Pressure Transient Formation and Well Testing, Volume 5: Convolution, Deconvolution and Nonlinear Estimation (Developments in Petroleum Science)

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.

• 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

This book is a guide to the interpretation of pressure transient data obtained from both formation and well testing applications conducted in porous, permeable media for model identification and estimation of formation parameters. It offers comprehensive theoretical as well as practical information regarding the interpretation and analysis of pressure transient data from formation and well testing.

The first few chapter present the reader with the tools and hardware including pressure gauges and metrology used in formation and well testing, before focusing on the interpretation techniques and information obtainable from pressure transient measurements. Then, the basic theory and formulas used for interpretation, flow regime identification, and analysis are given. Later chapters present the reader with the advanced topics that are essential to pressure transient interpretation, namely: 1) the convolution integral or Duhamel’s theorem for dealing with simultaneously measured pressure and flow rate data sets acquired at the same wellbore or in any other spatial locations; 2) Deconvolution techniques including the more recent robust techniques to reconstruct the influence (impulse) function of the well and reservoir system from pressure-rate and pressure-pressure data sets; 3) Nonlinear estimation methods based on the least-squares estimation (LSE) and maximum likelihood estimation (MLE) with or without prior knowledge on the formation parameters to be estimated. Unlike the LSE method, the MLE offers a simple way to estimate formation parameters by simultaneous history matching of spatially distributed pressure data sets with different, but unknown noise levels. A comprehensive theoretical treatment of both methods including gradient based optimization methods and statistical inference about the estimated parameters is also provided. The book ends with a chapter on the general principles of pressure transient test design and interpretation process.

The book is aimed at undergraduate and graduate students, and researchers in petroleum engineering, and can be used as useful reference book for undergraduate and graduate courses on pressure transient interpretation. It includes many worked simulated and field test examples to demonstrate the methodologies, and an extensive bibliography on pressure transient formation and well testing.