Laplace Transform Deconvolution and Its Application to Perturbation Solution of Non Linear Diffusivity Equation

8/9/2019 Laplace Transform Deconvolution and Its Application to Perturbation Solution of Non Linear Diffusivity Equation

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LAPLACE TRANSFORM DECONVOLUTION AND ITS APPLICATION

TO PERTURBATION SOLUTION OF NON-LINEAR

DIFFUSIVITY EQUATION

by

Mahmood Ahmadi


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A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial

fulfillment of the requirements for the degree of Doctor of Philosophy (Petroleum Engineering).

Golden, Colorado

Date: ______________________

Signed: ______________________________

Mahmood Ahmadi

Signed: ______________________________

Dr. Erdal Ozkan

Thesis Advisor

Signed: ______________________________

Dr. Luis Tenorio

Thesis Advisor

Golden, Colorado

Date:_________________________

Signed: ______________________________

Dr. Ramona M. Graves

Professor and Petroleum Engineering

Department Head


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ABSTRACT

The primary objective of this dissertation is to extend the conveniences of deconvolution to non-linear problems

of fluid flow in porous media. Unlike conventional approaches, which are based on an approximate linearization of

the problem, here the solution of the non-linear problem is linearized by a perturbation approach, which permits

term-by-term application of deconvolution. Because the proposed perturbation solution is more conveniently

evaluated in the Laplace-transform domain and the standard deconvolution algorithms are in the time-domain, an

efficient deconvolution procedure in the Laplace domain is a prerequisite. Therefore, the main objective of the

dissertation is divided into two sub-objectives: 1) the analysis of variable-rate production data by deconvolution in

the Laplace domain, and 2) the extension of perturbation solution of the nonlinear diffusivity equation governing gas

flow in porous media presented by Barreto (2011) into the Laplace domain.

For the first research objective, a new algorithm is introduced which uses inverse mirroring at the points of

discontinuity and adaptive cubic splines to approximate rate or pressure versus time data. This algorithm accurately

transforms sampled data into Laplace space and eliminates the Numerical inversion instabilities at discontinuities or

boundary points commonly encountered with the piece-wise linear approximations of the data. The approach does

not require modifications of scattered and noisy data or extrapolations of the tabulated data beyond the end values.

Practical use of the algorithm presented in this research has applications in a variety of Pressure Transient

Analysis (PTA) and Rate Transient Analysis (RTA) problems. A renewed interest in this procedure is inspired from

the need to evaluate production performances of wells in unconventional reservoirs. With this approach, we could

significantly reduce the complicating effects of rate variations or shut-ins encountered in well-performance data.

Moreover, the approach has proven to be successful in dealing with the deconvolution of highly scattered and noisy

data.

The second objective of this research focuses on the perturbation solution of the nonlinear gas diffusivity

equation in Laplace domain. This solution accounts for the nonlinearity caused by the dependency of gas properties

(viscosity-compressibility product and gas deviation factor) on pressure. Although pseudo-pressure transformation

introduced by Al-Hussainy et al. (1966) linearizes the diffusivity equation for compressible fluids (gas), the pressure

dependency of gas properties is not completely removed. Barreto (2011) presented a perturbation-based solution

using Green’s functions to deal with the remaining non-lineraities of the gas diffusion equation after pseudo-

pressure transformation. The presented work is an extension of the work of Barreto (2011) into Laplace domain. The

extension of the solution into Laplace domain is an advantage as less effort is required for numerical integration.

Moreover, solutions of different well and reservoir geometries in pressure transient analysis are broadly available in

Laplace domain. Field application of the solution will involve analysis of gas-rate data after deconvolution.


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TABLE OF CONTENTS

ABSTRACT .................................................................................................................................................. iii

LIST OF FIGURES ....................................................................................................................................... vi

LIST OF TABLES......................................................................................................................................... ix

ACKNOWLEDGEMENTS ............................................................................................................................ x

CHAPTER 1 INTRODUCTION ................................................................................................................ 1

1.1

Organization of the Thesis .......................................................................................................... 2

1.2

Motivation of Research .............................................................................................................. 3

1.3

Objectives ................................................................................................................................... 4

1.4

Method of Study ......................................................................................................................... 4

CHAPTER 2 LITERATURE REVIEW ..................................................................................................... 6

2.1

Convolution and Deconvolution ................................................................................................. 6

2.1.1

Superposition and Convolution ............................................................................................. 6

2.1.2 Rate Normalization ............................................................................................................... 8

2.1.3

Deconvolution ....................................................................................................................... 8

2.2 Solution of Non-Linear Diffusivity Equation ........................................................................... 10

CHAPTER 3

DEVELOPMENT OF THE CUBIC SPLINE BASED DECONVOLUTION METHOD . 12

3.1

Convolution and Deconvolution ............................................................................................... 12

3.2 Laplace Transformation of Sampled Functions Using Cubic Splines ...................................... 13

3.3

Inverse Mirroring at Boundaries ............................................................................................... 16

3.4

Adaptive Cubic Spline .............................................................................................................. 16

3.5 The Iseger Algorithm ................................................................................................................ 19

3.5.1

Verification Examples Using Iseger Algorithm .................................................................. 22

3.5.2 Discontinuous and Piecewise Differentiable Functions ...................................................... 23

3.5.3

Wellbore Pressure Solution for a Combined Drawdown and Buildup................................ 25

3.6

Adaptive Nonparametric Kernel Regression ............................................................................ 25

3.7 Deconvolution of Pressure Responses for a Sequence of Step-Rate Changes ......... ........... ...... 28


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3.8

Field Examples ......................................................................................................................... 32

3.9

Sandface Rate Deconvolution – Muenuier et al., (1985) Example ........................................... 32

3.10

Sandface Rate Deconvolution – Fetkovich and Vienot (1984) Example.................................. 33

3.11

Deconvolution of Variable-Rate Data-Shale-Gas Well ............................................................ 33

CHAPTER 4

LAPLACE TRANSFORMATION SOLUTION TO THE NONLINEAR

DIFFUSIVITY EQUATION ............................................................................................................. 35

4.1

Mathematical Model ................................................................................................................. 35

4.2 Asymptotic Expansion (Perturbation) ...................................................................................... 39

4.3 Variable Gas Rate Deconvolution ............................................................................................ 45

4.4 Validation ................................................................................................................................. 45

4.4.1

Solution for Wells in Infinite Slab Reservoirs .................................................................... 45

4.4.2

Solution for Wells in Closed Cylindrical Reservoirs .......................................................... 56

4.5

Discussions ............................................................................................................................... 61

CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS.............................................................. 66

5.1

Conclusions .............................................................................................................................. 66

5.2 Recommendations .................................................................................................................... 66

NOMENCLATURE ................... .......... ........... .......... ........... .......... ........... .......... .......... ........... .......... .......... 68

REFERENCES ............................................................................................................................................. 71

APPENDIX A EXAMPLE OF CONVOLUTION INTEGRAL .......... .......... ........... .......... ........... .......... . 74


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LIST OF FIGURES

Figure 3.1 Interpolation of discontinuous pressure data using cubic spline. ....................................... 17

Figure 3.2 Inverse Mirroring at discontinuous points ( 50,100,200)t . ............................................ 17

Figure 3.3 Application of inverse mirroring at discontinuous points using cubic spline

interpolation. ............................................................................................................................. 18

Figure 3.4 Inverse mirroring at both ends. .......................................................................................... 20

Figure 3.5 Application of adaptive cubic spline using simulated data. Circles show where the

adaptive cubic spline is required. ......... ........... .......... ........... .......... ........... .......... ........... .......... . 20

Figure 3.6 Application of adaptive cubic spline using field data. Circles show where the

adaptive cubic spline is required. ......... ........... .......... ........... .......... ........... .......... ........... .......... . 21

Figure 3.7 Numerical Inversion of Unit Step Function by the Iseger algorithm; Effect of thenumber of inversion points. ...................................................................................................... 22

Figure 3.8 Effect of nrp in the numerical inversion of a function by the Iseger algorithm. ................ 22

Figure 3.9 Numerical Inversion of a unit step function; comparison of Iseger and Stehfest

algorithms. ................................................................................................................................ 23

Figure 3.10 Numerical Inversion of a function with multiple step changes; comparison of Iseger

and Stehfest algorithms. ........................................................................................................... 24

Figure 3.11 Numerical Inversion of a piecewise differentiable function; comparison of Iseger

and Stehfest Algorithms. .......................................................................................................... 24

Figure 3.12 Numerical Inversion of wellbore pressure change for a combined drawdown and

buildup sequence. .................. ........... .......... ........... .......... ........... .......... ........... .......... ........... .... 26

Figure 3.13 Optimum fixed bandwidth obtained from cross-validation method. ............................... 28

Figure 3.14 Local bandwidth factor, i , applied in adaptive Nadaraya-Watson kernel estimator.

Higher lambda means higher dispersion at particular time. ......... ........... .......... ........... .......... ... 29

Figure 3.15 Nadaraya-Watson kernel estimator using Gaussian kernel regression using fixed and

adaptive bandwidth ( 0.5 ) showing improvement over ordinary kernel regression. .......... 29

Figure 3.16 Step-rate sequence for data in Table 2. ............................................................................ 30

Figure 3.17 Pressure changes corresponding to step-rate sequence shown in Figure 3.16. ................ 31

Figure 3.18 Deconvolution of pressure responses in Figure 3.17 for the step-rate sequence in

Figure 3.16. ............................................................................................................................... 31

Figure 3.19 Sandface-rate deconvolution to remove wellbore storage effect; Muenuier et al.

(1985) example. ........................................................................................................................ 32


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Figure 3.20 Sandface- Numerical inversion of pressure drop from tabulated data; Fetkovich and

Vienot (1984) example. ............................................................................................................ 33

Figure 3.21 Example of flowing gas rate and corresponding pseudo-pressure. .................................. 34

Figure 3.22 Numerical inversion of pressure drop from tabulated data: Shale gas application for

data shown in Figure 3.21. ........... .......... ........... .......... ........... .......... .......... ........... .......... .......... 34

Figure 4.1 Omega factor ( )

( )( )

t t i

t i

c cw

c

versus dimensionless pressure in an infinite acting

reservoir. ................................................................................................................................... 52

Figure 4.2 Omega factor ( )

( )( )

t t i

t i

c cw

c

versus dimensionless time in an infinite acting

reservoir. ................................................................................................................................... 53

Figure 4.3 Dimensionless pseudo-pressures ( (0) Dm ) for the data set presented in Table 4.1

(Infinite Acting Reservoir). ...................................................................................................... 53

Figure 4.4 Dimensionless first non-linear term of pseudo-pressure ( (1) Dm ) for the data set

presented in Table 4.1 (Infinite Acting Reservoir). .................... .......... ........... .......... ........... .... 54

Figure 4.5 Dimensionless pseudo-pressure ( (0) (1) D D Dm m m ) for the data set presented in

Table 4.1 (Infinite Acting Reservoir). ...................................................................................... 54

Figure 4.6 Dimensionless pseudo-pressure ( (1) Dm ); comparison of two different rates for the

reservoir and gas ....................................................................................................................... 55

Figure 4.7 Dimensionless pseudo-pressure ( (0) Dm ) for data set presented in Table 4.2 (Closed

Cylindrical Reservoir). ............................................................................................................. 62

Figure 4.8 Dimensionless first non-linear term of pseudo-pressure ( (1) Dm ) for data set presented

in Table 4.2 (Closed Cylindrical Reservoir). ............................................................................ 62

Figure 4.9 Dimensionless pseudo-pressure ( (0) (1) D D Dm m m ) for data set presented in Table

4.2 (Closed Cylindrical Reservoir). .......................................................................................... 63

Figure 4.10 Dimensionless pseudo-pressure ( (1) Dm ) comparing two different rates for reservoir

and gas properties presented in Table 4.2 (Closed Cylindrical Reservoir). The results

show that, (1) Dm is rate dependent. ........................................................................................... 63

Figure 4.11 Comparing Green’s function for both Laplace and time domain at 1 Dr . .................... 65

Figure 4.12 Comparing(0) '

'

'

( )( ) D D D

D

m t w t

t

for both Laplace and time domain at 1 Dr . .................. 65

Figure A.1 In an infinite acting reservoir ( ) D D

D

m t

t

is a decreasing function as time increases. ......... 75


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Figure A.2 In an infinite acting reservoir Greens function as a decaying function as time

increases. .................................................................................................................................. 75

Figure A.3 In an infinite acting reservoir'

' '

'

0

( )( )

Dt

D D D D D D

D

m t G t t dt

t

is a decaying function as

time increases. .......................................................................................................................... 76


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LIST OF TABLES

Table 3.1 Data for drawdown followed by build up ............................................................................ 25

Table 3.2 Data for deconvolution example ......................................................................................... 30

Table 4.1 Data for infinite acting reservoir. ........................................................................................ 55

Table 4.2 Data for closed cylindrical reservoirs. ................................................................................. 64


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ACKNOWLEDGEMENTS

First and foremost, I give thanks to the One above all of us, the omnipresent God, for giving me the strength to

march on and complete my PhD – thank you so much Dear Lord.

I would also like to express my gratitude to my compassionate advisor Dr. Erdal Ozkan for his continuous

support and valuable advice throughout my studies at Colorado School of Mines. I am highly grateful for his

immense gentleness, enthusiasm, endurance, inspiration, and motivation. His exceptional guidance and counsel has

changed my life course several times throughout my PhD research journey. “Thank You” is just not enough - thank

you very, very much.

A special thanks to my co-advisor Dr. Luis Tenorio and my thesis committee member Dr. Paul Martin for their

support, help, and guidance during my research.

I thank my thesis committee members: Dr. Hossein Kazemi, Dr. Vaughan Griffiths, and Dr. John Humphrey for

their help and guidance.

I would like to thank Dr. Mahadevan Ganesh from the Mathematics Department for all the help and insightthroughout my research.

I would also like to thank MI3 Petroleum Engineering for strong support and my colleague Mr. Oscar G.

Gonzalez for his assistance and direction.

The Marathon Center of Excellence for Reservoir Studies (MCERS) at Colorado School of Mines has been

invaluable as well as the camaraderie with my school colleagues: Ali, Ayyoub, Elham, John, Mehdi, Reza, Nasser,

Shirin, Mojtaba, Najeeb, Farshad, and Younki.

I am grateful for Denise Winn-Bower, for helping me with A LOT of administrative issues during my current

study at CSM.

Finally, I want to give a special thanks to my wife Elham for her support and motivation to accomplish this

endeavor and to my great father, mother, and family who have always been my guiding light. Their prayers have

always been my inspiration. I love you all.


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CHAPTER 1

INTRODUCTION

This dissertation presents the results of a study for a Doctor of Philosophy degree conducted at Marathon Center

of Excellence for Reservoir Studies (MCERS) in the Petroleum Engineering Department at Colorado School of

Mines.

The main objective of the dissertation is to extend the conveniences of the deconvolution to non-linear problems

of fluid flow in porous media.

The usual to approach to extend deconvolution procedures to non-linear problems in oil and gas reservoirs is to

linearize the non-linear diffusion equation in terms of a pseudo-pressure and then to apply the deconvolution. It is

well known that the pseudo-pressure approach does not completely remove the nonlinearity, but, for practical

purposes, the remaining nonlinearity is assumed to be weak and ignored. In this dissertation, the solution of the non -

linear problem is obtained by a perturbation approach, which presents the solution as a series of solutions of linear

problems. This approach permits term-by-term application of deconvolution. One practical problem still remains:The proposed perturbation solution is more conveniently evaluated in the Laplace-transform domain. The standard

deconvolution algorithms, however, are in the time-domain. Thus, the development of an efficient deconvolution

procedure in the Laplace-transform domain is a prerequisite to accomplish the main objective of this dissertation.

Therefore, the main objective of the dissertation can be expanded into two sub-objectives.

The first sub-objective of the dissertation appertains to the analysis of variable-rate reservoir performance data

in Laplace domain by deconvolution. The approach taken in this work leads to the deconvolution of variable-rate

data in the Laplace domain. Specifically, an approximate function is required to take sampled (tabulated) production

rate and pressure data into Laplace domain. Furthermore, the step-changes in the production rate during shut-in

periods lead to inaccuracy in approximating functions and instability in numerical Laplace inversion algorithms. In

this study, a cubic-spline method with piecewise linear interpolation and boundary mirroring is developed in

Laplace domain to approximate and transform the production rate and bottom-hole pressure into the Laplace

domain. This algorithm accurately transforms sampled (tabulated) data into Laplace domain and eliminates the

numerical inversion instabilities at discontinuous points or boundaries commonly encountered in the piecewise

linear approximations of the data. The developed approach does not require modifications of scattered and noisy

data or extrapolations of the tabulated data beyond the end values.

Rate and pressure measurements of wells usually include some level of noise, and due to the nature of the

deconvolution process (more specifically, deconvolution in Laplace domain), the computed underlying constant-rate

response will display oscillations, which requires some degree of smoothing. To smooth the deconvolved pressure

response, an adaptive approach using a Gaussian and Epanechnikov kernel regression is proposed. The adaptive

kernel regression proposed herein is shown to be more successful than the normal kernel regression.

Since the deconvolution algorithm and the approximating function for tabulated data are in the Laplace domain,

the solution requires a numerical Laplace inversion algorithm. Common Laplace inversion algorithms usually face

accuracy problems in dealing with functions including contributions from step-changes and discontinuities. For this


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particular problem, a numerical Laplace inversion algorithm developed by Peter Iseger (2006) and introduced to the

Petroleum Engineering field by Al-Ajmi et al. (2008) is used. The application and the accuracy of Iseger ’s numerical

Laplace inversion algorithm for functions including step-changes and discontinuities in pressure-transient analysis

have been validated in the work presented by Al-Ajmi et al. (2008). The algorithm presented by Iseger (2006)

removes the restriction of continuity (that is, it tolerates piecewise continuous functions) and provides opportunities

for many practical applications. The Iseger algorithm has been tested for several common conditions requiring the

use of piecewise-differentiable and discontinuous functions including:

the use of tabulated data in the Laplace transform domain

problems involving step-rate changes

build-up tests following a drawdown period

Mini-DST tests with a sequence of drawdown and buildup periods.

Due to the significant contribution to this research, one section of this study will be devoted to Iseger’s numerical

Laplace inversion algorithm and its verification examples. The critical parameters of the Iseger algorithm will also

be reviewed.

The second sub-objective of the dissertation involves the fundamental perturbation solution of the nonlinear gas

diffusivity equation in Laplace domain. The fundamental perturbation solution presented here will be an extension

of the solution developed by Barreto (2011), which is in terms of Green’s functions but accounts for the nonlinearity

caused by the dependency of gas properties on pressure.

The original work of Barreto (2011) was presented in the time domain. The objective of the present work is to

enhance the utility of Barreto’s approach by using the conveniences offered by the properties of Laplace transforms.

The expected benefits from extending Barreto’s solution to Laplace domain include:1) solutions of different well

and reservoir geometries in pressure transient analysis are broadly available in Laplace domain, 2) less effort is

required for numerical integration, and 3) the solution will be utilized in convolution and deconvolution in Laplace

domain as presented herein.

1.1 Organization of the Thesis

This dissertation is presented in five chapters. The preliminary results are also discussed.

Chapter 1, the introduction, describes the research objectives, the methodology, and the motivation of the

study.

Chapter 2 is the literature review of publications and papers along with the proposed research.

Chapter 3 addresses mainly the first sub-objective of the dissertation; the derivation of a cubic spline based

deconvolution method. It includes the following:

o The mathematical derivation for cubic spline method adapted with piecewise linear interpolation

in Laplace domain used in the deconvolution process.

o Inverse mirroring at both boundaries to reduce the oscillation of the response function recovered

from the deconvolution process.


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o Smoothing of the constant pressure response recovered from the deconvolution process using

adaptive kernel regression.

o A brief review of Iseger’s algorithm, along with verification examples and critical parameters. For

demonstration purposes, simulated and field data are also examined.

In Chapter 4 a perturbation solution to the nonlinear diffusivity equation is presented in the Laplace-

transform domain. Although the non-linearity displayed in the gas diffusivity equation in terms of psudo-

pressure is week, it is used for the demonstration of the solution approach in this chapter because it is the

best-known, non-linear flow problem in porous media.

Chapter 5 reviews the results of the research accomplished in this PhD dissertation and proposes

recommendations for future work.

1.2 Motivation of Research

Interest in the analysis of variable-rate production data (rate and pressure measurements as function of time) has

increased in the oil and gas industry over the last two decades due to the popularity of mini-DST tests and the

increased utilization of unconventional, tight oil and natural-gas resources. In mini-DST tests, a sequence of

production and shut-in periods gives rise to a piecewise-continuous rate and pressure behavior. In the case of

unconventional tight reservoirs, well performance data are usually available in the form of daily or monthly

production and the corresponding tubing-head pressures. However, the standard theory of pressure-transient and

well-performance analysis is based on the assumption of constant production rate or bottom-hole flowing pressure.

Duhamel’s principle expresses the bottom-hole pressures of a variable-rate production case in the form of a

convolution relationship between the variable flow rate and the pressures for the unit constant-rate production (the

influence function). Therefore, due to the immense utility in data analysis, recovering the influence function of the

convolution integral; that is, deconvolution of variable-rate responses to obtain the underlying constant-rate

responses, is of great interest in petroleum engineering.

van Everdingen and Hurst (1949) introduced the application of convolution/deconvolution in Laplace domain for

the solution of common transient-flow problems in porous media. Specifically, they highlighted the conveniences

due to the fact that the convolution of two functions turns into an algebraic product of the functions in the Laplace

transform domain. However, although convolution/deconvolution in Laplace domain provides a convenient means

of generating analytical solutions for many variable-rate problems, its use for the measured (tabulated) data is not

straightforward due to the requirement that discretized (measured and tabulated) rate and pressure data be

transformed into Laplace domain.

Several approaches have been tested regarding the deconvolution of tabulated variable-rate and pressure data in

Laplace domain (Roumboutsos-Stewart (1988) and Onur-Reynolds, (1998)). These algorithms have suffered from

problems in the numerical Laplace inversions of piecewise continuous functions. This was a problem due to the

limitations of the existing numerical inversion algorithms. Another obstacle was the construction of approximating

functions representing sampled (tabulated) data in Laplace domain. The approaches of these algorithms to this

problem required extrapolation of the sampled data beyond the limits of the sampling interval to evaluate the


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Laplace integral over the semi-infinite positive domain. This usually led to an oscillatory deconvolution result at the

late portion of the tabulated data (usually referred to as tail effect).A regularization algorithm or some type of

regression procedure was also needed to smooth the constant pressure response in all scenarios.

Moreover, complications exist in the application of Duhamel’s principle (convolution/deconvolution) regarding

the flow of real gases in porous media. Application of Duhamel’s principle requires a linear system but the

diffusivity equation of compressible fluids (gas) is non-linear. The conventional approach of using a pseudo-

pressure definition does not completely remove the pressure dependency of the gas properties and the

convolution/deconvolution of gas-well response remains non-rigorous. Therefore, a new approach is required to

extend Duhamel’s principle to variable-rate problems of gas-well performances.

1.3

Objectives

This research aims to improve the applications of convolution/deconvolution in Laplace domain and the

extension of these applications to the non-linear problems of gas flow. The specific objectives for the improvement

of Laplace-domain convolution/deconvolution include the following:

1.

Develop alternatives to existing approaches to transform tabulated data to Laplace domain.

2.

Introduce techniques to reduce oscillatory deconvolution behavior caused by the tail effect or

discontinuities of the input functions.

3.

Consider the impact of the numerical inversion algorithms on the success of Laplace-domain

deconvolution for different methods of transforming tabulated data into Laplace domain.

4.

Examine the possibility of smoothing deconvolved responses without interfering with the physical

signatures of the data.

5. Rigorously verify the theoretical basis of the ideas used in the above developments.

6.

Demonstrate the use of Laplace-domain deconvolution with simulated and field data and compare

the applications with the existing algorithms.

For the extension of Duhamel’s principle to the non-linear problems of gas flow in porous media, the following

objectives are defined:

1. Consider the use of the perturbation approach introduced by Barreto (2011) to obtain an

approximate Laplace transformation of the non-linear gas-diffusivity equation.

2.

Demonstrate the use of the deconvolution algorithm for the analysis of variable gas-rate field data.

1.4

Method of Study

The main method of this study is analytical. Existing mathematical-physical techniques are used to formulate the

problems mathematically and to obtain solutions at a desirable form and level of accuracy. Perturbation and infinite

series expansion are used to approximate solutions to non-linear diffusion equation. The non-linear solutions are

obtained in the form of small additions from a series of linearized problems. The solutions of the linearized diffusion

equation are expressed in terms of appropriate Green’s functions. Semi-analytical and numerical techniques are used

to compute solutions and approximating functions are invoked to deal with tabulated data.


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Because the linearized problems encountered in this research lend themselves to convolution-type solutions,

Laplace transformation is preferred due to the simplification of convolution relationship in the transform domain.

This also brings the necessity to deal with the numerical inversion of the results in the Laplace transform domain.

Various approximating functions are used to represent tabulated data and kernel regression is used to smooth the

solutions.

To validate the developed solutions both simulated and field data are used. The results are then compared to

commercial software results. For computational purposes, Stehfest and Iseger numerical Laplace inversion

algorithms are employed. The computational codes are written in both Fortran 77 and Fortran 90.


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CHAPTER 2

LITERATURE REVIEW

This chapter is divided into two parts: the first part discusses the convolution and deconvolution of variable-rate

data for well-test analysis and the second part discusses the development of a solution for the non-linear gas-

diffusivity equation and application of deconvolution to variable-gas-rate data in the Laplace domain.

2.1

Convolution and Deconvolution

The solution to variable-rate problems has been widely popular in petroleum engineering literature since the

seminal work of van Everdingen and Hurst (1949). In regards to this subject, various approaches and references can

be found in the literature. The approaches available in the literature have been listed and categorized by Ilk (2005).

The three main categories are as follows:

I.

Superposition and Convolution

II.

Rate Normalization and Material Balance Deconvolution

III.

Deconvolution

2.1.1 Superposition and Convolution

The term superposition simply states that, for all linear systems, the net response of the total system at a given

point in space and time is the summation of all the individual parts/stimulus that contributes to the total system.

Petroleum engineers apply superposition in space to construct solutions for bounded reservoirs, complex well

geometries, and multiple-well problems. These applications are also known as the method of images. Superposition

in time is used to construct solutions for pressure buildup and variable-rate problems from constant rate solutions.

The governing equation for fluid flow in porous media is the diffusivity equation. For the flow of liquids, due to

the small and constant fluid compressibility and small pressure gradients, the diffusion equation can be linearized for

most practical purposes and the superposition principle becomes valid. In the case of gas flow, however, fluid

properties are strong functions of pressure and the nonlinear terms in the diffusivity equation cannot be removed by

physically acceptable assumptions. A standard approach is, then, to recast the diffusivity equation in terms of a

pseudo-pressure. The pseudo-pressure definition groups the pressure and pressure-dependent fluid properties in a

new pseudo-variable in which the nonlinearity of the diffusion equation becomes weaker. For some applications,

such as pressure-transient analysis, the remaining nonlinearity of the diffusion equation may be ignored if the

viscosity-compressibility product does not change considerably over the infinite-acting period of the well test. This

assumption, however, has been shown not to hold during boundary-dominated flow of high-flow-rate wells

(Raghavan 1993). One of the major consequences of this condition is the lack of true pseudo-steady state flow in gas

wells, which makes many of the standard performance prediction tools for gas wells questionable. To remove the

nonlinearity completely, a pseudo-time definition has also been proposed, but its validity could not be agreed upon

except for pressure-buildup applications (Raghavan 1993). The details of the use of pseudo-pressure and pseudo-

time can be found in the works of Al-Hossainy et al. (1966) and Agarwal (1980).


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In mathematics, particularly in functional analysis, convolution is defined as a mathematical operation on two

functions ( ) f t and ( ) g t producing a third function that is typically viewed as a modified version of one of the

original functions is translated. In other words, the overlap amount between two functions ( ) f t and translated

function of ( ) g t can be represented by convolution. The mathematical form of the convolution of two functions ( ) f t

and ( ) g t is defined as:

0 0

( ) ( )* ( ) ( ) ( ) ( ) ( )

t t def

z t f t g t f g t d f t g d (2.1)

A discrete form for the right hand side of Eq. (2.1) over a finite interval can be written as:

1 1

1

( ) ( ) ( )

n

i i i

i

z t f g t

(2.2)

where ( ) z t is the response of the system or the convolution of the functions ( ) f t and ( ) g t and is a dummy variable.

This is also called the superposition equation, which was previously defined as the total system response and is

equal to the summation of individual stimulus that contributes to the total system.

In regards to petroleum engineering applications utilizing linearized diffusivity equation, the pressure response

of the reservoir to a sequence of fluid withdrawals at a source location can be obtained by the superposition of the

pressure responses for instantaneous withdrawals of variable volumes of fluid (variable-strength instantaneous

sources) over an interval of time. This operation can be written in the form of the convolution of the rate variations

(input) with the system’s response to an impulse function (the unit instantaneous source), which is also known as

Duhamel’s principle. In petroleum engineering terms, Duhamel’s principle states that the observed wellbore

pressures drop for a variable rate can be written as the convolution of the input rate function and the derivative of the

impulse function or constant-rate pressure response. It is also assumed that the system is in equilibrium initially

( , 0) i p r t p . The convolution integral in dimensionless form is defined in the petroleum engineering literature

as:

0 0

( )( )( ) ( ) ( )

D Dt t

sD D DwD D sD D D

d p t dq p t p t d q d

d d

(2.3)

where wD p is the dimensionless pressure for variable-rate production (system response), Dq is the dimensionless

variable production rate, and sD p is the dimensionless pressure for the constant-rate solution (influence function),

which is equal to

( ) sD D D p p t s (2.4)

Here s represents the mechanical skin damage around wellbore. The equivalent equation to Eq.(2.2) or the

discretized form for the right-hand side of Eq.(2.3) in terms of superposition can also be written as:

1 1

1

( ) ( ) ( )

n

wD D i i D i

i

p t q q p t t s

(2.5)


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van Everdingen and Hurst (1949) utilized Duhamel’s principle to obtain a dimensionless wellbore pressure-drop

solution for a continuously varying rate. The method of Odeh and Jones (1965) or the methods of Soliman (1981)

and Stewart et al. (1983) can be derived from van Everdingen and Hurst ’s solution. Generally, their approaches use

the exponential integral solution to the constant-rate wellbore response or its semi-log approximation. Fetkovich and

Vienot (1984) and Agarwal (1980) also used Duhamel’s principle to generate solutions for the variable-rate

production problems. All above methods, however, were limited to forward modeling applications since a particular

reservoir model had to be selected in advance for the constant-rate pressure function under the convolution integral.

The analysis of variable-rate production data by the modern regression analysis techniques requires backward

modeling; that is, the constant-rate pressure function under the convolution integral is determined by knowing the

field response.

2.1.2 Rate Normalization

Rate normalization is generally used when the rate is smoothly changing as a function of time. The rate

normalization method was introduced by Gladfelter et al. (1955) for the analysis of wellbore-storage-dominated

pressure buildup responses. Winestock and Colpitts (1965) employed the rate normalization for drawdown test

analysis. The rate normalization method is also used to remove the effect of wellbore storage when the sand face

rates are available. The appeal of the rate normalization method is in its simplicity (it only requires an algebraic

operation of the rate and pressure data). Raghavan (1993) showed that the rate normalization is valid if the

convolution integral can be approximated as follows:

( ) ( ) ( )c p t q t p t (2.6)

This approximation is shown to be valid when the rate and pressure are smoothly varying functions of time, which is

a restrictive condition for most production data analysis applications (Raghavan 1993).

2.1.3 Deconvolution

The analysis of pressure drop over time at constant production rate is a primary goal in well-test analysis.

Identifying reservoir characteristics from constant/unit rate pressure response by pressure-transient analysis is a

well-established practice. When variations exist in rate, it is desirable to convert the data to constant-rate production

responses before analysis; in other words, the first step in analysis becomes the identification of the unit-rate

pressure response from the convolution integral. In mathematics, the convolution integral is a linear Voltera integral

equation of the first kind. Deconvolution of variable-rate production, mathematically, is an ill-conditioned inverse

problem (Lamm, 2000), and a small error in the data or in the subsequent calculations results in much larger errors

in the answers. It should be re-emphasized that convolution/superposition and rate normalization requires that a

model to be utilized for the influence function while in the deconvolution of variable-rate production, finding the

influence function is the objective.

The deconvolution approaches of variable-rate production data proposed in the literature can be grouped into two

main categories; time domain methods and spectral methods. The most successful deconvolution algorithm in time


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9

domain is the one recently presented by von Schroeter et al. (2004) and Levitan (2003). In the progression of the

time-domain deconvolution algorithms for variable-rate data, instability of the influence function was an issue and

some artificial approaches were designed to make it stable. Coats et al. (1964) employed the prior knowledge of

constant-rate response (non-negativity) before applying constraints in their model. These additional constraints in

the model proposed by Coats (1964) increased the number of unknowns, which leads to an over-determined system.

An over-determined system requires some minimization. Kuchuk (1985) and Baygun (1997) attempted to improve

Coat’s method by applying a least squared approach and by using different constraints. Issues related to the

constraints were solved by the work presented by Schroeter et al. (2004). According to Schroeter et al. (2004),

instead of the rate-normalized pressure derivative itself, they estimated its logarithm, which makes explicit sign

constraints unnecessary. The deconvolution algorithm proposed by Schroeter et al. (2004) is a nonlinear, total least

squares problem, which means both pressure and rate data are influenced by noises. In the presence of a certain level

of noise in the input data, the deconvolved pressure response could still show some instability. Regularization or

eliminating irregular response behavior is also proposed by Schroeter et al. (2004). Various ways for the

regularization of the impulse function may be found in literature. The regularization in the work by von Schroeter etal. (2004) is based on a measure of overall curvature. Levitan (2004) presented some modifications on Shroeter et

al.’s work proposing some critical recommendations to obtain accur ate results.

Spectral methods more specifically, the Laplace transform, became a standard tool in petroleum engineering for

the analysis of transient flow problems after the classical paper of van Everdingen and Hurst (1949). Performing

deconvolution in the Laplace domain simplifies the convolution integral to the product of the transforms of the

functions. Eq. (2.4) in the Laplace domain can be written as follows:

( )( )

( )

wwc

p s p s

s q s

(2.7)

Once the tabulated pressure and the rate data are transformed into Laplace domain, the impulse function for

constant-production rate can be obtained by the simple division of the functions as shown in Eq .(2.7). The results

can be numerically inverted back to the time domain using a numerical Laplace inversion algorithm, such as those

proposed by Stehfest (1970) or Iseger (2006). In spite of the simplicity of using Eq.(2.7), Laplace transformation

suffers from two issues: firstly, Laplace transformation is defined on the entire positive axis of time from zero to

infinity, which requires some extrapolation techniques. Roumboutsos et al. (1988) and Onur and Reynolds (1998)

proposed piecewise linear and polynomial functions for transforming the data into Laplace domain with some

extrapolation techniques; but neither of them was successful to overcome the foregoing issue. Secondly, Laplace

transform of piecewise-continuous functions is well defined; however, discontinuities in the Laplace domainfunction cause problems in the main stream numerical inversion algorithms. For example, the numerical inversion

algorithm of Stehfest (1970), which is commonly used in petroleum engineering, cannot handle the discontinuities in

the function to be inverted. In the approach proposed by Ilk (2005), a combination of B-splines and numerical

inversion of the Laplace transform is used. The difference between their methods and other methods in the Laplace

domain is in the fact that B-spline was used to represent the unknown response solution. Similar to the other

methods, a limitation in their method is that the non-negativity of the response function was not ensured.


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Additionally, their regularization approach seems weaker, which makes their algorithm less tolerant to the noise in

pressure and rate data. A comparative study of recent deconvolution algorithms can be found in the work by Cinar

(2006).

As previously explained, solutions to numerous pressure transient analysis problems in the field of petroleum

engineering are provided in the Laplace transform domain. Usually these solutions do not lend themselves to an

analytical inversion, and numerical inversion is normally the only resort. The common numerical inversion

algorithm presented by Stehfest (1970) is applicable only to continuous functions and fails where discontinuities

emerge. Some other algorithms, such as Bellman (1966), Crump (1976), and Talbot (1979), were tried but their

complexity and inflexibility limited their application. The simple form of the convolution integral in Laplace

transformation and the high interest in deconvolution algorithms encouraged many researchers to look for new

Laplace inversion algorithms. Ilk (2005) applied Gaver-Wynn-Rho’s algorithm presented by Valko and Abate

(2004) in the B-spline deconvolution.

Iseger (2006) presented a new algorithm, which removes the necessity of the function to be continuous. In

addition, the algorithm provides remarkable results in the case of piecewise continuous and piecewise differentiablefunctions. In well test analysis, such functions can be tabulated as data in the Laplace transform domain,

deconvolution algorithms, and solutions that include step-rate changes in the mini-DST tests. Adapting cubic spline

with piecewise linear approximation in the Laplace domain and employing inverse boundary mirroring and kernel

regression using a Gaussian or Epanechnikov kernel function is shown in this dissertation to improve the behavior of

constant pressure response obtained by deconvolution.

2.2

Solution of Non-Linear Diffusivity Equation

The general diffusivity equation, which governs fluid flow in porous media is a non-linear equation. For slightly

compressible fluids (liquids) the variation of fluid viscosity and compressibility with pressure can be assumed

negligible. However for compressible fluids (gas) this is not a valid assumption.

The diffusivity equation for the flow of a real gas in porous media is given by:

.( ) t c p p p

p z k z t

(2.8)

Eq.(2.8) is non-linear due to the fact that the gas properties (viscosity, compressibility and real gas deviation factor)

are pressure dependent. The pseudo-pressure transformation introduced by Al-Hussainy et al. (1966) attempts to

linearize the diffusion equation and allows an analytical solution of Eq.(2.8). The pseudo-pressure proposed by Al-

Hussainy et al. (1966) is defined by

'( ) 2 '

( ') ( ')b

p

p

pm p dp

p z p (2.9)

where b p is a reference pressure. Upon applying the pseudo-pressure transformation, the gas diffusivity equation can

be written as


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2 1=( )

t c m mmk t m t

(2.10)

where the diffusivity term is defined as:

( )( ) ( )

t

k m

m c m

(2.11)

Eq.(2.10) is not completely linear since the diffusivity term given in Eq.(2.11) is a function of pressure. For

practical purposes, the quantity ( ) ( )t m c m is approximated as constant at the average pressure of the drainage area.

Some other approaches have been tried for the linearization of the gas diffusivity equation. Kale and Mattar

(1980) derived an approximate solution using a perturbation technique for radial flow with constant rate. An exact

solution using a perturbation approach was developed for constant-rate production in an infinite reservoir by Serra

and Reynolds (1990a, 1990b). In their derivation the Boltzmann transformation is employed.

In the case of variable-rate production, Duhamel’s principle can be applied if the governing equation of fluid

flow is linear. For slightly compressible fluids (liquid) this is a routine approach to well test problems. However, for

gas flow, superposition in time cannot be applied, as the gas diffusivity equation is non-linear. If the diffusivity term

is assumed to be constant, then the theory of superposition can be applied as in the works of Gupta and Andsage

(1967), Samaniego and Cinco-Ley (1991), and recently von Schroeter and Gringarten (2007).

Recently, Barreto (2011) introduced a solution to the diffusivity equation with a pressure-dependent diffusivity

term. He obtained the solution of the nonlinear diffusion equation using a perturbation approach and Green’s

Functions. Barreto’s solution is in the time domain, which requires double and/or triple numerical integrations. In

this research, Barreto’s approach is extended to Laplace domain. There are several advantages to extending

Barreto’s solution to Laplace domain: 1) solutions for different reservoir geometries in petroleum engineering are

broadly available in the Laplace domain (Ozkan and Raghavan, 1991a, b), 2) because the time integral of the

Green’s functions drops off in Laplace domain, less effort will be required for the numerical evaluation of the

solution, 3) the Laplace-domain deconvolution method proposed in this work can be used with the solution of the

nonlinear diffusion equation in Laplace domain. The solution proposed in this work may be applied to

deconvolution of gas well-test data, especially in shale-gas reservoirs.


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CHAPTER 3

DEVELOPMENT OF THE CUBIC SPLINE BASED DECONVOLUTION METHOD

Despite unequivocal advantages of using sampled well-performance data in the Laplace transform domain, time-

domain analyses of pressure and production data have been more popular lately. This is due to the fact that

unresolved problems in the transformation of sampled data into Laplace domain as opposed to the demonstrated

success of the recent real-time deconvolution algorithms. However, the transformation of sampled data into Laplace

domain has a broader range of applications than deconvolution and the limited success of the past approaches to

transforming tabulated data to Laplace domain, such as piece-wise linear approximations, is an algorithmic issue;

not a fundamental defect. Specifically, an adequate algorithm to transform the piecewise-continuous sampled data

into the Laplace space and an appropriate numerical Laplace inversion algorithm capable of processing the

exponential contributions caused by the tabulated data are essential to exploit the potential of Laplace domain

operations.

In this chapter, we introduce a new algorithm, which uses inverse mirroring at the points of discontinuity andadaptive cubic splines to approximate rate or pressure versus time data. This algorithm accurately transforms

sampled data into Laplace space and eliminates the Numerical inversion instabilities at discontinuities or boundary

points commonly encountered with the piece-wise linear approximations of the data. The approach does not require

modifications of scattered and noisy data or extrapolations of the tabulated data beyond the end values.

Practical use of the algorithm presented in this chapter has applications in a variety of Pressure Transient

Analysis (PTA) and Rate Transient Analysis (RTA) problems. A renewed interest in this procedure has arisen from

the need to evaluate the production performances of wells in unconventional reservoirs. With this approach, we

could significantly reduce the complicating effects of rate variations or shut-ins encountered in well-performance

data. Moreover, the approach has proven to be successful in dealing with the deconvolution of highly scattered and

noisy data. To illustrate the applications, typical field examples including shale-gas wells are presented in this

chapter.

3.1 Convolution and Deconvolution

Laplace transformation of a function, f t , defined for all 0t , is given by

0

st f t f s e f t dt

L (3.1)

where, f s , is the Laplace transformation of f t and s denotes the Laplace transform parameter. Convolution of

two functions yields the algebraic product of the functions in Laplace domain as follows:

1 2 1 2 1 20

t

f f t f f t d f s f s L = L (3.2)


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Eq.(3.2) has important applications in transient flow in porous media associated with variable-rate production. The

solution of pressure drop, , p M t , at a point M and time t due to variable production rate, q t , is known as

Duhamel’s principle (Duhamel, 1833) and given by the convolution of flow rate, q t , and ,c p M t as follows:

0

, ,

t

c p M t q p M t d (3.3)

In Eq.(3.3), ,c p M t is the time derivative of the pressure drop at a point M and time t for a constant (unit)

production rate. In many applications of pressure transient analysis, the constant-rate pressure response at the

wellbore, ,c

p M t , is of interest. To recover the constant-rate response from the measured data, , p M t and

q t , the deconvolution of Eq.(3.3) is required.

Application of Eq.(3.2) on Eq.(3.3) yields the following algebraic expression for deconvolution in Laplace space:

c

p s p s

sq s

(3.4)

In the field application of the Laplace-domain deconvolution (Eq.(3.4)), the input data, , p M t and, q t need to

be transformed to the Laplace domain. This requires an approximating function to properly transform sampled

(tabulated) data into Laplace space. In addition, a numerical Laplace inversion algorithm, which handles

discontinuities and singularities, is entailed. All these requirements need to be addressed to develop a successful

deconvolution algorithm in Laplace domain as deconvolution described by Eqs.(3.3) and (3.4) is an ill-condition

problem. Moreover, production-rate data frequently show step-change behavior, which causes instability in the

numerical Laplace inversion.

The objective of this chapter is to contribute to the solution of the following problems:

3.2

Laplace Transformation of Sampled Functions Using Cubic Splines

Interpolation by polynomials of degree n is widely used in practice. For various functions, the higher quality of

the interpolation might be expected with the increasing degree of the polynomials. Unfortunately, this is not always

true and the interpolation may yield oscillatory results by higher-degree of polynomials. Losing the quality of the

interpolation by using higher-degree polynomials has been discussed by Kreyszig (1999). To avoid such

oscillations, spline methods are widely applied. The mathematical idea of the spline is to replace a single high-

degree polynomial over the entire interval by several low-degree polynomials (De Boore, 2001). This is expected to

reduce the oscillation of the interpolation. In general, a spline function is a function that consists of piecewise

polynomials joined together with certain smoothing conditions. A spline function also utilizes the thn degree

piecewise polynomials to preserve ( 1)thn order derivatives at the data points. A spline function is defined by knots

and the order of the spline. In the theory of splines, the points0 1, ,..., nt t t , at which the character of the function

changes, are called knots.


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In this work, the most popular piecewise polynomial, called the natural cubic spline, is utilized. The cubic spline

preserves the first and second derivative continuity at knots. If ( )S t is given over the interval a t b with knots

defined by

0 1 2 ... na t t t t b (3.5)

then the cubic spline in each subinterval can be written in the following form:

2 3

0 1 2 3( ) , 1, 2,...,

i i i i iS t A A t A t A t i n (3.6)

The coefficientsij

A need to be determined for each subinterval.

The simplified cubic splines function for each subinterval 1i it t t with values 1i y and i y at 1it and it respectively

can be address in the form of (Atkinson, 1985)

3 3

1 1

1

1 1

1 1 1

1

( ) ( )( )

6( )( ) ( ) 1

+ ( ) ( ) ( ) 2,3,...,( ) 6

i i i i

i

i i

i i i i

i i i i i i

i i

t t M t t M S t

t t t t y t t y

t t t t M t t M i nt t

(3.7)

In Eq.(3.7), M represents the second derivative and1i it t t . Eq. (3.7) can also be rearranged as follows:

3 21 1 1

1 1

2 2

1 1 1

1 1 1 1

1 1

3 3

1 1 1 1

1 1

2

1

3 3( ) +

6( ) 6( )

3 3 1+ + +

6( ) 6

6( )

1

6

i i i i i i

i

i i i i

i i i i i i

i i i i i i i i

i i i i

i i i i i i i i

i i i i

i i i

M M t M t M S t t t

t t t t

t M t M y yt M t M t M t M t

t t t t

t M t M t y t y

t t t t

t M t t

-1

2

1 1 1 1

2,3,..., with i i

i i i i i i i

i n t t t

M t t M t M

(3.8)

Applying Eq.(3.1) on cubic spline over the interval 1 nt t t for tabulated data, such as pressure or production rate,

yields

1

1 10 0

( ) ( ) ( ) ( ) ( ) ( ) = ( )

n n

n

t t t

st st st st st

t t t

L S t S s S t e dt S t e dt S t e dt S t e dt S t e dt

(3.9)

For the set of tabulated data from 1t to nt both terms

1

0

( )

t

st S t e dt

and ( )n

st

t

S t e dt

vanish because the cubic spline

becomes zero outside the interval. Application of Eq.(3.9) to individual terms of Eq.(3.8) yields:

The Laplace transform of the first term:


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0

1 2

0 1 1

3 31 1

1 1

3 3 31 0 12 1

1 0 2 1 1

3 2

2 3 4

1

12

=6( ) 6( )

...6( ) 6( ) 6( )

3 6 6

6( )

n

n

n

t

st i i i i

i i i it

t t t

st st st n n

n nt t t

i i i

n

i i

i ii

M M M M L t t e dt

t t t t

M M M M M M t e dt t e dt t e dt

t t t t t t

t t t

s s s s M M

t t

1

-13 2

1 1 1

2 3 4

2,3,... , with3 6 6

i

i

st

i i

st i i i

e

i n t t t t t t

e s s s s

(3.10)

The Laplace transform of the second term:

0

1

0 1

2 21 1 1 1

1 1

2 21 0 0 1 1 1

1 0 1

2

2

1 1

12

3 3 3 3 =

6( ) 6( )

3 3 3 3 ...

6( ) 6( )

2 2

3 3

6( )

n

n

n

t

st i i i i i i i i

i i i it

t t

st st n n n n

n nt t

i i

n

i i i i

i ii

t M t M t M t M L t t e dt

t t t t

t M t M t M t M t e dt t e dt

t t t t

t t

s s st M t M

t t

1

3

-12

1 1

2 3

2,3,..., with

2 2

i

i

st

i i

st i i

e

i n t t t

t t e

s s s

(3.11)

The Laplace transforms to the third and the fourth terms, respectively:

2 2

1 1 1

1 1

1 1 1 1

2 2

1 1 1

1 1

2

1 1 1 1

3 3+

6( )

1

+ 6

3 3+

16( )

1+

6

i i i i i i

i i i i

i i i i i i i i

i i i i i i

ii i i i

i i i i i i i i

t M t M y y

t t t t L t

t M t M t M t M

t M t M y y

t t t t t

s st M t M t M t M

11

2

1

-1

1

2,3,..., with

i i

n st st i

i

i i

t e e

s s

i n t t t

(3.12)


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3 3

1 1 1 1

1 1

2 21 1 1 1 1

3 3

1 1 1 1

1 1

2 21 1 1 1 1

6( )

1

6

+6( )

1

6

i i i i i i i i

i i i i

i i i i i i i i i i

i i i i i i i i

i i i i

i i i i i i i i i i

t M t M t y t y

t t t t L

t M t t M t t M t M

t M t M t y t y

t t t t

t M t t M t t M t M

1

2

-1

1

2,3,..., with

i i st st

n

i

i i

e e s

i n t t t

(3.13)

In all of above equations, L denotes Laplace operator and s denotes the Laplace transform parameter.

3.3 Inverse Mirroring at Boundaries

Discontinuous points in sampled data, such as step changes in production rate or build up in the pressure data,

may cause oscillations in the approximation functions obtained (for example, by cubic spline). Transforming the

data into the Laplace domain with such oscillations in the approximation function increases the error in the

deconvolved constant pressure response. Figure 3.1 shows an example of the oscillatory behavior around

discontinuous points caused by the application of cubic-spline approximation of the data. In addition, sampled

functions are normally available over a finite interval while Laplace transformation requires the function be defined

over the positive semi-infinite domain. To take the sampled data to Laplace domain, extrapolation from zero to the

first sampling point and from the last sampling point to infinity is required. If the behavior of the tabulated function

beyond the endpoints is known [e.g. constant wellbore storage, radial flow, pseudo steady state, etc. (Al-Ajmi et al.,

2008)], then the Laplace transform of the function can be generated by the procedures suggested by Roumboutsos et

al. (1988) and Onur and Reynolds (1998). As previously noted, however, due to the property of the cubic spline, the

Laplace integration over the regions of extrapolation vanishes. To remove the remaining oscillations of the

approximating function at discontinuity points, we propose the use of inverse mirroring at these points. In this

approach, the function is extended beyond the points of discontinuity by using its inverse mirror image and the

cubic-spline interpolation is applied to each extended function obtained by individual inverse mirroring. This

reduces the oscillations of the function known as the tail effect. Figure 3.2 shows an example of inverse mirroring at

discontinuous points ( 50,100,200)t . Figure 3.3 shows the interpolation results from the application of the inverse

mirroring and cubic spline.

3.4 Adaptive Cubic Spline

Inverse mirroring at discontinuous points can be also used to extend the sampled data at both ends of the table

where the behavior of the function is unknown. In this case, the two discontinuous points are the two ends of the

data set. Figure 3.4 shows an example of inverse mirroring at the data boundaries of an arbitrary function. The

inverse mirroring at the first data point may create negative values. These data points are rejected while

transforming the extended function into the Laplace domain.


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Figure 3.1 Interpolation of discontinuous pressure data using cubic spline.

Figure 3.2 Inverse Mirroring at discontinuous points ( 50,100,200)t .


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Figure 3.3 Application of inverse mirroring at discontinuous points using cubic spline interpolation.

Field data may not be as smooth as was shown in Figure 3.3. The oscillatory nature of the field data may cause

the same effect as discontinuities on the cubic spline interpolation. The application of inverse mirroring for the

entire set of data in these cases would make the procedure impractical. Using inverse mirroring only at the two ends

of the data set as shown in Figure 3.4 may be a partial solution for this problem. Inverse mirroring at the first data

point is used to fill the gap from zero to first data point. However, It is used in the last data point to shift the tail

effect may occur during Laplace inversion to the right.

An alternative solution for data with discontinuity at any point between the first and last data points is to use an

adaptive cubic spline approach. In the adaptive cubic spline approach, piecewise linear approximations are

substituted for the function in intervals where the cubic spline approximation causes large oscillations. The

mathematical expression of the Laplace transformation of the piecewise linear approximations used in selected

segments are as follows:

The equation of a straight line over a segment can be written as:

y mx b (3.14)

where m is the slope of the line passing through knots over a particular segment, and b represents the intercept of the

line.

Slope m for a particular line between knots can be found from the following relation

1

1

i i

i i

y ym

t t

(3.15)

Similarly, the intercept of the line can also be found as follow


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1

1 1 1 1

1

i i

i i i i

i i

y yb y mt y t

t t

(3.16)

The Laplace transformation of a piecewise-linear function over an interval from 1it to it is then written as follow:

1 1 1 1

1 11

2 2

( ) ( ) ( ) ( )

1 1

i i i i

i i i i

i i i i

t t t t

st st st st

t t t t

st st st st i i

L y t y s y t e dt mt b e dt mte dt be dt

t t bm e e e e

s s s s s

(3.17)

where m and b are given in Eq. (3.15) and Eq.(3.16), respectively. A cutoff value is used to decide where the switch

between the cubic-spline and piecewise-linear approximations should occur. The cutoff value is defined as the

absolute value obtained from the difference between the cubic spline approximation and the linear approximation in

each individual segment in time domain. For practical purposes, the adaptive cubic spline approach combined with

inverse mirroring at data-set boundaries has proven to be an excellent combination for deconvolution purposes in

Laplace domain. Figure 3.5 and Figure 3.6 illustrate the application of the adaptive cubic spline with linear

approximation for both simulated and field examples.

3.5 The Iseger Algorithm

As previously noted, the Stehfest (1970) algorithm is a standard numerical Laplace inversion algorithm used in

petroleum engineering. The Stehfest algorithm, however, cannot handle discontinuous functions. As previously

discussed, in the application of the deconvolution method in transient flow problems, step-wise behavior of

production rate and other discontinuities make the Stehfest algorithm inapplicable. A newer algorithm presented by

Iseger (2006) removes the restriction on discontinuities. The Iseger algorithm is based on Poisson’s summation

formula in the form of Fourier series. In this algorithm, Poisson’s summation relates an infinite sum of Laplacetransform values to Z-transforms of the function’s values. The infinite sum is approximated by a finite sum based on

the Gaussian quadrature rule, and the time domain values of the function are computed by a Fourier Transform

algorithm (Al-Ajmi et al., 2008). The practical application of Iseger’s algorithm in transient-flow problems was

introduced by Al-Ajmi et al. (2008). In this work, the Iseger algorithm is used for deconvolution applications.

The appeal of the Iseger algorithm is in its ability to compute inverse Laplace transforms of functions with

discontinuities, singularities, and local non-smoothness, even if the points of discontinuity and singularity are not

known a-priori. The implementation of the algorithm may be slightly more involved than the Stehfest (1970)

algorithm, but it is comparable to the implementation of the other algorithms with similar capabilities.

A critical parameter used in the Iseger algorithm is computed from the following relationship:

M

T (3.18)

where T is the period for which the inversions are computed and M is the number of points at which the inverse

Laplace transforms are computed. To obtain accurate results, Al-Ajmi et al. (2008) found that 1 should be an

integer 1 ( ) Î N éë ùû and the following condition must be satisfied:


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Figure 3.4 Inverse mirroring at both ends.

Figure 3.5 Application of adaptive cubic spline using simulated data. Circles show where the adaptive cubic

spline is required.


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Figure 3.6 Application of adaptive cubic spline using field data. Circles show where the adaptive cubic spline is

required.

M M 10 ; that is, T M 10 (3.19)

To demonstrate the use of these conditions, we consider the inversion of the unit step function for a period of 20 hr

as shown in Figure 3.7. By Eq. (3.19), the minimum number of inversion points required for this case is 200 M .

If 220 points are used, the condition in Eq. (3.19) is satisfied and N 111 . Figure 3.7 shows how we obtain

an accurate inversion of the unit step function with 220 data points. If we use 103 data points, for example, the

condition in Eq. (3.19) is not satisfied and if we use 223 data points, the condition in Eq. (3.19) is satisfied but

N 1 . For both cases, inversions are not sufficiently accurate. We also note from our experience, that if the

characteristics of the function change in short intervals (e.g., rate and pressure variations due to short drawdown and

buildup periods in a mini-DST test), a large M may be required for accurate inversions.

The Iseger algorithm uses M nrp M 2 over sampling points to compute inversions at M data points. It was

recommended by Iseger (2006) to use 8nrp especially for well-behaved functions. This particular choice of nrp

was dictated by the Fast Fourier Transform originally used by Iseger, which required M and nrp to be a power of 2.

For our applications, however, we did not obtain stable inversions with 8nrp . We have found that using 3nrp

with Discrete Fourier Transforms generated more stable inversions for our applications. Figure 3.8 shows the effect

of nrp on numerical inversion of a typical pressure function encountered in fluid flow problems by using the Iseger

algorithm.


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Figure 3.7 Numerical Inversion of Unit Step Function by the Iseger algorithm; Effect of the number of inversion

points.

3.5.1 Verification Examples Using Iseger Algorithm

In this section, we present examples to verify the success of the Iseger algorithm in the inversion of functions

that are piecewise differentiable or discontinuous. We compare the results to the Stehfest algorithm to delineate the

differences from standard algorithms.

Figure 3.8 Effect of nrp in the numerical inversion of a function by the Iseger algorithm.

-1

-0.5

0

0.5

1

1.5

2

0 5 10 15 20Time, hr

U n i t S t e p F u n c t i o n ( O n e S t e p

C h a n g e )

220 data

103 data

233 data

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20

Time, hr

F u n c t i o n

nrp=3nrp=7


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3.5.2

Discontinuous and Piecewise Differentiable Functions

Figure 3.9 shows the inversion of a unit step function (Heaviside function) with the pole at 10t hr using the

Iseger and Stehfest algorithms. For the Iseger algorithm, inversion was performed with 220 M and 3nrp . Three

values of the parameter, N 6, 8, and 12, were used for the Stehfest inversions ( N controls the number of

functional evaluations in the Stehfest algorithm and, theoretically, higher N values yield better inversions). Figure

3.9 shows that the Iseger algorithm recovers the true step-wise character whereas the Stehfest algorithm smears the

function around the point of discontinuity ( 10t hr).

In Figure 3.10, we consider a function with multiple step changes. This function corresponds to the rate sequence

used in a synthetic deconvolution example shown below. For the results in Figure 3.10, we used 512 M and

3nrp in the Iseger algorithm and N 6, 8, and 12 in the Stehfest algorithm. The smearing of the function at the

points of discontinuity with the Stehfest algorithm completely destroys the local and global characteristics of the

original function. The success of the Iseger algorithm in this example is remarkable.

To show that special algorithms, such as the Iseger algorithm, may be required even when the function is

continuous but only piecewise differentiable, we present the results in Figure 3.11. To generate the function in this

example, we used the function in Figure 3.10 as the rate sequence and generated the corresponding pressure

changes. The Laplace transformation of this tabulated pressure-change function was obtained by the Roumboutsos-

Stewart (1988) algorithm (the Onur-Reynolds, 1998, algorithm generated the same results) and inverted back by the

Iseger and Stehfest algorithms. As shown in Figure 3.11, while the Stehfest algorithm returns a smooth,

continuously differentiable function, the Iseger algorithm accomplishes an accurate inversion that preserves all the

characteristics of the piecewise differentiable function.

Figure 3.9 Numerical Inversion of a unit step function; comparison of Iseger and Stehfest algorithms.

-0.5

0.0

0.5

1.0

1.5

2.0

0 5 10 15 20

Time, hr

U n i t S t e p F u n c t i o n

Original Function

Iseger

Stehfest, N=6

Stehfest, N=8

Stehfest, N=12

8/9/2019 Laplace Trans

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