Pressure Transient Formation and Well Testing (eBook)

Convolution, Deconvolution and Nonlinear Estimation

Florian Hollaender, Fikri J. Kuchuk, Mustafa Onur (Autoren)

eBook Download: EPUB

2010 | 1. Auflage
414 Seiten
Elsevier Science (Verlag)
978-0-08-093174-6 (ISBN)

Infos

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

Inhalt

Front cover 1
Half title page 2
Previous volume contents 3
Title page 4
Copyright page 5
Dedication 6
Contents 8
Preface 12
Introduction 16
Nomenclature 22
Chapter 1. Formation and Well Testing Hardware and Test Types 28
1.1. Testing Hardware 28
1.2. Pressure Transient Test Types 50
Chapter 2. Mathematical Preliminaries and Flow Regimes 54
2.1. Introduction 54
2.2. Point-Source Solutions 58
2.3. Line-Source Solutions 62
2.4. Skin Factor 68
2.5. Wellbore Storage 69
2.6. Flow Regime Identification 73
Chapter 3. Convolution 78
3.1. Introduction 78
3.2. Convolution Integral 80
3.3. Discrete Convolution 85
3.4. Duhamel's (Superposition) Theorem and Pressure-Rate Convolution 86
3.5. Wellbore Pressure for Certain Variable Sandface Flow-Rate Schedules 93
3.6. Logarithmic Convolution (Superposition or Multirate) Analysis 97
3.7. Rate-Pressure Convolution 103
3.8. Pressure-Pressure Convolution 104
Chapter 4. Deconvolution 142
4.1. Introduction 142
4.2. Analytical Deconvolutions 146
4.3. Discrete Numerical Deconvolution without Measurement Noise 148
4.4. Deconvolution with Constraints 151
4.5. Nonlinear Least-Squares Pressure-Rate Deconvolution 153
4.6. Practicalities of Deconvolution 168
4.7. Pressure-Rate Deconvolution Examples 189
4.8. Pressure-Pressure ( p - p ) Deconvolution 203
4.9. Pressure-Pressure Deconvolution Examples 206
Chapter 5. Nonlinear Parameter Estimation 224
5.1. Introduction 225
5.2. Parameter Estimation Problem for Pressure-Transient Test Interpretation 226
5.3. Parameter Estimation Methods 230
5.4. Likelihood Function and Maximum Likelihood Estimate 232
5.5. Extension of Likelihood Function to Multiple Sets of Observed Data 240
5.6. Least-Squares Estimation Methods 241
5.7. Maximum Likelihood Estimation for Unknown Diagonal Covariance 245
5.8. Use of Prior Information in ML Estimation: Bayesian Framework 255
5.9. Simultaneous vs. Sequential History Matching of Observed Data Sets 266
5.10. Summary on MLE and LSE Methods 271
5.11. Minimization of MLE and LSE Objective Functions 273
5.12. Constraining Unknown Parameters In Minimization 278
5.13. Computation of Sensitivity Coefficients 279
5.14. Statistical Inference 280
5.15. Examples 284
Chapter 6. Pressure Transient Test Design and Interpretation 330
6.1. Introduction 330
6.2. Pressure Transient Test Design and Interpretation Workflow 332
6.3. Multiwell Interference Test Example 344
6.4. Horizontal Well Test Interpretation of a Field Example 364
References 388
Subject Index 400

Textauszug

Introduction

Fikri J. Kuchuka; Mustafa Onurb; Florian Hollaendera, a Schlumberger, b The Technical University of Istanbul

A pressure transient test is a field experiment that is, like any experiment, only partially controlled. It cannot be repeated under the same conditions, but can be rerun using the results from earlier tests (experiments). There are many ways to interpret pressure transient test data; there are many models with a set of parameters that may match the observed data, but there is only one correct and more than a few probable answers.

The primary objective of pressure transient formation and well testing is to obtain the productivity of a well and properties of the formation from downhole and/or surface pressure and flow-rate measurements. The formation and reservoir information obtained from pressure transient measurements are essential because they reflect the in situ dynamic properties of the reservoir under realistic production conditions. When pressure transient test data are incorporated with geoscience data such as geophysical, geological, core, log, etc., it considerably improves reservoir characterization. Particularly, when long-term production data are not available for undeveloped reservoirs, it is necessary to complement the volumetric estimate of oil or gas in-place with long-duration well tests to estimate well productivity and reservoir size before the optimization of the field development.

The rate change at the surface or subsurface creates pressure diffusion (transient) in porous but permeable formations. The pressure diffuses away from the wellbore deep into the formation and brings information about the properties and characteristics of the reservoir. This process is traditionally called pressure transient well testing. Pressure transient tests are also conducted with Wireline Formation Testers (WFT). Such tests are called formation pressure transient tests.

Pressure transient formation and well testing (reservoir testing) for oil, gas, and/or water exploration, and production and injection wells are two of the most powerful tools for determining well and reservoir parameters under dynamic conditions. Because they are dynamic and direct, pressure measurements provide essential information for well productivity and dynamic reservoir description and hold critical importance for exploration as well as production and reservoir engineering. Introduced in the 1920s, pressure transient well testing was first used for taking fluid samples and obtaining average reservoir pressure. Gradually, in addition to pressure and samples, formation permeability and skin (wellbore damage or stimulation) have been also obtained from transient pressure measurements. Innovation and refinements in testing hardware have made it possible to measure pressure accurately across the sandface (downhole). Although measuring downhole pressure remains one of the fundamental functions of reservoir testing, today it is possible to measure downhole flow rate, fluid density, and temperature simultaneously with pressure as a function of time and depth in the wellbore as well as taking fluid samples.

Acquiring accurate downhole pressure data is the most critical part of pressure transient testing for the interpretation. The first downhole well-test system was introduced by the Johnston brothers in the 1920s and was called the formation tester. This system was basically a packer system that temporary isolated the zone to be tested from the well hydrostatic pressure. After the packer setting, the downhole valve was opened to produce the formation fluids through the drillstring. In this system, both flow rate and pressure were measured at the surface, and bottomhole pressure was obtained from the hydrostatic pressure of the fluid in the drillstring and surface pressure measurements. O’Neill (1934) reported that the Johnston formation tester was used the first time in the Mid-Continent, Texas in 1926. In parallel, Geophysical Research Corporation of Amerada introduced the first bottomhole pressure gauge in 1929, and it was called the Amerada gauge or bomb. In 1930, Millikan and Sidwell (1931) reported that the Amerada gauge was used in several wells in Oklahoma in 1930.

Since its introduction in the 1920s, pressure transient testing has held a great promise for drilling, production, and reservoir engineers. It offers a potential to assess well condition, and to obtain formation transmissibility, reservoir pressure, and inhomogeneities, such as faults and fractures, and heterogeneities. Circular reservoirs with a constant pressure or no-flow boundary condition have been well studied since the beginning of the industry. In fact unsteady-state (transient) solutions for both constant pressure and no-flow boundary circular reservoirs as a function of time and outer radius were presented by Moore et al. (1933) and Hurst (1934), based on earlier works on heat conduction.

Furthermore, Moore et al. (1933) also presented a history match, as shown in Figure 1, of a pressure transient test to their infinite-acting 1D radial solution to estimate formation permeability. The Moore et al. (1933) pressure transient test consisted of a drawdown test, during which both pressure and flow rate were measured simultaneously, and a subsequent buildup test. Moore et al. (1933) described that the flow rate was measured from the changing annulus liquid level by using a sonic tool. Perhaps this was the first downhole flow rate measurements obtained with the downhole transient pressure. Furthermore, they attributed the change in the downhole flow rate to the wellbore storage effect during the production period. As can be seen from Figure 1, the match is excellent [digitized from a 2-by-2.5-inch graph given by Figure 2 of Moore et al. (1933)]. It should be pointed out that the match was obtained manually by trial-and-error. It should be also noticed that this is a very short test, about 2 hr, and has a few measured data points (about 10). Their infinite-acting 1D radial solution did not include skin and wellbore storage effects because van Everdingen and Hurst (1949) formulated the wellbore storage and van Everdingen (1953) and Hurst (1953) introduced the concept of damage skin about 20 years later. The five important contributions of the work of Moore et al. (1933) are:

Figure 1 The history match of drawdown and buildup tests given by Moore et al. (1933).

1. The first transient (unsteady-state) solution of pressure diffusion in 1D radial porous media,

2. The type-curves for dimensionless transient pressure versus dimensionless time for infinite acting, and both constant-pressure and no-flow boundary circular reservoirs, and also as a function of outer reservoir radius,

3. The first downhole flow rate measurements and their usage in well test interpretation,

4. The first history matching for parameter estimation, and

5. The first realization of wellbore storage effects (Ramey, 1976b).

In his classic book on unsteady-state flow problems, Muskat (1937a) presented many analytical solutions to both incompressible and compressible single-phase fluid flow in porous media and the relationship between the flow rate (input) and pressure (output) as a convolution integral (Duhamel’s principle). Furthermore, Muskat (1937b) presented a trial-and-error procedure to determine both reservoir pressure and formation permeability from downhole pressure buildup data.

Many of the modern developments in pressure transient test interpretation and the understanding of the theoretical reservoir and well behaviors have been made by applications of Laplace transforms and Green’s functions to fluid flow problems. In reservoir engineering, van Everdingen and Hurst (1949) were the first to apply Laplace transforms to solve compressible single-phase fluid flow problems in 1D radial infinite and bounded reservoirs with both constant-pressure and constant-rate inner boundary conditions on a finite-radius cylindrical wellbore and no-flow and constant-pressure outer boundary conditions.

They also presented an equation describing the wellbore storage phenomenon. Muskat (1937a) used Green’s functions to solve a few steady-state and transient flow problems.

Horner (1951) applied the superposition (Duhamel’s) principle to the constant-rate line-source solution to obtain a pressure buildup equation similar to the one described by Theis (1937). He presented an interpretation technique (now called the Horner method) to estimate both the formation permeability, the distance to a sealing fault, and to obtain the static reservoir pressure (extrapolated) from pressure buildup test data. At around the same time, Miller et al. (1950) presented a different semilog interpretation technique (now called the MDH method), where the shut-in pressure was plotted as a function of the logarithm of the shut-in time for buildup tests. The concept of damage skin was not known to Horner (1951), but fortunately, both the extrapolated reservoir pressure and the permeability obtained from the Horner and MDH methods are independent of...

Produkt-Details

• Erscheint lt. Verlag: 4.8.2010
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geologie
• Themenwelt: Technik ► Bergbau
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Wirtschaft
• ISBN-10: 0-08-093174-X / 008093174X
• ISBN-13: 978-0-08-093174-6 / 9780080931746