Pressure Transient Formation and Well Testing: Convolution, Deconvolution and Nonlinear Estimation

1 Convolution

1.1 Introduction

1.2 Convolution In tegral

1.3 Discrete Convolution

1.4 Duhamels (Superp osition) Theorem and Pressure-Rate Convolution

1.5 Wellbore Pr essure 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 Ra te-Pressure Convolution

1.8 Pressure-Pressure Convolution

< br>2 Deconvolution

2.1 Introduction

2.2 Analytica l Methods

2.3 Direct Numerical Deconvolution

2.4 Dec onvolution with Constraints

2.5 Nonlinear Least-Squares Pressu re-Rate Deconvolution

2.5.1 Deconvolution, Gauge Resolution, a nd Radius of Investigation

2.5.2 Practicalities of Deconvoluti on Implementation

2.5.3 Deconvolution Parameters Selection

2.6 Pressure-Rate Deconvolution Examples

2.6.1 Simulate d Well Test Example

2.6.2 Horizontal Field Test Example

2.6.3 Wirelin Formation Tester Example

2.7 Pressure-Pressu