Pressure Transient Formation and Well Testing, Volume 57

1st Edition

Convolution, Deconvolution and Nonlinear Estimation

Authors: Fikri Kuchuk Mustafa Onur Florian Hollaender
Hardcover ISBN: 9780444529534
eBook ISBN: 9780080931746
Imprint: Elsevier Science
Published Date: 4th August 2010
Page Count: 414
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Table of Contents

1 Convolution

1.1 Introduction

1.2 Convolution Integral

1.3 Discrete Convolution

1.4 Duhamels (Superposition) Theorem and Pressure-Rate Convolution

1.5 Wellbore Pressure for Certain Variable Sandface Flow Rate Schedules

1.5.1 Polynomial Rate Functions

1.5.2 Exponential Flow Rate

1.6 Logarithmic Convolution (Multirate Well Testing)

1.7 Rate-Pressure Convolution

1.8 Pressure-Pressure Convolution

2 Deconvolution

2.1 Introduction

2.2 Analytical Methods

2.3 Direct Numerical Deconvolution

2.4 Deconvolution with Constraints

2.5 Nonlinear Least-Squares Pressure-Rate Deconvolution

2.5.1 Deconvolution, Gauge Resolution, and Radius of Investigation

2.5.2 Practicalities of Deconvolution Implementation

2.5.3 Deconvolution Parameters Selection

2.6 Pressure-Rate Deconvolution Examples

2.6.1 Simulated Well Test Example

2.6.2 Horizontal Field Test Example

2.6.3 Wirelin Formation Tester Example

2.7 Pressure-Pressure Deconvolution

2.8 Example Pressure-Pressure Deconvolution Examples

2.8.1 Logarithmic Convolution Example: Limited-Entry Well A

3 Nonlinear Parameter Estimation

3.1 Introduction

3.2 Overview

3.3 Parameter Estimation Methods

3.4 Likelihood Function and Maximum Likelihood Estimate

3.4.1 Single-Parameter Linear Model

3.4.2 Single-Parameter Nonlinear Model

3.5 Extension of Likelihood Function to Multiple Sets of Observed Data

3.6 Least-Squares Estimation Methods

3.7 Maximum Likelihood Estimation for Unknown Diagonal Covariance Case

3.7.1 MLE, WLS, and UWLS Estimates for a Single-Parameter Linear Model

3.7.2 A Synthetic Example Application Based on A Two-Parameter Linear Model with Three Sets of Observed Data

3.8 Use of Prior Information In MLE Estimation: The Bayesian


This reference presents a comprehensive description of flow through porous media and solutions to pressure diffusion problems in homogenous, layered, and heterogeneous reservoirs. It covers the fundamentals of interpretation techniques for formation tester pressure gradients, and pretests, multiprobe and packer pressure transient tests, including derivative, convolution, and pressure-rate and pressure-pressure deconvolution. Emphasis is placed on the maximum likelihood method that enables one to estimate error variances in pressure data along with the unknown formation parameters.

Key Features

  • Serves as a training manual for geologists, petrophysicists, and reservoir engineers on formation and pressure transient testing
  • Offers interpretation techniques for immediate application in the field
  • Provides detailed coverage of pretests, multiprobe and packer pressure transient tests, including derivative, convolution, and pressure-rate and pressure-pressure deconvolution


Reservoir engineers, geologists, petrophysicists


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© Elsevier Science 2010
Elsevier Science
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About the Authors

Fikri Kuchuk Author

Affiliations and Expertise

Schlumberger SRPC, F-92142 Clamart, France

Mustafa Onur Author

Affiliations and Expertise

Technical University of Istanbul, 34469 Maslak, Turkey

Florian Hollaender Author

Affiliations and Expertise

Schlumberger Middle East SA, Abu Dhabi, UAE