Volume 57. Pressure Transient Formation and Well Testing

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