu-

soidal steady state space for the three multilayered models with two skin values.

The storage effect does not reflect on the steady-state models and thus omitted on

the plot.


With skin effect, estimating both permeability distribution and skin factor be-

comes a challenge for the frequency method.

Estimation of Wellbore Storage and Skin

The same approach in estimating the storage and skin factor is used for multilay-

ered reservoirs as introduced in Section 5.5 and 5.5 for radial reservoir models.

6.6 Effect of Boundary Conditions

The periodically steady-state solutions formed by attenuation and phase shift at

multiple frequencies are defined differently for different extended boundary con-

ditions. The parameter estimation should be formulated differently. The cross-

plots in Figure 6.11 illustrate attenuation and phase shift data at multiple frequen-

cies when the outer boundary is located at reD = 500 and reD = 5000. When

the outer boundary is not too far, among the three boundary conditions, the no-

flow boundary case demonstrated the strongest observation signal, whereas the

constant pressure boundary demonstrated the weakest. With the outer boundary

close to infinity, only low frequencies can distinguish the frequency attributes.

The three conditions are as follows:

- Infinite reservoir: limrD→∞

pjD(rD, tD) = 0

- No flow boundary:∂pjD∂rD

∣∣∣rD=reD

= 0

- Constant pressure boundary: pjD(reD, tD) = 0

The formulation applies the same as in Section 5.6, except that the index j means

a layer for a multilayered model and a ring for a radial ring model. For a detailed

formulation, refer to Appendix C. The transmission of pressure with the three dif-

ferent boundary conditions is illustrated with the attenuation and phase shift val-

ues over depth.


0 5 10 15 20 25 300

10

20

30

40

50

60

Time (hr)

Inje

ctio

n pr

essu

re c

hang

e (p

si)

CD = 0

CD = 100

CD = 1000

CD = 10000

0 5 10 15 20 25 30−1

0

1

2

3

4

5

6

Time (hr)

Inje

ctio

n pr

essu

re c

hang

e (p

si)

CD = 0

CD = 100

CD = 1000

CD = 10000

(a) (b)

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Frequency, rad/hr

Atte

nuat

ion


0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency, rad/hr

Pha

se s

hift


(c) (d)

Figure 6.5: Storage effect on multilayered model: (a) injection pressure, (b) obser-vation pressure; ten harmonic frequency attributes with steady-state: (c) attenua-tion, (d) phase shift.


0 5 10 15 20 25 300

5

10

15

Time (hr)

Pre

ssur

e ch

ange

(ps

i)

s = 0s = 1s = 5

10−4

10−3

10−2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Attenuation

Pha

se s

hift


Figure 6.6: Skin effect on multilayered model with CD = 0: (a) injection pressure,(b) cross-plot of attenuation and phase shift with varying skin factors


0 5 10 15 20 25 300

2

4

6

8

10

12

14

Time (hr)

Pre

ssur

e ch

ange

(ps

i)

s = 0s = 1s = 5

0 5 10 15 20 25 30−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (hr)

Pre

ssur

e ch

ange

(ps

i)

s = 0s = 1s = 5

Figure 6.7: Skin effect on multilayered model with CD = 100: (a) injection pressure,(b) observation pressure

10−4

10−3

10−2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Attenuation

Pha

se s

hift


Figure 6.8: Effect of skin factors on attenuation and phase shift with CD = 100


10−5

10−4

10−3

10−2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Attenuation

Pha

se s

hift

Steady state(s=0)Steady state(s=5) Square pulses(C

D =0, s=0)

Square pulses(CD =0, s=5)



Figure 6.9: Cross-plot of attenuation and phase shift with storage and skin (sum-mary).

0 2 4 6 8 10 12 14 16

x 10−3

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Phase shift

Atte

nuat

ion

Model 4, s=0Model 5, s=0Model 6, s=0Model 4, s=5Model 5, s=5Model 6, s=5

Figure 6.10: Attenuation and phase shift of three models with skin factors


0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

Attenuation (reD

=500)

Pha

se s

hift

(reD

=50

0)


0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

Attenuation (reD

=5000)

Pha

se s

hift

(reD

=50

00)


Figure 6.11: Sensitivity of attenuation and phase shift with different boundary con-ditions: reD = 500 (left) and reD = 1000 (right).


6.7 Application to Field Data

Two field data examples were investigated to estimate the interlayer vertical per-

meabilities. The field data values were modified in time and pressure to disguise

the actual field tests. The frequency information from the last two pulses, after

the detrending process, was used for the estimation in the steady-state space. The

radial permeability was assumed to be constant with 100 md. The initial guess for

the vertical permeabilities was 20 md.

6.7.1 Field Data 3

The frequency method was used for the Field data 3 in estimating the five un-

known interlayer vertical permeabilities. The thickness of a layer is 3 ft. The

pressure data consist of 13410 time points, and have five square pulses which are

sourced every 2 hr. Figure 6.12 (a) are the retrieved attenuation and phase shift data

pair at 20 harmonic frequencies. The figure also shows the reconstructed frequency

attributes according to the permeability estimates as shown in Figure 6.12 (b) by

using 5, 10, and 20 harmonic frequencies. Except a few high frequency points, the

attenuation and phase shift pairs follow an ideal behavior, that is, the attenuation

decreases and the phase shift increases over frequencies. The frequency data from

the lowest five harmonics best fit the retrieved attenuation and phase shift. Fig-

ure 6.12 (c) and (d) describe the detailed trend over the 20 harmonic frequencies

by using the permeability estimates by five frequency data. The attenuation esti-

mate at a dominant frequency matches almost perfectly to the measured one; the

trend of the phase shift is well matched to the measured one.

When compared with the permeability estimate by history matching, the flow

rate was estimated as 0.85 STB/day. The estimated permeabilities are similar for

both methods (Figure 6.13 (a)). Figure 6.13 (c) and (d) show the match for the re-

constructed injection pressure and the observation pressure, respectively. Table 6.7

summarizes the mismatch of pressure. The history matching shows a closer match

to the measured injection pressure, while the frequency method shows a closer

match to the observation pressure. This is because the objective function for the


history matching favors reducing the gap for the data with a larger magnitude,

which is injection pressure. When reconstructing the pressure data pair, storage

and skin effects are evaluated from the constant rate pressure response by apply-

ing the findings in Section 3.6.2. Figure 6.13 (b) compares the extracted constant

rate pressure response and the reconstructed one by the two methods. The dimen-

sionless storage was calculated to be CD = 500 at early time. The skin factor is

estimated to be s = −0.3 using the relation in Equation 5.2. The skin factor was

applied after the permeability estimate is obtained in the steady state space with

no skin effect. It is assumed that the absolute value of the skin factor is small that

the effect is insignificant for the steady-state space. It was checked that the last

two pulses where the attenuation and phase shift are gathered take place when the

wellbore and storage effect become negligible.


0.01 0.02 0.03 0.04 0.050.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Attenuation

Pha

se s

hift


3 6 9 12 150

2

4

6

8

10

12

14

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


(a) (b)

0 20 40 60 80 100 120 1400.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

Frequency, rad/hr

Atte

nuat

ion

MeasuredEstimate

0 20 40 60 80 100 120 1400.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frequency, rad/hr

Pha

se s

hift

MeasuredEstimate

(c) (d)

Figure 6.12: Field data 3- Vertical permeability estimates by the frequency method:(a) attenuation and phase shift data at ten harmonics from the last two squarepulses compared with the frequency result by permeability estimates, (b) five in-terlayer vertical permeabilities; comparison of measured and estimated frequencydata over 20 harmonics- (c) attenuation, (d) phase shift.


3 6 9 12 150

1

2

3

4

5

6

7

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


103

104

105

106

0

1

2

3

4

5

6

7

8

9

Dimensionless time

Con

stan

t rat

e pr

essu

re (

psi)


(a) (b)

1 2 3 4 5 6 7 8 9 10

−4

−3

−2

−1

0

1

2

3

4

Time (hr)

Inje

ctio

n pr

essu

re c

hang

e (p

si)


1 2 3 4 5 6 7 8 9 10

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time (hr)

Obs

erva

tion

pres

sure

cha

nge

(psi

)


(c) (d)

Figure 6.13: Field data 3- Reconstruction of pressure data in comparison with his-tory matching: (a) vertical permeability estimate result, (b) comparison of recon-structed constant rate pressure responses, (c) comparison of injection pressure, (f)comparison of observation pressure.


6.7.2 Field Data 4

The frequency method was used for the Field data 4 in estimating the six unknown

interlayer vertical permeabilities. The thickness of a layer is 2 ft. The pressure data

consist of 8094 points, and have three square pulses which are sourced every 2 hr.

The source of injection is located at the perforated layer at the wellbore, and the ob-

served point is 150 ft away in the radial direction, 12 ft down from the perforated

layer, that is, pobs(t) = p7D(r = 150 f t, t). Figure 6.14 (a) are the retrieved attenu-

ation and phase shift data pair at 20 harmonic frequencies. The figure also shows

the reconstructed frequency attributes according to the permeability estimates as

shown in Figure 6.14 (b) by using 5, 10, and 20 harmonic frequencies. Except a few

high frequency points, the attenuation and phase shift pairs follow a favorable be-

havior for frequency attributes. The frequency data from the lowest five harmonics

best fit the retrieved attenuation and phase shift. Figure 6.12 (c) and (d) describe

the detailed trend over the 20 harmonic frequencies by using the permeability es-

timates by five frequency data. The estimated attenuation and phase shift is one

of Pareto optima, in that adjusting the permeability values to fit the attenuation

at the dominant frequency better with the measured attenuation would result in

the phase shift point that is worse off. Despite the fluctuating trend in attenuation

and phase shift over frequencies, the frequency method generates the permeability

distribution that fit the given frequency attributes in the least square sense.

The flow rate is projected to be 0.162 STB/day. Figure 6.15 (a) shows the per-

meability estimate result by the frequency method and history matching. The his-

tory matching results in a closer match to the measured injection pressure, while

the frequency method results in a closer match to the observation pressure (Fig-

ure 6.15 (c) and Table 6.7). When reconstructing the pressure data pair, storage and

skin effects are evaluated from the constant rate pressure response by applying the

findings in Section 3.6.2. Figure 6.15 (b) compares the extracted constant rate pres-

sure response and the reconstructed one. The dimensionless storage is calculated

to be CD = 400 at early time. The skin factor is estimated to be s = −1.3 using the

relation in Equation 5.2. It is assumed that the absolute value of the skin factor is


small that the effect is insignificant for the steady-state space. It was checked that

the last two pulses where the attenuation and phase shift are gathered take place

when the wellbore and storage effect become negligible.


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Attenuation

Pha

se s

hift


2 4 6 8 10 120

5

10

15

20

25

30

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


(a) (b)

0 20 40 60 80 100 120 1400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Frequency, rad/hr

Atte

nuat

ion

MeasuredEstimate

0 20 40 60 80 100 120 1400.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency, rad/hr

Pha

se s

hift

MeasuredEstimate

(c) (d)

Figure 6.14: Field data 4- Vertical permeability estimates by the frequency method:(a) attenuation and phase shift data at 20 harmonics from the last two squarepulses, (b) six interlayer vertical permeabilities; comparison of measured and esti-mated frequency data over 20 frequencies- (c) attenuation, (d) phase shift.


2 4 6 8 10 120

5

10

15

20

25

30

Depth, h, ft

Ver

tical

per

mea

bilit

y, k

v, md


103

104

105

106

0

1

2

3

4

5

6

7

8

9

10

Dimensionless time

Con

stan

t rat

e pr

essu

re (

psi)


(a) (b)

1 2 3 4 5 6

−4

−2

0

2

4

6

8

Time (hr)

Inje

ctio

n pr

essu

re c

hang

e (p

si)


1 2 3 4 5 6

−0.6

−0.4

−0.2

0

0.2

0.4

Time (hr)

Obs

erva

tion

pres

sure

cha

nge

(psi

)


(c) (d)

Figure 6.15: Field data 4- Reconstruction of pressure data in comparison with his-tory matching: (a) vertical permeability estimate result, (b) comparison of recon-structed constant rate pressure responses, (c) comparison of injection pressure, (f)comparison of observation pressure.


Table 6.7: Performance of frequency method in comparison with history matchingfor Field data 3 and 4

Frequency method History matchingField 3

∥∥ pinj,measured − pinj,est∥∥

2 /Np 0.0040 0.0078∥∥pinj,measured − pinj,est∥∥


Field 4∥∥ pinj,measured − pinj,est

∥∥2 /Np 0.0019 0.0037∥∥pinj,measured − pinj,est∥∥


6.8 Summary

The performance of the permeability estimation by frequency method was demon-

strated with three synthetic multilayered models and two field data sets. The ro-

bustness of the frequency method to noise is examined both from the perspective

of additive pressure noise and direct perturbation in the frequency space. At least

two frequency points were required to estimate the trend of vertical permeabili-

ties. The proximity to the true distribution is not as good as the history matching

technique. The wavelet performs as well as the frequency method, but the com-

putational cost is similar to the pressure history matching in this study. In terms

of depicting the permeability distribution trend, the frequency method wins over

the wavelet thresholding. Storage effect does not alter the periodic steady-state

solutions, but the discrepancy at high frequencies increases with a large storage

coefficient. The skin factor shifts the periodic steady-state space, therefore it is

hard to obtain skin and permeability distribution at the same time. The periodic

steady-state space should be adjusted with different boundary conditions.

Chapter 7

Conclusions and Future Work

Signal processing was conducted to obtain the attenuation and phase shift from

pressure pulse data from injection and observation points. Detrending was ap-

plied as a preprocessing to aid in better estimation of accurate frequency infor-

mation which is free of upward transient trend. Then, permeability estimation in

sinusoidal steady-state was performed and two reservoir models were examined.

The two cases, one with a radial ring model and another with a partially pene-

trating well in a multilayered reservoir, were established as a demonstration for

estimating heterogeneous permeability using frequency contents as conditioning

input data. The studies provide valuable insights into the behavior of flow rate and

pressure in general heterogeneous systems. A nonlinear optimization was used to

infer the permeability distribution that satisfies the given frequency response in-

formation. After analyzing sinusoidal tests with multiple frequencies to homoge-

neous and heterogeneous reservoir cases, the usage of multiple frequency compo-

nents from the square pulse was investigated in comparison with other methods.

The study also examined the benefits and limitations of using multiple frequencies

in estimating permeability distributions.

7.1 Conclusions

The main conclusions observed in this study are as follows:

209

CHAPTER 7. CONCLUSIONS AND FUTURE WORK 210

1. The periodic steady-state solutions were derived for a radial ring model and

a partially penetrating well in a multilayered reservoir with cross flow. The

formulations address the basis for analyzing producer and observation pair

connectivity.

2. Attenuation and phase shift pairs at a specific frequency provide the key

to characterize the frequency response of the reservoir. Such harmonic fre-

quency contents can reveal the heterogeneous character of the permeability

distribution between the point of injection and point of observation. The as-

sessment of petrophysical properties can be improved by incorporating har-

monics beyond the dominant frequency. Previously, only one dominant fre-

quency component has been utilized to calculate the average permeability

for a radial and linear flow model.

3. When the frequency range is sourced to cover the differentiating character for

the heterogeneity, the heterogeneous permeability can be revealed. The nu-

merical study using various scenarios implies that the signals from different

frequencies can be used to reveal permeabilities by reflecting different radii

of influence from the point of injected signal. When the frequency is sourced

to cover the distance with varying cyclic influence, the heterogeneous per-

meability can be revealed.

4. The lower part of the frequency domain spectrum, including the dominant

frequency component, is robust to noise because the measurement error in

the pressure data appears in the high frequency range. The higher resolution

or the sampling rate for the pressure data, the greater is the robustness to

noise. Only the lower frequency attenuation and phase shift data are reliable

for estimating permeability in the presence of noise. However the inverse

problem framework, converting attenuation and phase shift information to

permeability, shows the permeability estimate is sensitive to small variations

in the attenuation and phase shift values and termination criteria used in

parameter estimation. Obtaining frequency attributes accurately determines

the performance of the permeability estimate.


5. Compared to the direct history matching method or wavelet compression

method, the attenuation and phase shift method takes less computational ef-

fort in extracting the heterogeneous permeability without having to know

the flow rate information or match the whole pressure history. Performance-

wise, the history matching approach sets the upper bound for the quality of

permeability estimation in general. Wavelet compression is proved useful as

well for estimating permeability with the same number of input data as fre-

quency method, although no computational gain is achieved. Thresholding

is done in abstracting the signal. At the expense of having more data from

approximation and detail coefficients, a good replication of pressure data is

possible.

6. A heuristic way of detrending is established and brings closer match to the

attenuation and phase shift values in sinusoidal space than the case without

detrending. The process takes advantage of the periodic structure from the

pressure signal without having to know the value of flow rate.

7. In a square pulsing scenario, only the lower harmonic frequency contents are

useful for the frequency method to work. This is due to the fact that more dis-

crepancies to the sinusoidal steady-state solutions are observed as the pres-

sure signal content decays more for the higher frequency part. This suggests

that there exists a limitation to describe the heterogeneity of the reservoir at

a one time pulsing, however the dominant frequency part is set appropri-

ately. Based on the scope of this study, a more efficient way of obtaining the

heterogeneity can be devised, by sourcing multiple times, for instance.

8. The higher the sampling rate, the more robust the frequency method is to

measurement noise in pressure. Greater number of pressure pulses is also

helpful because the frequency information matches closer to the steady-state

solution.

9. The wellbore storage attenuates and delays the pressure response at injec-

tion and observation points. In the frequency domain, the storage effect is


not discernable in the sinusoidal steady-state space. The skin factor shifts the

attenuation and phase shift data such that with an unknown skin factor mul-

tiple permeability distributions can be possible. The attenuation and phase

shift can be used to estimate the same permeability distribution regardless

of storage and skin effect, but only a few low frequency points are reliable

in steady-state space. The larger the storage and skin, the more discrepancy

with the steady-state model is observed at the high harmonic frequencies.

The estimation of storage and skin can be done in the same manner as in

conventional well testing with a constant rate pressure response. For fre-

quency analysis, the pressure data should be gathered from the time that is

beyond the wellbore storage and skin effects, tD > CD(60 + 3.5s).

10. This method suffers from the shortcoming that the permeability estimation

depends largely on the precision of attenuation and phase shift measure-

ments. However this in many cases can be offset by greater number of fre-

quency measurements, higher sampling rate, and more pulses. The success

of the proposed procedure of using multiple attenuation and phase shift is

dependent on the proximity to the steady-state solutions.

7.2 Recommendation for Practical Pulse Test Design

and Analysis

The following recommends the overall procedure to obtain permeability distribu-

tions from pulse tests based on the study. To apply the frequency method effec-

tively, selecting the appropriate sourcing frequency whose odd-multiple frequen-

cies penetrate the reservoir with different propagation lengths should be consid-

ered beforehand.

1. Based on an expected permeability value of the reservoir and a fixed distance

between the sourcing and observation point, plot the attenuation and phase

shift from periodically steady-state solutions. For the radial flow model, refer


to Figures 2.9, 2.10, and for the multilayered model, refer to Figures 2.15, 2.16.

Determine a dominant sourcing frequency such that the attenuation values

at several odd harmonic frequencies are larger than the precision of the mea-

surement device, and phase shift values at several odd harmonic frequencies

fall within one cycle.

2. When the sourcing frequency is determined, generate several cycles of pulses

by alternating the flow and shut-in periods. Record pressure at the injection

and observation points.

3. From pressure data, detrend the data by extracting a constant rate pressure

data and deducting it from the pressure data using the heuristic method de-

scribed in Chapter 3.

4. Gather attenuation and phase shift at odd harmonic frequencies by applying

the Fourier transform. Use pressure pulses at later time. The pressure pulses

at later time are closer to meeting the periodic steady-state condition and are

beyond the wellbore storage and skin effects.

5. Estimate permeability distributions by matching periodically steady-state so-

lutions to the extracted attenuation and phase shift data. When the data show

an aberrant trend, assign unequal weights to the misfit at different frequen-

cies. The low frequency data points are honored when large weights are

assigned at low frequencies.

6. Using the permeability estimates, calculate storage and skin factor from the

constant rate pressure response using the conventional well testing method.

7.3 Future Work

Further studies based on the results and findings of the work may lead to a bet-

ter understanding and design of pressure pulse testing for various other types of

reservoirs that are not covered in the study. The series of subsequent work that


may be required to build a stronger and more practical pressure pulse testing tech-

nique with multiple frequencies could be as follows:

1. A case study will be helpful in revealing specific stratigraphic characteristics

such as a type of rock for a layer by investigating the problem with functional

geologic relationships in mind.

2. The newly developed frequency method interpretation for pressure pulse

testing can be extended to cover a wider range of topics in interference test-

ing. The scope of reservoirs are limited to the ones where analytical solutions

for a periodically steady state can be established. For instance, a horizontal

well testing environment with an active well and an observation well can be a

good candidate to apply pressure pulse testing and estimate the permeability

distribution in between. Another case to be explored is for uniformly frac-

tured reservoirs. Najurieta (1979) applied pulse testing with consideration of

interaction among fractures and the matrix rock and estimated transmissivity

of fractures.

3. The proposed frequency method could be applied to estimate water and oil

relative permeabilities. Nanba and Horne (1989) formulated the multicom-

posite system for water injection well at discretized water saturation inter-

vals and used nonlinear regression algorithm in solving the inverse problem

of water and oil relative permeabilities (krw and kro). The radius of investi-

gation used in the study was rinv =√

4η∆t, where ∆t is falloff time and η

is the average diffusivity coefficients in the water invaded zone. Once peri-

odic steady state solution is established for this application, the two relative

permeabilities can be explored at multiple frequencies as well.

4. Another possible extension would be to implement various weighting least

squares objectives scheme for the estimation to be more robust to highly non-

linear attenuation and phase shift data.

Chapter 8

Nomenclature

ct = compressibility, psi−1

fs = sampling frequency, which is the same as 1/∆t, hour−1

h = layer thickness, ft

hw = layer thickness of a perforated layer, ft

kr = permeability in radial direction, md

kv = permeability in vertical direction, md

kD = dimensionless permeability

p = constant rate pressure response, psi

pinit = initial pressure, psi

pw = wellbore pressure, psi

pwD = dimensionless wellbore pressure

q = flow rate, STB/day

r = radius, ft

rw = well bore radius, ft

rD = dimensionless distance

reD = dimensionless distance of outer boundary

rDj = dimensionless inner radius of the jth block

s = skin factor

t = time, hour

∆t = sampling time interval, hour

215

CHAPTER 8. NOMENCLATURE 216

x = attenuation

xss = attenuation for periodically steady-state

C = storage, STB/psi

CD = dimensionless storage

H(ω) = transfer function according to frequency ω

Np = number of time-series pressure data

Ns = number of pressure data per cycle

Pinj(ω) = Laplace transform in ω domain of pinj(t) in time domain

Pobs(ω) = Laplace transform in ω domain of pobs(t) in time domain

I0(·) = modified Bessel function of first kind, zero order

I1(·) = modified Bessel function of first kind, first order

K0(·) = modified Bessel function of second kind, zero order

K1(·) = modified Bessel function of second kind, first order

Tp = time of periodicity, hour

φ = porosity, fraction

µ = viscosity, cp

θ = phase shift, normalized over 2π

θss = phase shift for periodically steady-state

ω = frequency, hour−1

ωD = dimensionless frequency

ωh = harmonic frequency

Appendix A

Frequency Range and Permeability

Distribution

The frequency characteristic of a reservoir remains unchanged when the condition

is met for the frequency range and the permeability distribution. The following is

the frequency response equation for the radial ring model.

H(ωD) =

C2j−1K0(

√iωD

ηDobsrDobs) + C2j I0(

√iωD

ηDobsrDobs)

C1K0(√

iωDηD1

) + C2 I0(√

iωDηD1

)(A.1)

H(αk, ω) = H(ηD,1α·ωD) (A.2)

H(k,1α·ω) = H(ηD,

1α·ωD) (A.3)

When k j → αk j for all j blocks, ηDobs and ηD1 remains unchanged, but ωD

changes such that ωD → 1α ωD. This has the same effect as ω → 1

α ω with the

same k j. Therefore, H(αk, ω) = H(k, 1α ·ω).

The relationship holds the same for the multilayer reservoir model, as long as

kv/kr ratio is kept the same. In other words, H(αkv, αkr, ω) = H(kv, kr, 1α · ω).

217

APPENDIX A. FREQUENCY RANGE AND PERMEABILITY DISTRIBUTION218

This is because the following equations hold:

H(αkv, αkr, ω) = H(kv, kr,1α·ωD) (A.4)

H(kv, kr,1α·ω) = H(kv, kr,

1α·ωD) (A.5)

The same applies for the attenuation and phase shift, since H(ωD) = x(ωD)eiθ(ωD)

hold.

Appendix B

Tridiagonal Matrix Algorithm

(TDMA)

In solving the coefficients for the pressure, the inversion can be obtained in ob-

tained in O(n) instead of O(n3) by utilizing the structure of the matrix. The tridi-

agonal matrix algorithm, also known as the Thomas algorithm, is introduced here

for the analytical solution for the radial ring model.

The algorithm is aiming at solving the equation aixi−1 + bixi + cixi+1 = di,

where a1 = 0, cn = 0 and xn = Cn for n = 2N + 1 (William et al., 2007).

b1 c1 · · · 0

a2 b2 c2

a3 b3 · · ·0 · · · · · · cn−1

0 an bn

x1

x2

· · ·· · ·xn

=

d1

0

. . .

. . .

0

(B.1)

With modified coefficients denoted with primes, the forward step is as follows:

c′i =

c1b1

; i = 1ci

bi−c′i−1ai; i = 2, 3, ..., n− 1

(B.2)

219

APPENDIX B. TRIDIAGONAL MATRIX ALGORITHM (TDMA) 220

d′i =

d1b1

; i = 1di−d′i−1aibi−c′i−1ai

; i = 2, 3, ..., n− 1(B.3)

Then a back substitution is as follows:

xn = d′n (B.4)

xi = d′i − c′ixi+1 ; i = n− 1, n− 2, ..., 1 (B.5)

Appendix C

Different Boundary Conditions

There are three different outer boundary conditions: infinite-acting, no flow and

constant pressure. Each boundary condition generates different set of pressure

solutions, thus a different behavior of the attenuation and phase shift at multiple

frequencies as well.

C.1 Radial Composite Model

For infinite-acting condition, C2N+2 = 0 holds (Section 2.3) by which the number of

unknown coefficients is reduced to 2N + 1. For the other two boundary conditions,

the degree of freedom will be the same as the infinite acting case by stating C2N+2

in terms of C2N+1.

For no flow outer boundary conditions, we can calculate C2N+2 from C2N+1 by

the following:[∂gDN+1

∂rD

]r=reD

= 0 (C.1)

=

√iwD

ηDN+1

(−C2N+1K1(reD

√iwD

ηDN+1

) + C2N+2 I1(reD

√iwD

ηDN+1

)

)

221

APPENDIX C. DIFFERENT BOUNDARY CONDITIONS 222

Replacing C2N+2 from the above relationship, C2N+2 = C2N+1

K1(reD

√iwD

ηDN+1)

I1(reD

√iwD

ηDN+1)

, we

have the following coefficients:

a2N,3 = −K0(rDN+1

√iwD

ηDN+1

)−K1(reD

√iwD

ηDN+1)

I1(reD

√iwD

ηDN+1)

I0(rDN+1

√iwD

ηDN+1

) (C.2)

a2N+1,3 = λDN+1

√iwD

ηDN+1

K1(rDN+1

√iwD

ηDN+1

)−K1(reD

√iwD

ηDN+1)

I1(reD

√iwD

ηDN+1)

I1(rDN+1

√iwD

ηDN+1

)

(C.3)

For constant pressure outer boundary condition, the following holds:

gDN+1(reD) = 0 (C.4)

= C2N+1K0(reD

√iwD

ηDN+1

) + C2N+2 I0(reD

√iwD

ηDN+1

)

Replacing C2N+2 by the relationship as above, C2N+2 = −C2N+1

K0(reD

√iwD

ηDN+1)

I0(reD

√iwD

ηDN+1)

,

we have the following coefficients.

a2N,3 = K0(rDN+1

√iwD

ηDN+1

) +

K0(reD

√iwD

ηDN+1)

I0(reD

√iwD

ηDN+1)

I0(rDN+1

√iwD

ηDN+1

) (C.5)


a2N+1,3 = λDN+1

√iwD

ηDN+1

K1(rDN+1

√iwD

ηDN+1

) +

K0(reD

√iwD

ηDN+1)

I0(reD

√iwD

ηDN+1)

I1(rDN+1

√iwD

ηDN+1

)

(C.6)

Figure C.1 shows the trend of attenuation and phase shift over distance with a

homogeneous radial ring model with 100 md. The boundary (re) is located at 70 ft

away from the wellbore.

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Atte

nuat

ion


0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pha

se s

hift


Figure C.1: Frequency attributes with attenuation (left) and phase shift (right) overdistance with different boundary conditions for radial ring model.


C.2 Multilayered Model

For multilayered models, Park (1989) demonstrated that the outer boundary con-

ditions have only small effect on the wellbore response as long as the transition

terminates before the system reaches the boundary. The other two boundary con-

ditions:

For closed boundary [∂gjD

∂rD

]r=reD

= 0 (C.7)

For constant pressure boundary

gjD(reD, ω) = 0 (C.8)

A functional relationship Bkj = bk Ak

j applies for gjD(rD, ωD)

=N∑

k=1Aj,k[K0(σk(ωD)rD) + bk I0(σk(ωD)rD)]:

bk =

0 infinite boundaryK1(σkreD)I1(σkreD)

no flow boundary

−K0(σkreD)I0(σkreD)

constant boundary

(C.9)

An illutration of pressure transmission over depth is shown in Figure C.2, with

a periodicity of Tp = 2 hr, and ∆h = 3 ft over nine layers.


0 5 10 15 20 25 300.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Depth, h, ft

Atte

nuat

ion


0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Depth, r, ft

Pha

se s

hift


Figure C.2: Attenuation (left) and phase shift (right) over depth with differentboundary conditions for multilayered model

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Pressure Pulse Testing in Heterogeneous Reservoirs

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