# Pressure Transient Formation and Well Testing

## Convolution, Deconvolution and Nonlinear Estimation

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.

Audience
Reservoir engineers, geologists, petrophysicists

Hardbound, 414 Pages

Published: August 2010

Imprint: Elsevier

ISBN: 978-0-444-52953-4

## 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 Framework

3.8.1 Single-Parameter Linear Model Case

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

3.9 Simultaneous vs. Sequential History Matching of Multiple IPTT Data Sets

3.10 Summary on MLE and LSE Methods

3.11 Minimization of MLE and LSE Objective Functions

3.12 Constraining Unknown Parameters In Minimization

3.13 Computation of Sensitivity Coefficients

3.14 Statistical Inference for MLE

3.15 IPTT Example Applications

3.15.1 Example 1

3.15.2 Example 2

3.15.3 Example 3

3.15.4 Example 4

3.15.5 Example 5

3.15.6 Example 6