Masters Thesis: Well Testing in the Framework of System Identification

Delft Center for Systems and Control Delft University of Technology

Master of Science Thesis

Well Testing in the Framework of

System IdentificationAn exploratory study

I.T.J. Kuiper

February 17, 2009

Well Testing in the Framework of

System IdentificationAn exploratory study

Master of Science Thesis

For obtaining the degree of Master of Science in Applied Physics at

Delft University of Technology

I.T.J. Kuiper

February 17, 2009

Supervisors:

prof.dr.ir. P.M.J. Van den Hof DUTprof.dr.ir. J.D. Jansen DUT, Shelldr.ir. A.J. den Dekker DUTdr.ir. X.J.A. Bombois DUTdr.ir. S.G. Douma Shell

Faculty of Applied Sciences · Delft University of Technology

The work in this thesis was supported by Shell International E&P, Exploratory Research.

Delft University of Technology

Copyright c© Delft Center for Systems and ControlAll rights reserved.

Abstract

Well testing has been the subject of research for many decades. Well tests are performed inorder to estimate reservoir properties. These estimations are needed to predict the amountof oil that can be produced, and to determine a strategy how to produce this predictedamount. The current method to estimate the reservoir properties seems cumbersome andinefficient. Also, uncertainty regions of the estimated properties are not yet available. Inthis report the focus is on evaluating and improving the current well testing methods froma system identification point of view. The latest and best performing well testing methodin the literature is evaluated by simulations and a new well testing method using PredictionError Identification (PEI) is introduced. PEI is a black box identification method. The twomethods are compared for various reservoirs. Both methods contribute to improvements inthe field of well testing. The new PEI based method estimates a full expression for the system,consisting of the reservoir and wellbore. A number of recommendations for further researchare given.

Keywords: well testing, system identification, uncertainty regions

MSc. Thesis I.T.J. Kuiper

iv Abstract

I.T.J. Kuiper MSc. Thesis

Table of Contents

Abstract iii

Nomenclature xiii

Acronyms xvii

Index xix

1 Introduction 1

1-1 Oil Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2 Well Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1-2-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-2-2 The Purpose of Well Testing . . . . . . . . . . . . . . . . . . . . . . . . 2

1-2-3 What is a Well Test? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-2-4 Relevant Reservoir Properties . . . . . . . . . . . . . . . . . . . . . . . . 5

Skin Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Wellbore Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Reservoir Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-3 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1-4 Well Testing in the Framework of System Identification . . . . . . . . . . . . . . 6

1-5 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Oil Reservoir Model 11

2-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112-2 Derivation of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . 11

2-3 Solution of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . 13

2-3-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132-3-2 Conditions of Analytical Solutions . . . . . . . . . . . . . . . . . . . . . 14

2-3-3 Analytical Solutions of the Diffusivity Equation . . . . . . . . . . . . . . 15

2-4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18


vi Table of Contents

3 Current Well Testing Practice 19

3-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3-2 Well Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3-3 Well Test Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3-4 Drawbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3-5 Improvement Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3-6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 System Identification 25

4-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4-2 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4-3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4-4 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4-5 Identification by Von Schroeter . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4-5-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4-5-2 Detailed Method Description . . . . . . . . . . . . . . . . . . . . . . . . 30

4-5-3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4-5-4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4-6 Prediction Error Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4-6-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4-6-2 Detailed Method Description . . . . . . . . . . . . . . . . . . . . . . . . 34

4-6-3 Uncertainty of PEI Estimator . . . . . . . . . . . . . . . . . . . . . . . . 37

4-6-4 Different Model Structures . . . . . . . . . . . . . . . . . . . . . . . . . 39

4-6-5 Prediction Error Identification Applied to Well Testing . . . . . . . . . . 40

Preparatory experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4-6-6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4-7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Results 45

5-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5-2 Model Order Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5-3 Infinite Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5-3-1 Estimation of Reservoir Properties . . . . . . . . . . . . . . . . . . . . . 52

5-4 Finite Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5-5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Conclusions and Recommendations 57

6-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6-2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6-3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Bibliography 63


Table of Contents vii

A The Diffusion Equation and Solutions 65

A-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A-2 Derivation of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . 65

A-3 Solutions of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . 68

B Errors in SPE 77688 73

C Reservoir Parameters and Frequency Domain 75

C-1 Reservoir Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

C-1-1 Skin and permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

D Well Test Simulation: MoReS 79

D-1 Why Using a Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

D-2 Demands for Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

D-3 Alternatives for Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

E Software 81


viii Table of Contents


List of Figures

1-1 Schematic representation of an oil reservoir. . . . . . . . . . . . . . . . . . . . . 2

1-2 Reservoir modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1-3 Well testing in broader perspective . . . . . . . . . . . . . . . . . . . . . . . . . 3

1-4 Schematic representation of a wellbore. . . . . . . . . . . . . . . . . . . . . . . 4

1-5 Reduction of production loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2-1 Pressure distribution for transient regime. . . . . . . . . . . . . . . . . . . . . . 16

2-2 Illustration of the transient regime. . . . . . . . . . . . . . . . . . . . . . . . . . 16

2-3 Example of a simulated step response. . . . . . . . . . . . . . . . . . . . . . . . 17

2-4 Pressure distribution for semi steady state regime. . . . . . . . . . . . . . . . . . 17

2-5 Pressure distribution for steady state regime. . . . . . . . . . . . . . . . . . . . . 18

3-1 Semi log representation of a shut in well test . . . . . . . . . . . . . . . . . . . . 21

3-2 The derivative type curve illustrated. . . . . . . . . . . . . . . . . . . . . . . . . 22

4-1 The identification procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4-2 Multiple flow periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4-3 Estimating type curve parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4-4 Used parameters of the identification method . . . . . . . . . . . . . . . . . . . 34

4-5 Schematic overview of the PEI procedure. . . . . . . . . . . . . . . . . . . . . . 36

4-6 LTI property of a reservoir without boundaries. . . . . . . . . . . . . . . . . . . 40

4-7 Nonlinear behaviour of reservoir with closed boundaries. . . . . . . . . . . . . . . 41

4-8 Overview research method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5-1 The output for different model order. . . . . . . . . . . . . . . . . . . . . . . . . 46

5-2 Flow rate without uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5-3 Results for an infinite reservoir and data without uncertainty. . . . . . . . . . . . 48


x List of Figures

5-4 Flow rate with 2 percent uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . 49

5-5 Results for an infinite reservoir and data with 2 percent uncertainty. . . . . . . . 50

5-6 Flow rate with 5 percent uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . 50

5-7 Results for an infinite reservoir and data with 5 percent uncertainty. . . . . . . . 51

5-8 Results for a finite reservoir and data without uncertainty. . . . . . . . . . . . . 53

5-9 Examples of used basis functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6-1 The basic setup for a dynamic error-in-variables (EIV) and a false EIV problem. . 60

C-1 The sensitivity of the bode plot for a changing skin factor and permeability. . . . 77


List of Tables

4-1 Five different modelstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5-1 Estimation of permeability and the skin factor after input shown in Figure 5-2. . 52

5-2 Estimation of permeability and the skin factor after complete shut in. . . . . . . 52


xii List of Tables


Nomenclature

Symbol Description Unit Page

A(z, θ) Polynomial 39As Surface flux area m2 13aj Starting time flow period j s 28B(z, θ) Polynomial 39bj End time flow period j, s 28C Well bore storage coefficient m4 · s2/kg 15C(z, θ) Polynomial 39C(z) System Matrix 31Cu Constant 20c Isothermal compressibility m · s2/kg 12co Isothermal compressibility of oil m · s2/kg 12cp Isothermal compressibility of the pore volume m · s2/kg 12ct Total isothermal compressibility m · s2/kg 12cρ Isothermal compressibility as function of density m · s2/kg 61D(z, θ) Polynomial 39E Expectation operator 36E Generalized expectation operator for quasi-

stationary signal36

e(t) White noise 35F (z, θ) Polynomial 39f(x) Probability density function 36G0 True transfer function of system 34G(s) Laplace transform of impulse response g(t) 27G(z, θ) Parametric model for G0 35g(t) Impulse response of system 21


xiv Nomenclature


H0 True transfer function of disturbance 34H(z, θ) Parametric model for H0 35h Height m 12h(t) Step response of system 21

h(t) Time derivative of step response of system 30K Dimensionless constant, number of nodes 15k Absolute permeability m2 12M Number of pressure response measurements 47M Chosen model structure (space) 35m Slope infinite flow regime 20N Number of flow periods 28n Order of model 39na Number of parameters in the function A(z, θ) 39nb Number of parameters in the function B(z, θ) 39nc Number of parameters in the function C(z, θ) 39nd Number of parameters in the function D(z, θ) 39nf Number of parameters in the function F (z, θ) 39ng Number of parameters in the function G(z, θ) 39nk Number of samples time delay 39Pθ Covariance matrix 37p(r, t) Pressure kg/(m · s2) 3p0 Initial pressure in reservoir kg/(m · s2) 32pe Pressure at the boundary of the reservoir kg/(m · s2) 16pi Initial equilibrium pressure in reservoir kg/(m · s2) 14pwf Pressure at bottom of the well bore, oil flow rate

non zerokg/(m · s2) 3

ptf Pressure at surface, oil flow rate non zero kg/(m · s2) 3q(t) Flow rate at the top of the well bore BBL/day 8q(t) Time derivative of the flow rate BBL/day2 28qj Flow rate of flow period j at the top of the well

boreBBL/day 28

qw Flow rate at bottom of well bore BBL/day 3qt Flow rate at top of the well bore BBL/day 3q∗ Flow rate per unit volume s−1 13R Measure of curvature rad 32Rn Toeplitz matrix 39r Radius measured from the center of the well m 12re Radius of the reservoir m 14rw Radius of the well bore m 14S Skin factor 16S True system 34s Laplace variable 27T Reservoir temperature, length of well test K 12, 31t Time s 12u Input 35


Nomenclature xv


V Volume m3 12Vo Volume of oil m3 12Vp Pore volume m3 12V (θ) Costfunction 36W Costfunction 32X Random or deterministic variable 36y Output 35y Time average of pressure response 45ym Simulated output using model 45Z Estimated type curve 30z Variable of Z-transform 34zk Nodes used to estimate type curve Z 30

αk Coefficient 31βk Coefficient 31γ Dimensionless constant 16δ Measurement error in flow rate 32∆p Pressure difference 14ε Measurement error in pressure 32ǫ Prediction error 35η tg(t) 30ηq Uncertainty percentage flowrate (input) 47ηp Uncertainty percentage pressure (output) 47θ Vector of variables in PEI 35θ0 Vector of true variables in PEI 36θj Interpolating function 31θ∗ Vector of optimal variables in PEI 36

θ Vector of estimated variables in PEI 37Λ Derivative matrix 38λ Factor to weight curvature in cost function 32µ Viscosity kg/(m · s) 12ν Factor to weight relative errors 32~ν Darcy velocity m/s 12τ The logarithm of time 30φ Porosity (dimensionless) 13Φν Power density distribution of disturbance 38Φu Power density distribution of input 38ρ Density kg/m3 13σe Standard deviation of e(t) 37Υ Vector of length M containing ones 32χ2 Chi squared distribution 38ψ Derivative matrix 37


xvi Nomenclature


Acronyms

ARMAX Autoregressive moving average model with exogenous inputs

ARX Autoregressive model with exogenous inputs

BIBO Bounded-Input Bounded-Output

BJ Box Jenkins

DUT Delft University of Technology

EIV Errors In Variables

FIR Finite Impulse Response

FOH First Order Hold

MC Monte Carlo

ODE Ordinary Differential Equation

OE Output Error

PEI Prediction Error Identification

SI System Identification

SISO Single Input Single Output

TLS Total Least Squares

ZOH Zero Order Hold


xviii Acronyms


Index

AAquifer . . . . . . . . . . . . . . . . . 6, 15

BBarrel . . . . . . . . . . . . . . . . . . . . 2Barrier . . . . . . . . . . . . . . . . . . . . 1Boundary Conditions . . . . . . . . . . . 13Bulk Reservoir Properties . . . . . . . . . 5

CCausality . . . . . . . . . . . . . . . . . . 27Compressibility . . . . . . . . . . . . . . . 12Conservation of Mass . . . . . . . . . . . 13Convolution . . . . . . . . . . . . . . . 7, 27

DDarcy’s Law . . . . . . . . . . . . . . 5, 11Deconvolution . . . . . . . . . . . 7, 29, 57Deliverability . . . . . . . . . . . . . . . . 1Diffusion . . . . . . . . . . . . . . . . . . 11Diffusion Equation . . . . . . . . . . . . . 11

Analytical Solution . . . . . . . . . . 15Conditions . . . . . . . . . . . . . . . 14Derivation . . . . . . . . . . . . . . . 11

Diffusive Process . . . . . . . . . . . . . . 4Disturbances . . . . . . . . . . . . . . . . 35Dynamical System . . . . . . . . . . . . . 6

EExploitation . . . . . . . . . . . . . . . . . 1

FFault . . . . . . . . . . . . . . . . . . . . . 1Finite Reservoir . . . . . . . . . . . . . . 52Flow rate . . . . . . . . . . . . . . . . . . 2

Fluid Dynamics . . . . . . . . . . . . . . . 12Fluid front . . . . . . . . . . . . . . . . . 1Fold . . . . . . . . . . . . . . . . . . . . . 1Formation . . . . . . . . . . . . . . . . . . 1Fracture . . . . . . . . . . . . . . . . . . . 1

GGas . . . . . . . . . . . . . . . . . . . . . 1

HHeterogeneities . . . . . . . . . . . . . . . 1Homogeneous . . . . . . . . . . . . . . . . 13

IImpermeable medium . . . . . . . . . . . 1Impulse Response . . . . . . . . . . . . . 57Infinite Flowing Regime . . . . . see RegimeInfinite Reservoir . . . . . . . . . . . . . . 46Initial Conditions . . . . . . . . . . . . . . 13Injection . . . . . . . . . . . . . . . . . . . 13Injection condition . . . . . . . . . . . . . 1Input . . . . . . . . . . . . . . . . . . 2, 35

LLaplace Transform . . . . . . . . . . . . . 27Laplace Variable . . . . . . . . . . . . . . 27Linearity . . . . . . . . . . . . . . . . . . 27

MMeasurement Location . . . . . . . . . . . 4Model . . . . . . . . . . . . . . . . . . . . 6

Nonparametric Model . . . . . . . . 6Parametric Model . . . . . . . . . . . 6

Model Order . . . . . . . . . . . . . . . . 45Model Structure . . . . . . . . . . . . . . 39


xx Index

Modeling . . . . . . . . . . . . . . . . . . 6Multi-Phase Flow . . . . . . . . . . . . . 5

OOil . . . . . . . . . . . . . . . . . . . . . . 1

Reservoir . . . . . . . . . . . . . . . 1Output . . . . . . . . . . . . . . . . . 2, 35

PPermeability . . . . . . . . . . . . . . . 2, 5Pore Volume . . . . . . . . . . . . . . . . 12Porosity . . . . . . . . . . . . . . . . . . . 4Prediction Error Identification . . . . . . 33Preparatory Experiments . . . . . . . . . 27Pressure Response . . . . . . . . . . . . . 2Production . . . . . . . . . . . . . . . . . 1

Condition . . . . . . . . . . . . . . . 1Production Strategy . . . . . . . . . . . . 3

RRegime . . . . . . . . . . . . . . . . . . . 14

Infinite Flowing . . . . . . . . . . . . 14Semi Steady State . . . . . . . . . . 14Steady State . . . . . . . . . . . . . . 15Transient . . . . . . . . . . . . . . . 14Wellbore Dominated . . . . . . . . . 14

ReservoirBoundaries . . . . . . . . . . . . . . 6Boundary . . . . . . . . . . . . . . . 3Model . . . . . . . . . . . . . . . . . 11Parameters . . . . . . . . . . . . . . 2Property . . . . . . . . . . . . . . . . 1

Reservoir Modeling . . . . . . . . . . . . . 3Response . . . . . . . . . . . . . . . . . . 1Rock type . . . . . . . . . . . . . . . . . . 1

SSemi Steady State Regime . . . . see RegimeShut In . . . . . . . . . . . . . . . . . . . 4Single-Phase Flow . . . . . . . . . . . . . 5Skin Factor . . . . . . . . . . . . . . . . 2, 5Stability . . . . . . . . . . . . . . . . . . . 27Steady State Regime . . . . . . . see RegimeStep Function . . . . . . . . . . . . . . . . 19Step Response . . . . . . . . . . . . . . . 19Surface . . . . . . . . . . . . . . . . . . . 1System Identification . . . . . . . 1, 6, 25, 58Systems Theory . . . . . . . . . . . . . . 2

TTime Invariance . . . . . . . . . . . . . . 27Transient Regime . . . . . . . . . see RegimeType Curve . . . . . . . . . . . . . . 7, 8, 45

UUncertainty . . . . . . . . . . . . . . . . . 37

estimations . . . . . . . . . . . . . . 58Uncertainty Bound . . . . . . . . . . . . . 8

VViscosity . . . . . . . . . . . . . . . . 12, 13

WWater . . . . . . . . . . . . . . . . . . . . 1Well Test . . . . . . . . . . . . . . . . . 1, 3

Analysis . . . . . . . . . . . . . . . . 58Interpretation . . . . . . . . . . . . . 19

Well Testing . . . . . . . . . . . . 2, 19, 40Current Practice . . . . . . . . . . . 19

WellboreParameters . . . . . . . . . . . . . . 2

Wellbore Dominated Regime . . see RegimeWellbore Storage . . . . . . . . . . . . . 2, 5


Chapter 1

Introduction

Geological formations hosting oil, gas, water are complex in their dynamics, and may containdifferent rock types, barriers and fluid fronts. To reach a decision about whether and howto exploit a given reservoir best, the ‘ability to produce’ (deliverability) of the reservoirand properties such as size have to be known. Well tests are performed to get part of thisinformation. During a well test, the pressure response of a reservoir to changing production(or injection) conditions is monitored. Production is the process of producing oil. Sincethe pressure response depends on the properties of the reservoir, it is possible to infer somereservoir properties from the pressure response.

The goal of this first chapter is to introduce the various aspects of well testing. The firstsection, section 1-1, is about the origin of oil and what is considered to be an oil reservoir.The second section will elaborate on well tests and their purpose. In section 1-3, the generalidea of system identification is briefly explained. In section of this chapter, section 1-4, theimportance of the research topic: ‘Well Testing in the Framework of System Identification’ ismotivated. In this section also the research questions are stated. Section 1-5, the last sectionof this chapter, gives the outline of this report.

1-1 Oil Reservoir

Oil and gas are formed from the remains of plants and animals that lived hundreds of millionsof years ago. Organisms, mostly algae and tiny animals called plankton, died and sank tothe bottom of the seas in which they lived. They were buried deep under mud, sand andother sediments for millions of years [Schlumberger, 2008]. Because oil and gas are less densethan water, they tend to migrate towards the surface and will form a reservoir if they aretrapped by a layer of impermeable medium along the way [Zandvliet, 2008]. A schematicfigure of a reservoir is Figure 1-1. The medium below the surface is generally deformed overlong periods of geological time, leading to folds, faults and fractures. Reservoirs thereforehave spatially varying properties, due to chemical and biological processes over time. Theseproperty differences are referred to as heterogeneities.


2 Introduction

Impermeable

layer

surface

Oil in porous medium

Well

Figure 1-1: Schematic representation of an oil reservoir.

1-2 Well Testing

1-2-1 Introduction

During a well test, the pressure response of a reservoir to changing production (or injection)conditions is monitored. Since the pressure response depends on the properties of the reser-voir, it is possible to estimate some reservoir properties from the pressure response. This isthe goal of well testing.

Several steps can be distinguished in well testing. At first, the flow rate is varied at thewell, see Figure 1-1, and the variation causes a change in the existing pressure p(r, t) in thereservoir. Note that the pressure is location dependent in the reservoir. The variation inpressure is measured at the well as a function of time. The flow rate is usually measured inbarrels1 per day (BBL/day). The pressure response is usually measured in psi.

The measurements are interpreted and from this interpretation knowledge is gained about thereservoir. In systems theory , the flow rate can be considered as the input of the system andthe variation in pressure can be considered as the output. The system consists of the reservoirand wellbore or only the reservoir, as will be shown in section 1-2-3. In systems theory, thesystem is often described by the impulse response, which is the output of the system whenthe input is an Dirac delta function (impulse).

In the following subsections, more details about well testing and its purpose are given.

1-2-2 The Purpose of Well Testing

When a oil reservoir is located, it has to be explored2 before it can be exploited3.

Well testing helps the reservoir engineers exploring the reservoir by being a tool to estimaterelevant reservoir and wellbore parameters . The estimations concerning the reservoir areneeded in order to make decisions about how to exploit the reservoir. The estimations con-cerning the wellbore are needed in order to examine whether a well is drilled satisfactory.The effective skin factor, information on the wellbore storage , the average permeability of

1An oil barrel corresponds with 159 liters.2Travel through (an unfamiliar area) in order to learn about it [Simpson and Weiner, 2008].3Make good use of (a resource) [Simpson and Weiner, 2008].


1-2 Well Testing 3

the reservoir and type and location of the boundaries are estimated. These properties will beexplained in more detail in section 1-2-4.

Well tests are done before exploiting the reservoir, but also after a period of production, tosee whether and how much the reservoir properties have changed.

1-2-3 What is a Well Test?

The estimations of the four properties mentioned above are achieved by comparing the pres-sure response with the expected pressure response based on a physical model. This procedureis shown in Figure 4-8. Figure 1-3 places well testing in a broader perspective. As can beseen, well testing is part of reservoir modeling . The results can be used in reservoir modelsto determine a production strategy for the reservoir. Based on this strategy modern wells canbe controlled. Since more and more data becomes available during production, it is possibleto improve the reservoir model by updating. In Figure 1-3 the direction of the arrows showsthe order in which these steps are taken.

Reservoir

Model

p

p

time

time

Model response

MATCH

Measurement input

Measurement input

Reservoir response

Figure 1-2: The way in which the reservoir properties are estimated: comparingand matching the pressure response with the expected pressure re-sponse based on a physical model. p is the pressure at the top of thewellbore. Based on [Bourdarot, 1998].

Reservoir

Control

Model

Well testing

Figure 1-3: Well testing is a part of the reservoir modeling.

The pressure and the flow rate can only be measured at the location of the wellbore. In Fig-ure 1-4 a wellbore is shown, with the locations where the rate and pressure can be measured:at the bottom of the well, flow rate qw and pressure pwf , or at the top of the well, at thesurface, flow rate qt and ptf . Measuring both the pressure and the flow rate at the top of


4 Introduction

the wellbore is most common, since then there is no need for expensive downhole shut-in andmulti phase flow measurement equipment. Downhole pressure measurements are becomingmore standard in new wells, but downhole flow meters are still relatively rare. In this researchit is assumed that all measurements are done at the top of the wellbore. Note that the locationof the measurements influences the system. In the situation that the flowrate and pressureresponse are measured at the bottom of the wellbore, the well itself is no longer part of thesystem. In the situation assumed in this research, the system consists of the wellbore and thereservoir, since it is assumed that all measurements are done at the top of the wellbore.

Surface rate qt

Pressure at top ptf

Pressure at

sandface pwf

Sandface

rate qw

Figure 1-4: Schematic representation of a wellbore. Pressure and flow rate canbe measured at the bottom of the well: flow rate qw and pressurepwf , or at the top of the well, at the surface: flow rate qt and ptf .Based on [Matthews and Russell, 1967].

There have been many advances in well testing the last thirty years, see e.g. [Earlougher, 1977,Gringarten, 2008, von Schroeter et al., 2004]. In most cases of current well testing, the wellis shut in (closed) and the pressure change at the top of the wellbore is measured as functionof time. The measured pressure change is called the pressure response and a plot of thispressure response can be represented in various ways. These plots are called type curves andare used to estimate the reservoir properties. During these shut-in well tests, the reservoircannot produce any oil. Therefore, well tests are expensive to perform. The reason for a totalshut in, and not a less expensive, partial closure of the well, is the high uncertainty in theflow rate, when the flow rate is nonzero. When the well is closed, there is no uncertainty inthe flow rate.

There is a limit to the level of detail that can be achieved in a reservoir description.This is because the pressure transmission is an inherently diffusive process , and henceis governed largely by average conditions rather than by local heterogeneities in prop-erties such as permeability and porosity. Well tests can be interpreted to estimate


1-2 Well Testing 5

bulk reservoir properties because they are insensitive to most local scale heterogeneities[Grader and Horne, 1988, Horne, 1995].

1-2-4 Relevant Reservoir Properties

In section 1-2-2 the reservoir properties that are estimated by well testing are already men-tioned. In this subsection more details will be given about these properties. The estimationof these four reservoir properties is the goal of well testing.

Skin Factor

Drilling a well is a practice that causes changes in the reservoir properties close to the well.Therefore the properties around the well are different from the properties further away. Totake the influence of this drill damage into account the skin factor is introduced. It representsthe increase or decrease of the pressure drop predicted with Darcy’s law (more details aboutDarcy’s law can be found in section 2-2). The skin factor can be positive and negative.Positive skin means an increase in pressure drop.

Wellbore Storage

As stated above, the measurements can be done at two locations: the first one is the bottomof the wellbore. In that case the system consists only of the reservoir, and the wellbore is notpart of the system. In the second case the measurements are done at the top of the reservoir.This has a large advantage, since there is no need for expensive down hole shut in equipment.However, in the latter case, the wellbore is part of the system considered. Wellbore storageis an effect that only needs to be taken into consideration when the wellbore is part of thesystem. When the well valve is opened, the fluid in the wellbore expands due to a pressuredrop in the wellbore. Due to this expansion, the production of the well in the begin periodis dominated by fluid that was already in the wellbore. The pressure drop therefore doesnot contain much information about the reservoir. One can imagine that in a multiple phaseflow4 situation, where there is also gas present, this effect is larger than the single phase flow5

that is considered in this research. Similarly, when a well is shut in, the flow rate will bezero at the top of the well, but not instantaneously zero at the bottom of the well, due to thecompressibility of the fluids in the well bore.

Permeability

The definition of ‘permeate’ is ‘to enter something and spread to every part’[Simpson and Weiner, 2008]. When something is permeable it means that it can be perme-ated by fluids or gas. Permeability is a measure for the level to which something is permeable.Oil has to flow through the reservoir, and the permeability of the reservoir determines how

4The simultaneous flow of more than one fluid phase through a porous medium. Most oil wells ultimatelyproduce both oil and gas from the formation, and often produce water. Consequently, multi-phase flow iscommon in oil wells. Most pressure-transient analysis techniques assume single-phase flow[Schlumberger, 2008].

5The flow of a single-phase fluid, such as oil, water or gas, through porous media [Schlumberger, 2008].


6 Introduction

much force is needed to get it through. The pressure response of the reservoir is very sensitiveto the permeability.

Reservoir Boundaries

When a reservoir is discovered, one tries to estimate the amount of recoverable oil from thisreservoir. The size of the reservoir (location of the boundaries) plays of course an importantrole in this estimation. Therefore, reservoir engineers are very interested in the locationof the boundaries of a reservoir. Also the type of the boundary is important. Two typesare most common: the open and closed boundaries. The closed boundaries imply no flowthrough the reservoir boundaries and the open boundaries imply a constant pressure at theseboundaries, that is, the reservoir is pressure supported by e.g. an aquifer6. More detailsabout these different boundaries are described in section 2-3. The more certain the typeand location of the boundaries can be predicted, the more accurate the predictions of theamount of recoverable oil are. In other words, when the radius of investigation is larger, theoil reserves that can be booked by the oil companies will probably be higher.

1-3 System Identification

System identification is about modeling dynamical systems for the purpose of e.g. simulation.System identification theory consists of mathematical tools and algorithms that build modelsof dynamical systems from measured data, and is used in many applications, for instance inthe field of closed loop control of all kinds of systems [Ljung, 1987]. System identificationuses measurements of the variables of the process. A model is constructed by identifyinga model that matches the measured data as well as possible. System identification is alsoreferred to as experimental modeling. Models can be parametric and nonparametric. Modelsare referred to as parametric whenever they are constructed from a model representation witha limited number of coefficients. In this research, the focus in on constructing a parametricmodel [Ljung, 1987].

In the field of well testing, a parametric model of the reservoir can be build based on flowrate and pressure measurements at the wellbore (see Figure 1-4). An advantage of using anidentification technique is that the flow rate does not have to be constant anymore for welltesting, and maybe the need for a shut in can be removed. In Chapter 4 details are givenabout system identification.

1-4 Well Testing in the Framework of System Identification

The quality of well test interpretations has improved considerably in the last fifty years.This is mainly caused by the availability of more accurate pressure data and the de-velopment of new software for computer aided analysis. In this research the well test-ing software package Saphir, developed by KAPPA7, is used as a reference of the cur-

6A water-bearing portion of a petroleum reservoir with a waterdrive [Schlumberger, 2008].7KAPPA ENG is a petroleum exploration and production software company created in 1987. Used version

of Saphir (Ecrin): 4.10.02.


1-4 Well Testing in the Framework of System Identification 7

rent state of well test analysis. An increasing number of theoretical interpretationmodels allow a more detailed definition of the flow behavior in the producing for-mation [van Everdingen and Hurst, 1949, Matthews and Russell, 1967, Bourdet et al., 1983,Raghavan, 1993, Horne, 1995, Lee et al., 2003, von Schroeter et al., 2004].

In a number of recent papers deconvolution is introduced in the field ofwell testing [Levitan et al., 2005, Gringarten, 2008, von Schroeter et al., 2001,von Schroeter et al., 2004]. In general terms deconvolution is the inverse of convolu-tion. Where convolution can be used to calculate the output of a system, when the inputand the system dynamics, often described by the impulse response, are known, deconvolutioncan be used to calculate the impulse response when the input signal and output signalare known. In the field of well testing deconvolution is often seen as a tool to estimatethe shut in pressure response of the reservoir based on the pressure response during avarying flow rate (e.g. [Bourdarot, 1998]). This latter definition is not used in this report.More details about deconvolution are given in Chapter 4. When the impulse responseis calculated, using deconvolution, the pressure response of the reservoir can be calcu-lated for an other flow rate history than the actual flow rate history, using convolution. In[Levitan et al., 2005, Gringarten, 2008, von Schroeter et al., 2001, von Schroeter et al., 2004]it is indicated that this technique improves the estimation of the type curve, by reshapingthe data, and therefore improves the estimation of the reservoir properties.

The goal of this research is to evaluate the current well test methods from a system identifica-tion point of view and to improve well testing by applying the theory of system identification.In order to reach this goal, five questions are stated. The first two questions are about evalu-ation of the current methods from the view point of system identification. The first questionhas to be dealt with before the second question can be answered. The last three questions arerelated to the second aspect of the research goal and can be dealt with separately: improvingwell testing by applying system identification. Each one of these three questions deals with apossible opportunity to improve well testing.

In order to evaluate the current well testing methods from a system identification point ofview, the current state of well test analysis has to be known. Therefore the first researchquestion is:

Q1 ‘How do the introduced deconvolution techniques work?’

Since the first goal of this research is to evaluate the current well testing methods from asystem identification point of view, it has to be investigated how the current well test analysistechniques relate to system identification using the answer to the first research question. Thisleads to a second research question:

Q2 ‘How do the already developed deconvolution algorithms relate to systemidentification. Are the used methods justified and reliable in terms of systemidentification?’

From the viewpoint of system identification, the procedure of estimating the reservoir param-eters using type curves can be questioned. The type curve is a representation of the system.When a dynamical model of the system is estimated using system identification, the inter-pretation of the system using type curves might be a redundant step in the well test analysis,that is not based on present mathematical or physical science, but a left over from early years.Type curves might be seen as overly simplistic, difficult to distinguish, and/or cumbersome


8 Introduction

Time

Oil flow rate[BBL/day]

Reduction production loss Current oil production

t1t2

q

0

Figure 1-5: The current well test results in an oil production loss of q(t2 − t1)[BBL]. q(t) denotes the flow rate. The shaded area is the potentialreduction of production loss, when shut in can be avoided, usingsystem identification.

to use. However, type curves are able to link the pressure response to the physical properties.If type curves are to be removed from the identification, there has to be an alternative linkbetween the measured pressure response and the properties that have to be estimated. Itmust be possible to express the reservoir properties as a function of the estimated model.The third research question is therefore:

Q3 ‘Is it possible to remove the type curves from well test analysis, using systemidentification techniques?’

A drawback of the type curve analysis methods is that uncertainty estimations are not avail-able. Uncertainty estimations are common in system identification. This leads to a fourthresearch question:

Q4 ‘Is system identification able to deliver reliable probabilistic uncertaintybounds for the estimations of reservoir properties?’

The current costs of well testing are high. During well tests, the well is shut in, and oilproduction is lost. In Figure 1-5 the oil production of the current situation is indicated bythe black areas. During well tests, there is no production. The production rate is the inputsignal of the system. Using the theory of system identification, the flow rate might be able tovary during the well test without influencing the estimations, and therefore the oil productionloss might be reduced. This is indicated by the shaded area.

The last research question is therefore:

Q5 ‘Is it possible to reduce the costs of well testing using deconvolution and/orsystem identification?’

Of course, the power of the input signal must be high enough to excite all dynamics of thereservoir and uncertainty in the flow rate measurement has to be taken into account.


1-5 Outline of Thesis 9

As said before, the first two research questions cover the first aspect of the research goal:evaluating the current well testing methods from a system identification point of view. Thelast three questions cover the second aspect of the research goal: improving well testing byapplying the theory of system identification. The answer of these questions can be found insection 6-2. Together, these answers are the conclusion of the research.

1-5 Outline of Thesis

The outline of this thesis is as follows. In this chapter, the various aspects of well testing havebeen explained very briefly. In Chapter 2 the way an oil reservoir is modeled is described.In Chapter 3, the current well testing practice is described. In this practice, there is noproduction of oil during the well test. The measured quantities and various ways to interpretthe data are explained.

Chapter 4 is about system identification. This term is briefly explained in the first sectionof this chapter. System identification is able to make an improvement in well testing. Twomethods to implement system identification in the field of well testing are also given in thischapter. The first method is introduced in recent well testing literature, and the secondmethod is based on Prediction Error Identification (PEI) and is newly introduced in thisresearch. Results of both methods are compared in Chapter 5. In Chapter 6 conclusions aredrawn and recommendations are given for future research.


10 Introduction


Chapter 2

Oil Reservoir Model

2-1 Introduction

As already stated in Chapter 1, the reservoir needs to be modeled, in order to interpret themeasurements of the well test. A model is derived in section 2-2. This model is used to linkthe measurements with the reservoir properties.

The fundament of all the present well testing is given by a physical equation, the diffusionequation1. The solutions give the expected outcome for the measurements of the well test.The diffusion equation describes the variation in space and time of pressure that is governedby diffusion and production.

In the next section the diffusion equation will be derived. To do this we need three laws offluid dynamics [Earlougher, 1977, Dake, 1978, Horne, 1995]. These laws will be introduced.The derived diffusion equation is nonlinear, but will be linearized. This linearized diffusionequation will be used in this report. In section 2-3 solutions of this linearized diffusion equationwill be given for various initial and boundary conditions. These solutions are important whenanalyzing the well test data. In the last section, section 2-4, a short summary of this chapteris given.

2-2 Derivation of the Diffusion Equation

As already stated, three laws of fluid dynamics are needed in order to derive the diffusionequation. These three laws are:

1. Darcy’s LawDarcy’s empirical flow law was the first extension of the principles of classical fluid

1diffusion is a net transport of molecules from a region of higher concentration to one of lower concentrationby random molecular motion.


12 Oil Reservoir Model

dynamics to the flow of fluids through porous media. The significance of Darcy’s lawis that it introduces flow rates into reservoir engineering. It implicitly introduces atime scale in oil recovery calculations. The law is stated in equation (2-1). Darcy’s lawexpresses the fact that the volumetric rate of flow per unit cross-sectional area at anypoint in a uniform porous medium is proportional to the gradient in potential in thedirection of flow at that point.

~ν =k

µ

∂p

∂r(2-1)

In this equation, ν represents the Darcy velocity2 of oil, µ the viscosity of the fluid andp the pressure. k is the absolute permeability of the porous medium, provided the latteris completely saturated with a fluid. This is assumed to be valid for the oil reservoir. kwill have a constant value irrespective of the nature of the fluid. The shortest distanceto the well is denoted by r. The law is valid for laminar flow at low Reynolds numbers3

and when gravitational effects are neglected [Bourdarot, 1998].

2. CompressibilityIn the diffusion equation, the pressure differences between two locations in the reser-

voir are essential. These differences would not be there without compressibility c asdefined in equation (2-2). All the information from well testing is obtained because theporous medium and the fluids are compressible. When the temperature is constant, thecompressibility of a medium is a function of the volume V and pressure p.

c(t) = − 1

V (t)

(∂V (t)

∂p(t)

)

T

(2-2)

In this equation, the volume V , the pressure p and the compressibility c can be a functionof time t. In our reservoir, two components can be compressible: oil (oil compressibilityis represented by co), and the pore volume4 (pore volume compressibility is representedby cp). The pore volume is balanced between the influence of the fluid pressure and thelithostatic5 pressure. The reservoir temperature T is assumed to remain constant. Thetotal isothermal compressibility is given by ct and is assumed to be constant in time.The equation for the total compressibility is given by:

ct = co + cp (2-3)

= − 1

Vo(t)

(∂Vo(t)

∂p(t)

)

T

− 1

Vp(t)

(∂Vp(t)

∂p(t)

)

T

(2-4)

The reservoir is modeled by an only slightly compressible liquid.

2The Darcy velocity is the fluid velocity that would occur if the entire cross section, and not just the pores,would be open to flow

3The Reynolds number Re is a dimensionless number that gives a measure of the ratio of inertial forces toviscous forces [van den Akker and Mudde, 2007].

4The pore volume Vp is the volume equal to the open space in rock or soil. The percentage of the porevolume in a reservoir is called the porosity.

5The pressure of the weight of overburden, or overlying rock, on a formation; also called geostatic pressure.


2-3 Solution of the Diffusion Equation 13

3. Conservation of massConsidering fluid flow in a porous medium, the most significant quantity conserved is

mass.

Increase in mass content of the region = Amount of mass input -Amount of mass output + Net amount of mass introduced by sources and sinks

Using a first order approximation, the mass balance can be written as:

As∂(νρ)

∂rdr − ρq = V φ

∂ρ

∂t, (2-5)

V = 2πrhdr is the volume of the small element of thickness dr with perpendicular tothe flow direction a surface of As. ν is the Darcy velocity, ρ is the density of the fluidand φ is the porosity of the medium. When the flow rate q is positive, fluid is produced.A negative value for q indicates injection [Dake, 1978, Jansen, 2007].

These three laws can be combined to the nonlinear diffusion equation:

1

r

∂

∂r

(kρ

µr∂p

∂r

)

+ ρq∗ = φctρ∂p

∂t. (2-6)

q∗ denotes the flow rate per unit volume [Jansen, 2007].

After linearization, equation (2-6) can be written as:

1

r

∂

∂r

(

r∂p(r, t)

∂r

)

+q∗µ

k=φµctk

∂p(r, t)

∂t(2-7)

Underlying assumptions and detailed derivations of equation (2-6) and equation (2-7) aregiven in Appendix A. The reservoir formation is assumed to be homogeneous and isotropic6,and drained by a fully penetrating well to ensure radial flow. The fluid itself must have aconstant viscosity and a small and constant compressibility.

2-3 Solution of the Diffusion Equation

2-3-1 Introduction

The diffusion equation is assumed to show the relation between the production flow rate andthe pressure in the reservoir. In order to solve the equation and to find an expression forthe pressure as function of r and t, one initial and two boundary conditions (assumptions)are needed. The initial and boundary conditions used to solve the diffusion equation implyseveral assumptions and are therefore only valid for specific time intervals. The assumptionsare described in the next sections. Note that in the situation of well testing, only the pressure

6When something is isotropic it is of equal physical properties along all axes.



at the wellbore as a function of time is of interest: p(r = rw, t). To evaluate p(r = rw, t) it isneeded to derive p(r, t) from equation (2-7) first.

In this section, the solution of the linearized diffusivity equation in radial coordinates is givenfor different conditions. First, the different initial and boundary conditions will be explainedin the next subsection.

2-3-2 Conditions of Analytical Solutions

As introduced in Chapter 1, the measured pressure response is compared to the modeled pres-sure response for the measured flow rate in order to estimate the properties of the reservoir.All modeled pressure responses are based on a constant production flow rate.

The diffusion equation (2-7) is solved for four different situations. These situations are oftenreferred to as regimes. For different regimes there are different boundary conditions to solvethe diffusion equation. These boundary conditions are assumptions and only hold in specifictime intervals. All regimes have thus a different solution of the diffusion equation (2-7).

- Transient regime. For the period that the transient regime is applicable it is assumedthat the pressure response of the reservoir at the top of the wellbore is not affectedby the presence of a boundary: the reservoir appears infinite in extent. The transientregime can be split up in two subregimes:

- Wellbore dominated regime. The wellbore dominated regime describes thevery early part of the well test where the dynamics of the wellbore is dominant overthe reservoir dynamics in the pressure response. It is assumed that the pressureresponse in this regime does not contain any information about the reservoir, butonly about the wellbore (storage).

- Infinite flowing regime. The infinite flowing regime describes the pressure re-sponse of the reservoir when the dynamics of the reservoir is assumed to be domi-nant over the wellbore dynamics and there is no significant influence of the bound-aries in the pressure response. Before producing, the pressure everywhere withinthe reservoir is equal to the initial equilibrium pressure pi: p = pi at t = 0 ∀r.The pressure at the infinite distance boundary is not affected by the pressure dis-turbance at the wellbore and vice versa: p = pi at r = ∞ ∀t. Assumed that thewellbore radius is negligible in comparison to the apparently infinite reservoir, thewellbore itself can be treated as a line in stead of a cylinder.

- Semi steady state regime. This regime is applicable when the influence of so calledclosed boundaries on the pressure response at the wellbore are significant. For a reser-voir without closed boundaries this regime is not applicable. A property of closed (or no

flow) boundaries is that there is no flow of fluids possible through the reservoir bound-aries. The first assumption made about these closed boundaries is that the pressurederivative with respect to time is constant everywhere in the reservoir: ∂p

∂tis constant,

∀r. A second assumption made is that the pressure gradient at the reservoir boundary(where r = re and p = pe) is zero: ∂p

∂r= 0 at r = re ∀t. The reservoir is assumed to be

radial.



Using the definition of compressibility, equation (2-2), the constant of the first assump-tion can be easily calculated:

∂p(r, t)

∂t= − q

ctV∀ r. (2-8)

In a radial reservoir with radius re, volume V = πr2ehφ. φ is the porosity of the porousmedium, and h is the height of the reservoir.

- Steady state regime. The steady state regimes is used when open boundaries havesignificant influence on the pressure response. For a reservoir without open boundariesthis regime is not applicable. Assumed is that at an open boundary of the reservoir(r = re), the pressure is constant pe, independent of time: ∂p

∂t= 0 ∀ r. It is assumed

that for a constant production flow rate, fluid withdrawal from the reservoir will beexactly balanced by fluid entry across the open boundary. This boundary condition canbe caused by an aquifer as already stated in section 1-2-4.

Note: Not all four regimes occur in each well test. The wellbore dominated regime onlyoccurs in a well test when the wellbore is part of the system. When performing down holeshut in, and down hole measurements, the wellbore is not part of the system, and the wellbore dominating regime will not occur.The semi steady state regime and the steady state regime do not occur for the same reservoirassuming that all boundaries of a specific reservoir are the same type: either closed or open.Depending on the type of boundary, the semi steady state regime (closed boundaries) orsteady state regime (open boundaries) are applicable.

2-3-3 Analytical Solutions of the Diffusivity Equation

Now the different conditions are known for the different regimes, the solution of the diffusionequation (2-7) can be derived for each of the regimes. The solutions for these regimes aredescribed in this section. As mentioned before, these solutions are used to interpret the welltest data.

- Transient regime. As stated earlier, the transient flow regime consists of two sub-regimes: the well bore dominated regime and the infinite flowing regime.

- Well bore dominated regime. As stated in the previous subsection, the veryearly time behaviour of the pressure response after shut in of the well bore isdominated by the well bore dynamics. In [Agarwal, 1970] it is shown that thisbehaviour can be approximated by:

p(rw, t) = pi −Kt

C(2-9)

K is constant, provided a constant production flow rate q. C represents the wellbore storage coefficient and is defined as the extra volume of oil V produced bythe well bore per unit pressure drop ∆p in the reservoir: C = V

∆p. pi is the initial

pressure in the reservoir.



- Infinite flowing regime. During the initial transient flow period, it has beenfound that the constant flow rate solution of the radial diffusion equation (2-7) canbe approximated using the assumption that in comparison to the infinite reservoirthe well bore radius is negligible and the well bore itself can be treated as a line instead of a cylinder (or a cylinder with radius zero). Since the detailed derivationof the solution of for the infinite flowing regime is complex and not used in theresearch, it is given in Appendix A. The resulting equation is:

p(r, t) = pi −qµ

4πkh

[

ln4kt

γφµctr2+ 2S

]

(2-10)

where µ is the viscosity of the fluid, h the height of the reservoir, S is the skinfactor and γ is a constant [Dake, 1978].

In Figure 2-1 a schematic plot is given of the infinite flow solution (2-10). In this pressureregime, the radius r of the reservoir is assumed to be infinity (re → ∞). Figure 2-2illustrates the pressure profile in a reservoir during the transient regime.

r

pi = pe pi = pe

rrere

t

p

Figure 2-1: Pressure distribution for transient regime.

d [m]d [m]

Pre

ssure

[PSI]

-5000 0 5000 -50000

50003700

3750

3800

3850

3900

3950

(a) Pressure at t1

d [m]d [m]

Pre

ssure

[PSI]

-5000 0 5000 -50000

50003700

3750

3800

3850

3900

3950

(b) Pressure at t2

d [m]d [m]

Pre

ssure

[PSI]

-5000 0 5000 -50000

50003700

3750

3800

3850

3900

3950

(c) Pressure at t3

Figure 2-2: Illustration of the transient regime. For increasing time (t1 < t2 <t3), the pressure profile of a reservoir is simulated for the transientregime.

Figure 2-3 shows a simulated pressure response for a shut in at the top of the well bore.In systems theory, this is referred to as the step response. In this well test there wasno influence of any boundaries, and therefore only the two transient regimes (the wellbore dominating regime and infinite flow regime) are applicable. The showed pressure



is the pressure at the top of the well bore. The production rate is given in BBL/day.1 BBL (Barrel) is a common used unit of volume in the field of well testing and is thesame as 159 liters.

Pro

duct

ion

rateq

[BB

L/D

AY

]

∆t[days]

∆t[days]

∆p[PSI]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

100

200

300

0

500

1000

Figure 2-3: Example of a simulated step response h(t). ∆p(t) is the pressureresponse at the top of the well. ∆t is the shut in time.

- Semi Steady State flow. The solution of the diffusion equation in the semi steadystate regime is given by:

p(r, t) − p(rw, t) =qµ

2πkh

(

lnr

rw− r2

2r2e+rw

2

2re2+ S

)

(2-11)

p(r, t) is the pressure at a distance r from the center of the well bore, and p(rw, t) is thepressure at the well bore radius rw at time t [Dake, 1978].

r

pi pi

r

p

t

rere

Figure 2-4: Pressure distribution for semi steady state regime.

In equation (2-11) rw2

2re2 can be considered to be negligible [Dake, 1978]. The difference

between p(r, t) and p(rw, t) is independent of time t, since ∂p∂t

is constant ∀ r, t. See Fig-ure 2-4 for an schematic plot of the semi steady state solution of the diffusion equation.In Appendix A a detailed derivation of this solution can be found. Note that equation(2-11) does not determines p(r, t), but only a pressure difference as function of radius r.However, the solutions give a clear impression of the pressure profile in the reservoir.

- Steady State flow. The solution of the diffusion equation for the steady state regime



is given by equation (2-12) [Dake, 1978]. Note that the solution is independent of timet.

p(r) − p(rw) =qµ

2πkh

(

lnr

rw+ S

)

(2-12)

r

pe = pi

rrere

p

pe = pi

Figure 2-5: Pressure distribution for steady state regime.

Again, as in the case with the semi steady state solution, equation (2-12) does notdetermine p(r, t) but only a pressure difference, again as function of the radius. However,p(r, t) is not dependent on time anymore for all r.

2-4 Summary

In this chapter the linearized diffusion equation, equation (2-7) is derived. This equationis the basis of the current well test analysis. In order to solve this equation and to get anexpression for p(r, t), one initial and two boundary conditions are needed. These conditionsare assumptions. A drawback of many assumed conditions is the limited time interval inwhich they are valid. This is the reason why flow regimes are introduced. The solutions inequation (2-9), (2-10), (2-11) and (2-12) show the different behaviour of p(r, t) in the variousregimes and the influence of the different types of boundaries. Note that in the situation ofwell testing, only the pressure at r = rw can be measured. There is no usable single expressionof the solution for p(r, t) of equation (2-7) for the duration of the whole well test because ofthe absence of suitable boundary conditions.

The solutions of equation (2-7) for the various regimes are used to interpret well test data.In the next chapter the current well test interpretation methods are described. These inter-pretation methods make extensive use of the solutions given in this chapter, in particular thesolution of the infinite flowing regime given in equation (2-10).


Chapter 3

Current Well Testing Practice

3-1 Introduction

Well tests exist already for a very long time. Since the first oil reservoir is discovered, oilcompanies have paid attention to estimating reservoir properties, such as size. In this chapter,the present well testing practice is briefly explained. The well testing practice consists ofmeasurements and analysis of the data. The part concerning the measurements of the welltest is shortly described in section 3-2. In section 3-3, two interpretation methods of well testdata are described (well test analysis). Drawbacks of these interpretation methods are givenin section 3-4. Section 3-5 addresses possible improvement opportunities of the current welltesting and in section 3-6 a short summary is given.

3-2 Well Testing

Data has to be gathered, before anything can be said about the reservoir. The productionrate of the well, q, is designed, and the pressure response is measured at the top of the samewell as where the flow rate is set (see Figure 1-4). In systems theory, the first is called theinput (signal), and the latter the output (signal). In the case of a conventional well test, theinput is chosen to be a shut in of the flow rate (flow rate is set to zero). A shut in of thewellbore corresponds in systems theory with a so called step function. The output (pressureresponse) as result of a step function as input is referred to as the step response of the system.A shut in is often chosen to be the input, since a flow rate of zero is the only flow rate thatcan be set without uncertainty.

3-3 Well Test Interpretation

In order to estimate the reservoir properties the step response has to be analyzed. This sectiondescribes two methods to estimate these properties: direct interpretation of the measured data


20 Current Well Testing Practice

and comparison of the measured data with other known type curves. For both methods, themeasured pressure (step) response is plotted. There are various ways to represent this curve.Reservoir parameters are estimated based on these type curves and are therefore indirectlyestimated. The estimations depend strongly on the quality of the plots. In the currentsituation, the reservoir properties are estimated without any indication of the uncertainty.

The second method, comparison of the measured data plot with known type curves, is atthe moment preferred by reservoir engineers. Direct interpretation of the measured data isnot used often anymore. Estimation of the reservoir properties using the second method isassumed to give less uncertain estimations (although no literature has been found in whichthis is quantified). The reason why the direct interpretation is explained, is that it givesa better idea of the link between the solution of the diffusion equation and the reservoirproperties. Furthermore, the direct calculation method is much easier to implement and istherefore used in Chapter 5 to give an indication of the quality of the estimated curves.

Direct Interpretation In order to perform direct interpretation of the measured data, theHorner plot is mostly used. The Horner plot is a plot of the pressure response ∆p(t) versusthe logarithm of time t. In Figure 3-1 a Horner plot is shown [Horner, 1951, Dake, 1978]. Inthis figure the two transient regimes and a boundary regime are appointed.

According to solution of the diffusion equation for the infinite flowing regime, equation (2-10),a plot of p(rw, t) versus the logarithm of t is linear with slope m:

m =Cuqµ

kh. (3-1)

In Figure 3-1 this semi log straight line is indicated by the solid line. This line can be usedto calculate the permeability k and skin factor S. Cu is a constant depending on the units ofthe other quantities used.

When slope m is found according to equation (3-1), the permeability k can be calculated:

k =Cuqµ

mh. (3-2)

In order to calculate the skin factor S, using equation (2-10) and the straight line with slopem, equation (3-3) can be derived. The value of p(rw, 1hr) is taken from the solid straight linewith slope m, in case the wellbore storage is still of influence after one hour.

S = K1

(pi − p(rw, 1hr)

m− log

k

φµcr2w+K2

)

(3-3)

Note that the direct interpretation method is an indirect estimation method, since the stepresponse has to be plotted first.


3-3 Well Test Interpretation 21

Storage Infinite flow Boundaries

∆p(t)

time [s]

100 101 102 104103 105 106 107

slope m

Figure 3-1: Semi log representation of a shut in well test. The ∆p(t)-axis is linear,the time-axis is logarithmic. This plot is also known as the Hornerplot [Earlougher, 1977, Dake, 1978]

.

Comparison The second method to estimate the reservoir properties is to compare the plotof the measured pressure response with type curves. These type curves are based on thesolutions of the linearized diffusion equation (2-7). As mentioned in the introduction of thissection, this is the method currently preferred by reservoir engineers. The data are plottedand this plot is then compared with plots derived from theory. By finding the best match,the reservoir properties are estimated. An uncertainty region is, again, not available or atleast not standard.

As mentioned in the introduction of this section, there are many kinds of type curves. Thesetype curves differ in their representation. A major improvement was the use of the derivativetype curve, introduced by [Bourdet et al., 1983]. In this research only the best performingmethod is presented, which is also the latest [Bourdet et al., 1983, Gringarten, 2008]. It isthe so called derivative LogLog type curve, in which log tdp

dtis plotted versus log t. Note that

this derivative type curve is plotted for a shut in of the flow rate. Therefore, the pressureresponse ∆p(t) is equal to the step response h(t) and the derivative of the step response isequal to the derivative of the measured pressure p(t). Furthermore, the time derivative of

h(t) is equal to the impulse response of the system denoted by g(t): dh(t)dt

= g(t). It follows

that tdpdt

= tg(t). From now on, the notation tg(t) will be used. In Figure 3-2 examples of thederivative type curves are given. Each curve is generated with different reservoir properties.

Until a few decades ago, the plotted data and the type curves were compared by hand. Theuse of computers and software developed for well testing such as Saphir, introduced in section1-4, made this process much easier, and enabled the opportunity to match the curves withleast squares methods. However, the fundamentals didn’t change, and uncertainty estimationsare still not standard.

Loglog derivative type curves are very sensitive on a LogLog scale for changes in the stepresponse. In other words, the tg(t) curve is very sensitive to variations in ∆p(t). Thisresult is a bit ambiguous. On the one hand, this is the reason why the derivative-type curveis presented in this way. The goal is to amplify the differences between various pressure



∆p(t)

tg(t)

time

Figure 3-2: LogLog representation of ∆p and tg(t) for various values of the reser-voir properties after a well is shut in. The solid line represents a plotof measured data, the dashed line shows the semi log straight linewith slope m.

responses. On the other hand it means that the pressure response has to be measured veryprecisely, since a small deviation from reality can already lead to a different derivative typecurve.

3-4 Drawbacks

There are four main drawbacks of the current well test analysis. The lack of reliable un-certainty estimation is the first major drawback. As stated above, there are no uncertaintybounds in the type curves, and therefore only the mismatch of the plotted data with thebest matching type curve gives an indication of the uncertainty. Uncertainty estimations area weak point of today’s well test analysis. The exploration strategy is partly based on theestimated properties. When the uncertainty appears to be much higher than is expected, itmight be possible that another strategy should have been chosen. A second major drawbackis the need for shut in. In order to estimate the properties, there is no production during thistime. This ‘dead time’ costs the oil companies a lot of money. The third drawback is thenumber of assumptions on which the analysis procedures are based, such as a constant flowbefore shut in. These assumptions can never be met exactly, and will cause an error in theestimate. The fourth and last drawback of the current use of type curves only applies to thelatest derivative type curve. This type curve plots a function of the derivative of the pressureresponse. Taking the derivative of measurement data in a reliable way is challenging fromnumerical perspective since it magnifies the measurement noise.

3-5 Improvement Possibilities

In the previous section a number of drawbacks are mentioned of the current well testingpractice. As stated in the introduction, these drawbacks are the main reason to look intowell testing from the viewpoint of systems theory. The use of a step as input is not the


3-6 Summary 23

only possibility to estimate a system. Also the use of type curves is questionable from theviewpoint of systems theory, provided an alternative method to link the estimated systemwith the reservoir properties. Standardization of uncertainty calculations would also definitelyimprove the current well testing practice. Since the flow rate is constant before and after theshut in, this is called analysis of a single flow period. An estimation method that can analyzemultiple flow periods of the well in order to estimate the reservoir properties is beneficial fortwo reasons: a well is never capable of keeping a perfect constant flow rate in time (the onlyexception is the zero rate after a shut in). Taking multiple flow rates into account is thereforemore realistic. A second reason is that when multiple flow periods can be taken into account,it might be possible to reduce the well testing costs by reducing the oil production loss, since atotal shut in of the well might be unnecessary, also when taking the measurement uncertaintyof the flow rate into account.

3-6 Summary

The well testing methods described in this chapter analyze the pressure response as result ofa shut in of the reservoir. There are two ways in which the properties can be estimated. Forboth methods the step response has to be plotted. The first method is a direct interpretationof the Horner plot. The second one is a method in which the plot of the well test data iscompared to plots of known properties, the type curves. By matching, the reservoir propertiesare estimated. There are a few drawbacks of these methods and also ways to improve thesemethods. In the next chapter, it is shown how system identification techniques are able toimprove the well test analysis.


Chapter 4

System Identification

4-1 Introduction

This chapter is about system identification. In this chapter two different methods are de-scribed to identify the system, that consists of the reservoir and wellbore. Both iden-tification methods assume that the pressure response can be described by a convolu-tion of the flow rate and the impulse response of the system. Section 4-2 describes thesystem identification concept. The details of convolution will be explained in section4-3. Recently, there is a lot of attention for deconvolution in the field of well testing[von Schroeter et al., 2004, Onur et al., 2008, Whittle and Gringarten, 2008]. In section 4-4,deconvolution is explained. According to the general definition the aim of deconvolution isto calculate the impulse response of the system, based on the pressure response and the flowrate. In well testing literature, there are however various different definitions for deconvolutionused.

The first identification method is introduced by [von Schroeter et al., 2004], developed atImperial College. This method is described in detail in section 4-5.

Prediction Error Identification (PEI) is an alternative method that can be used to model thesystem and to extract the impulse response based on the measured pressure response andflow rate (i.e. deconvolution) in order to identify the reservoir. Details about this newlyintroduced method in the field of well testing are given in section 4-6. In section 4-7 a shortsummary of this chapter is given.

4-2 Concept

System identification uses measurements of the variables of the process. A model is con-structed by identifying a model that matches the measured data a as well as possible.

Figure 4-1 shows three different aspects that are crucial in any identification procedure. Thesethree aspects are listed:


26 System Identification

prior knowledge / intended modelapplication

Intended model application

Data

Experimentdesign

Identificationcriterion

Model set

Construct model

Validate model

Figure 4-1: The identification procedure. Based on [Ljung, 1987]

.

- Data. Available from normal operating records, but it may as well be possible toperform tailor-made experiments.

- Model set. It has to be specified beforehand within which set of models one is goingto evaluate the most accurate model for the process. In the model set several ba-sic properties have to be fixed: linearity/nonlinearity, time variance/time invariance,discrete/continous-time.

- Identification criterion. Given the data and model set, the way to determine theoptimal model has to be specified.

After a model has been constructed using the aspects above, an important last step of systemidentification is the validation. Validation is used to check whether one is satisfied with theestimated model. The purpose of the estimated model plays an important role in determiningthe level of satisfaction [Ljung, 1987].

In many system identification methods there are four important assumptions made about thesystem. These four assumptions are linearity, time invariance, causality and stability:

Linear system Collection of one or more linear equations involving the samevariables.

Time invariant system The equations describing the system are not a function oftime.

Causality The output of the system at a certain time does not dependon future input. This is often called nonanticipative, as thesystem output does not anticipate future values of the input.

Stability Often Bounded-Input Bounded-Output (BIBO) stability.When offering a limited input, the output is also limited.


4-3 Convolution 27

The impulse response is denoted by g(t). The Laplace transform is denoted by a capital. TheLaplace transform of g(t) is thus G(s), where s denotes the Laplace variable.

In [Ljung, 1987] the implications of these assumptions on the dynamical system with transferfunction G(s) are explained:

Linearity of the dynamical system is induced by the linear Ordinary Differential Equation(ODE) that governs the dynamical system. Time-invariance of the dynamical system isinduced by the fact that the differential equation has constant coefficients, implying that G(s)is independent of the time t. Causality is the mechanism that g(t) = 0, t < 0. In termsof G(s) this is reflected by the condition lim|s|→∞G(s) < ∞. As already stated, stabilityoften implies BIBO-stability. BIBO-stability implies that the system is analytic. A system isanalytic if it has no poles in the right half plane, Re(s) > 0.

Before identifying the model using a specific identification method, it has to checked thatall the assumptions implied by the method are fulfilled. This is done by the preparatoryexperiments.

Preparatory Experiments The goal of preparatory experiments is to obtain information fromthe process that enables the proper design of an identification experiment. Basic knowledgeabout the process is gathered. Additionally it can be verified whether and to which extentthe process dynamics can be considered to be linear and time-invariant. Apart form possiblephysical restrictions one generally will be in favor of applying an input signal with a maximumachievable power, in order to increase the signal-to-noise ratio at the output. Considering apossible excitation of non-linear dynamics, the amplitude of the input signal may have to berestricted.

4-3 Convolution

As stated in section 3-5 it has many advantages when an estimation method can analyzemultiple flow periods at once. The underlying physical principle of the methods that canestimate the reservoir properties based on multiple flow periods is the principle of superpo-

sition. When generalized, this principle of superposition is similar to convolution. Using theassumed linearity of the reservoir and the known step response (in the situation that thesystem is known), it is possible to compute the pressure response of multiple flow periods bysuperposition of the known step responses. Convolution can be used to calculate the pressureresponse based on any flow rate function, when the system is known.

The pressure response as function of time of a particular multiple flow rate history can beeasily explained using systems theory. This will be done first. After that, attention is onthe physical interpretation of convolution. The same expression for the pressure response isderived in a way that levels better with current well testing literature.

In terms of systems theory, the output is the result of convolution of the impulse response ofthe system and the input. Since the pressure response ∆p(t) is the output, the flow rate q(t)the input and the reservoir is described by the impulse response g(t), the expression of the



pressure response as function of time is given by

∆p(t) = g(t) ∗ q(t) =

∫ t

0q(t′)g(t − t′)dt′ (4-1)

This equation is already introduced in the field of well testing in 1949[van Everdingen and Hurst, 1949], but it has not been used for many years. q(t) canhave multiple flow rates, as for example in Figure 4-2.

qN−1

qN

q5

q4

q3

q2

q1

b1

a2a1

b2

a3

b3

a4

b4 bN−1

aN

bN

a5

Time

Figure 4-2: Multiple (N) flow periods. Each flow rate j has a duration ∆tj = bj−aj

and flow rate qj.

The physical interpretation of the integral in equation (4-1) can be understood by the principleof superposition as already stated in the beginning of this section. This principle states thatany sum of individual solutions of a second order linear differential equation is also a solutionof the equation. The pressure behaviour as function of time can be obtained by simplysuperposing time-shifted pressure step responses.

Assume h(t) to be the pressure step response of the system. h(t) is equal to ∆p(t) when theflow rate q(t) is a unit step function. Assuming bN < t < bN−1, see Figure 4-2, equation (4-2)shows the expression for the N flow rates given in Figure 4-2:

∆p(t) = q1h(t) + (q2 − q1)h(t− b1) + . . .+ (qN − qN−1)h(t− bN−1). (4-2)

Equation (4-2) can be rewritten as:

∆p(t) =

N∑

j=1

∆qjh(t− bN−1), (4-3)

where ∆qj = qj − qj−1. When the number of flow periods N is going to infinity, N → ∞,and the time period ∆t that the flow rate is constant decreases towards zero for all periods,∆t→ 0, equation (4-2) can be restated as

∆p(t) =

∫ t

0q(t′)h(t− t′)dt′. (4-4)


4-4 Deconvolution 29

The dot in q(t) denotes the derivative with respect to time. Note the similar setup of equation(4-4) and equation (4-1).

When ∆p(0) = 0 and q(0) = 0, a general property1 for convolution of two function f(x) andw(x) dependent of variable x is given by:

df

dx∗ w(x) = f(x) ∗ dw

dx. (4-5)

This property can be used to rewrite equation (4-4) as equation (4-1) [Horne, 1995,von Schroeter et al., 2001, von Schroeter et al., 2004].

4-4 Deconvolution

Methods to extract the impulse response g(t) based on the input and output are deconvolution

methods. Deconvolution can be seen as the inverse of convolution. As already stated, there isa lot of attention for deconvolution in the field of well testing. However, not always the samedefinition is used. Sometimes estimating the step response h(t) of the system is considered tobe deconvolution, or, in the case of [von Schroeter et al., 2004], estimating tg(t) (the impulseresponse multiplied with time t) on a LogLog scale is called deconvolution. This last methodis described in section 4-5.

As mentioned earlier, deconvolution has already found its way into the world of well testingresearch papers, but is not yet fully accepted by the reservoir engineering community. Themajor reason for not being fully implemented is the fact that these insights are new. Oneother reason may be found in the academic background of most reservoir engineers. Most ofthem have a background in mining or civil engineering. They might be not familiar with thelatest mathematical tools to solve the deconvolution integral (equation (4-1)) and also mighthave a reserved stand towards these tools.

In section 4-5 an identification method developed at Imperial College, introduced in[von Schroeter et al., 2004], is described in detail, since this is the most advanced ‘de-convolution’ method for well testing currently available in literature [Onur et al., 2008,Whittle and Gringarten, 2008]. This method estimates nodes of the function tg(t) directly inthe LogLog domain. Remember that this is the domain that is described in section 3-3 as thelatest and best performing domain in existing well testing methods.

4-5 Identification by Von Schroeter

4-5-1 Introduction

In literature a few identification methods are described [von Schroeter et al., 2004,Levitan et al., 2005, Onur et al., 2008]. The one introduced in [von Schroeter et al., 2004]

1The Laplace transform of both sides of the equation is sF (s)W (s), the capitals denoting the Laplacetransforms, and s the Laplace variable.



shows the best results [Onur et al., 2008, Whittle and Gringarten, 2008] and is therefore re-searched in more detail and described in section 4-5-2. This identification method is alsoimplemented as a new option in Saphir (Software package introduced in section 1-4). Thegoal of this identification algorithm is to get a more accurate estimate of the LogLog repre-sentation of the tg(t) type curve, see section 3-3. It is not a identification method in the strictsense that it estimates g(t) but it directly estimates points of the LogLog representation of afunction of g(t) (time t multiplied with the impulse response g(t), the time derivative of thestep response h(t)).

Section 4-5-3 will consider the limitations of this identification technique and a short summaryof this method is given in section 4-5-4.

4-5-2 Detailed Method Description

As starting point, the convolution integral of equation (4-1) is stated. Note that the impulseresponse g(t) is the time derivative of the step response h(t):

g(t) = h(t) :=dh

dt. (4-6)

In current well testing, the pressure response ∆p(t) is equal to the step response h(t).

When assumed that q(0) = 0 and h(t) = 0∀ t ≤ 0 the convolution in equation (4-1) can berewritten as an integral:

∆p(t) = q(t) ∗ g(t) :=

∫ t

0q(t′)g(t − t′)dt′ =

∫ t

0q(t′)h(t− t′)dt′. (4-7)

The equality of the two integrals follows from equation (4-5). So far, the approach containsno new ideas. But the next step is innovative. η(t) is defined:

η(t) :=dh(t)

d ln t=dh(t)

d t

d t

d ln t= tg(t). (4-8)

To avoid the identification method to be numerically unstable, it is necessary to restrictthe solution space by adding constraints. This can be done explicitly, by adding ad-ditional equations or inequalities to be satisfied by the solution or its parameters. In[von Schroeter et al., 2004] however, the addition of these restrictions is done implicitly: thesolution space is parametrized in such a way that the constraints are automatically satisfied.Therefore, the goal is to get an estimate of Z(τ) = ln η(t) = ln tg(t), where τ = ln t. Theestimation of Z(τ) is done by estimating K points of this function. These points, or nodes,have the index k. Z(τk) is denoted by zk, as illustrated in Figure 4-3.

Applying these substitutions (and using the commutative property of convolution), equation(4-7) can be written as:

∆p(t) =

∫ t

0q(t− t′)t′g(t′)

dt′

t′=

∫ τ

−∞q(t− eτ )eZ(τ)dτ. (4-9)


4-5 Identification by Von Schroeter 31

Note that the expression on the right hand side of equation (4-9) has become nonlinear in theresponse function Z(τ).

Because the flow rate q(t) is not measured as often as the pressure response ∆p(t), but onlyat the start of a new flow period (see Figure 4-4), the flow rate has to be interpolated betweenthe measurements. A Zero Order Hold (ZOH) interpolation is used. The flow rate is thereforeassumed to be constant between the flow rate measurements, this is expressed by:

q(t) =N∑

j=1

qjθj(t), (4-10)

where qj is the flow rate over a time interval aj(t) ≤ t ≤ bj , with j = 1..N (see Figure 4-2).The interpolating function θj is defined as:

θj(t) =

{1 if aj ≤ t ≤ bj ,0 otherwise.

(4-11)

This is illustrated in Figure 4-4. The pressure response ∆p(t) is also measured and thereforediscrete. ∆p(t) is not interpolated. In the discrete situation, equation (4-7) can now berewritten as:

∆p(t) = C(Z)q(t) (4-12)

for all M time instants for which the pressure response ∆p(t) is measured. Matrix CM×N isdefined as:

Cij(Z) =

∫ lnT

−∞θj(ti − eτ )eZ(τ)dτ (4-13)

for i = 1..M the measurement times and j = 1..N the flow periods.

Estimating the derivative type curve Z(τ) is an iterative process. Z(τ) is approximated bya piecewise linear function in the LogLog domain. To start, K nodes at time instants tkfor k = 1..N are chosen without much information about the reservoir. As stated above,these nodes are denoted by zk. In between these points, the derivative type curve is linearlyinterpolated (First Order Hold (FOH)) in the LogLog domain. On the logaritmic x-axis thenodes are equally distributed: ∞ = τ0 < τ1 < τ2 < .. < τn = lnT . T is the total well testingtime, T = bN (see Figure 4-4).

With zk = Z(τk), αk = zk − βkτk and βk =zk−zk−1

τk−τk−1. Z(τ) is thus approximated by:

Z(τ) ≈ αk + τβk for τk−1 ≤ τ ≤ τk. (4-14)

The coefficients of the matrix C, Cij , have to be calculated for i = 1..M and j = 1..N . Cij

can be written as:

Cij(Z) =

K∑

k=1

eαkCijk(Z), (4-15)



where Cijk(Z) is given by

Cijk(Z) =

∫ τk

τk−1

θj(ti − eτ )eτβkdτ. (4-16)

Note that until now there are no measurement errors considered.

When ∆p(t) and q(t) are considered to be the measured signals, the measurement errors inthe pressure response ε and in the flow rate δ can be implemented by defining ∆p(t) + ε andq(t) + δ as the true, unobserved, signals.

In the presence of measurement errors, equation (4-12) can be written as:

∆p(t) + ε = C(Z)(q(t) + δ). (4-17)

The Total Least Squares (TLS) formulation of the identification problem is to find the Kcoefficients zk with the smallest perturbations ε and δ and the least overall curvature R suchthat equation (4-17) holds.

The cost function W can now be stated as:

W = ||ε||22 + ν||δ||22 + λR, (4-18)

where ||ε||2 = ||p0ΥM − p − C(Z)y||2 and ||δ||2 = ||y − q||2. p0 is the initial pressure in thereservoir and ΥM is a vector of length M containing ones. p is the measured pressure vectorof length M . y denotes the true, unobserved pressure response q + δ. The cost function ofequation (4-18) is minimized, subject to the constraint in equation (4-17). ||ε||2 is referredto as the pressure match and ||δ||2 is referred to as the rate match. Note that the true,unobserved signals have to be estimated before the cost function can be calculated.

R is a measure of the curvature of Z(τ), defined as the sum of the K − 1 angles betweenthe K segments visualized in Figure 4-3. ν is a constant factor that determines the relativeimportance between measurement error in the flow rate and pressure. Because the measure-ment errors in the flow rate are often known to be much larger then the measurement errorsin the pressure response, ν is often set to a value less then 1. The factor λ is a factor thatsets the required level of smoothness of Z(τ).

During the research, some minor errors were found in [von Schroeter et al., 2004]. Theseerrors are given in Appendix B.

4-5-3 Limitations

In the algorithm presented in [von Schroeter et al., 2004], the derivative LogLog type curveis estimated, and by doing that, the weight factors used by the representation of the LogLogderivative type curve are also used by the identification. A drawback is the weighting factorsthat have to be set. For example, the ‘punish factor’ λ for the curvature, equation (4-18)is set arbitrarily and might influence the result. Also, there is not a full expression for theimpulse response, but only a limited number of points of a function of g(t). Deconvolution


4-6 Prediction Error Identification 33

z1

zk

τ1 τk

Figure 4-3: Z(τ) given by K nodes zk = Z(τk). Between the nodes zk−1 and zk

Z(τ) is defined by αk + βkτ . β1 = 1 since the time span 0 < t < eτ1 isassumed to be dominated by wellbore storage, that is, for 0 < t < eτ1,Z(τ) = α1 + τ .

as described by [von Schroeter et al., 2004] is a tool to get a better estimate of tg(t) on theLogLog domain, used for the LogLog derivative type curve. Therefore, when the need forthis type curve is removed, this implementation of ‘deconvolution’ becomes useless. A secondlimitation is the number of nodes, this number is limited. Optimizing for many more nodes istime consuming and will increase the uncertainty. Another limitation is the non uniquenessof the solution. This is however essentially not a problem of this identification technique butis caused by multi interpretability of the well test data.

4-5-4 Summary

The subject of this section is the identification method described in[von Schroeter et al., 2004]. This method estimates K nodes of the derivative LogLogtype curve. The method removes the restriction for a shut in, but does not remove the needfor type curves: the final result of this identification method is an estimate of a type curve.This type curve can be used in order to estimate the reservoir properties.

4-6 Prediction Error Identification

4-6-1 Introduction

In the identification method described in the previous section, K nodes of the function tg(t)are estimated in the LogLog domain. In this section an analytic expression is found for theimpulse response g(t). In other words, the impulse response is identified completely. This isdone via the Z-transform. z is the variable of the Z-transform. The Z-transform converts adiscrete time-domain signal, which is a sequence of real or complex numbers, into a complexfrequency-domain representation.

In the section 4-6-2 a detailed method description is given. In section 4-6-3 theory aboutuncertainty calculation is presented. Section 4-6-4 presents different model structures for theestimation of the impulse response. In section 4-6-5, the PEI theory is applied to well testanalysis. In the last section, section 4-6-6, a summary is given.



1

1

1

1

1

1

1

0

0

0

0

0

0

0

θ1

θ2

θ3

θ4

θ5

θN−1

θN

qN−1

qN

q5

q4

q3

q2

q1

b1

a2a1

b2

a3

b3

a4

b4 bN−1

aN

bN

a5

Time

Figure 4-4: Parameters used by [von Schroeter et al., 2004]. qj represents theflow rate for aj < t < bj. During that time span, θj is 1, elsewhere θj

is 0. The values aj is the measured flow rate for flowing period j.

4-6-2 Detailed Method Description

PEI uses prediction error methods to identify parametric linear dynamic models. Predictionerror methods have become a wide-spread technique for system identification [Ljung, 1987].There are three types of signals that can be distinguished:

- Measurable output signals. These signals can be considered as responses of thesystem behaviour. In our situation the output signal is the pressure response at the topof the well. This signals are denoted by y(t).

- Measurable input signals. These measurable control signals that act as a case of thesystem response. In our case this signal is the flow rate at the top of the well. Noticethat in standard system identification theory, the input is assumed to be known. In oursituation there is a significant uncertainty of the input signal, with which is not dealtat the moment. This signals are denoted by u(t).



- Non-Measurable disturbances. These reflect non-measurable disturbances that acton the system. Assumed is that these disturbances are not exciting the system. Thesedisturbances are added to the response of the system. This signals are denoted by v(t).

When a system has only one input and one output, as is the case in well testing, the systemis called Single Input Single Output (SISO).

An assumption is that the true system S can be described as:

y(t) = G0(z)u(t) +H0(z)e(t)︸︷︷︸

v(t)

. (4-19)

The true system is denoted by:

S = {G0, H0}. (4-20)

G0 and H0 are unknown linear transfer functions in the Z-transform. A predictor model isdetermined by a collection of two rational functions G(z, θ) and H(z, θ). θ is a vector of allthe parameters. The model set M is defined as any collection of predictor models:

M = {G(z, θ),H(z, θ) ∀ θ ∈ R}. (4-21)

Note: The general goal of PEI is to find the best parametric models G(z, θ) and H(z, θ)for the unknown transfer functions G0 and H0 using a set of measured data u(t) and y(t)generated by the true system S.

The identification procedure can be best explained using Figure 4-5 and equation (4-19).y(t) is the measured output, the oil pressure measured at the top of the wellbore. G0 is thetrue systems transfer function. H0 is a (inversely) stable and monic linear transfer function.v(t) represents the measurement uncertainty. e(t) is a sequence of independent identicallydistributed zero mean random variables: a white noise signal. u(t) is the input, which is inthe this research the flow rate of oil at the top of the wellbore in the situation of well testing.When doing the measurements, the input must excite all the relevant dynamics of the system,that means that u(t) must contain all the frequencies that trigger a response of the system.Otherwise the estimated model is probably not capable of predicting the correct output forother input sequences.

A cost-function has to be defined in order to evaluate the performance of the estimated model.When this cost-function is minimized, the estimated model parameters are optimized. Thecost-function is defined as the power of the prediction error ǫ(t, θ). According to Figure 4-5,the prediction error ǫ(t, θ) can be written as equation (4-22). Note that the cost function isdefined in the time domain and not in the z-domain.

ǫ(t, θ) = H−1(θ) (y(t) −G(θ)u(t)) (4-22)

=1

H(z, θ)((G0(z) −G(z, θ))u(t) +H0e(t)) (4-23)

=G0(z) −G(z, θ)

H(z, θ)u(t) +

H0(z) −H(z, θ)

H(z, θ)e(t) + e(t) (4-24)



G0(z)

G(z, θ) +

+

-

e(t)

y(t)

ǫ(t, θ)

H0(z)

u(t)

1H(z,θ)

v(t)

Figure 4-5: Schematic overview of the PEI procedure.

G0(z) and H0(z) represent the true system and can be written as G(z, θ0) and H(z, θ0).

PEI assumes a quasi stationary signal, denoted by u(t). A quasi stationary signal is a finitepower signal that can be a stochastic signal, a deterministic signal or a summation of astochastic and deterministic signal. The expectation operator is denoted by E. When assumedthat the probability distribution of the variable X can be described by a probability densityfunction f(x), the expectation operator E is defined as:

E[X] ≡∫ ∞

−∞xf(x)dx. (4-25)

The generalized expectation operator for a quasi stationary signal, denoted by E, is given by:

E[·] ≡ limN→∞

1

N

N∑

t=1

E[·]. (4-26)

This generalized expectation operation combines a statistical expectation with a time-averaging.

The prediction error ǫ(t, θ) has four important properties:

- ǫ(t, θ) can be calculated, provided that the in- and output measurements are available,and a specific choice for θ is made.

- ǫ(t, θ0) = e(t).

- ǫ(t, θ) 6= white noise for all θ 6= θ0 (provided appropriate signal u(t)).

- θ0 minimizes the power E[ǫ2(t, θ)] of ǫ(θ).



The proofs of these properties can be found in [Ljung, 1987].

The theory above results into the ideal criterion for PEI. θ∗ is the solution of the minimizationof the cost function, which implies the minimization of the power of the prediction error. Theideal identification criterion is given by:

θ∗ = arg minθ

V (θ) (4-27)

with

V (θ) ≡ E[ǫ2(t, θ)] = limN→∞

1

N

N∑

t=1

E[ǫ2(t, θ)] (4-28)

V (θ) has a unique minimum θ∗. When S ∈ M, θ∗ = θ0.

However, it is not possible to use the criterion in equation (4-28) in practice: the power of theprediction error ǫ(t, θ) cannot be computed exactly with only N measurements. Therefore,the power of the prediction error is estimated using the N available data ZN :

VN (θ, ZN ) =1

N

N∑

t=1

ǫ2(t, θ) (4-29)

=1

N

N∑

t=1

((H(θ)−1(y(t) −G(θ)u(t))

)2. (4-30)

This criterion can be used in practice, and the estimation of the parameters is done byminimizing VN :

θN = arg minθ

VN (θ, ZN ), (4-31)

where θN is the identified parameter vector.

4-6-3 Uncertainty of PEI Estimator

For an identification experiment on S using an input signal u(t) and a number N of data, theparameter vector θN identified in such an experiment is the realization of a normal distribu-tion:

θN ∼ AsN (θ0, Pθ). (4-32)

The covariance matrix Pθ is defined as:



Pθ ≡ E((θN − θ0)(θN − θ0)T ) (4-33)

≃ σ2e

N(Eψ(t, θ0)ψ

T (t, θ0))−1 for N → ∞, (4-34)

with ψ(t, θ0) = ∂y(t,θ0)∂θ

∣∣∣θ=θ0

= −∂ǫ(t,θ)∂θ

∣∣∣θ−θ0

. σ2e denotes the variance of the white noise. It is

still assumed that S ∈ M.

This matrix gives an idea of the uncertainty region of θN . If Pθ is small, θN is close to θ0. Tomake this statement more quantitative, an uncertainty region is introduced:

θN ∼ AsN (θ0, Pθ) ⇐⇒ (θ0 − θN )TP−1θ (θ0 − θN ) ∼ Asχ2(k), (4-35)

with k the dimension of θN .

The unknown true parameter vector θ0 lies therefore in the ellipsoid U with probability β:

U ={

θ ∈ Rk|(θ − θN )TP−1

θ (θ − θN ) 6 α}

with an α such that Pr(χ(k) < α) = β.

The covariance matrix Pθ is a function of the input u(t) and N but also a function of the truesystem S via σ2

e and θ0. A reliable estimate Pθ of Pθ can be deduced using the data and θN .An expression for Pθ is given by:

Pθ =σ2

e

N(

1

N

N∑

t=1

ψ(t, θN )ψ(t, θN ))−1, (4-36)

with σ2e = 1

N

∑Nt=1 ǫ(t, θN )2.

Also G(z, θN ) and H(z, θN ) are random variables. G(z, θN ) and H(z, θN ) are asymptoticallyunbiased estimators of G(z, θ0) and H(z, θ0), respectively. The variance of G(z, θN ) is definedin the frequency domain as:

cov(G(ejω , θN )) ≡ E(|G(ejω , θN ) −G(ejω, θ0)|2) (4-37)

Similar as Pθ, also cov(G(ejω , θN )) is a function of the unknown true system S. However, itcan be estimated by using the data and θN :

cov(G(ejω , θN )) = ΛG(ejω, θN )PθΛ∗G(ejω, θN ), (4-38)

with ΛTG(z, θ) = ∂G(z,θ)

∂θ.

Due to the limited data length and stochastic properties of the disturbances, θN is differenteach experiment. When S ∈ M, θN has an asymptotically normal distribution AsN (θ0, Pθ)with θ0 the true parameters and Pθ the covariance matrix. Another property is that Pθ → 0when the data length N → ∞. In other words, for N → ∞, θN → θ0.



Table 4-1: Five different modelstructures

Modelstructure G(z, θ) H(z, θ)

ARX z−nk B(z,θ)A(z,θ)

1A(z,θ)

ARMAX z−nk B(z,θ)A(z,θ)

C(z,θ)A(z,θ)

OE z−nk B(z,θ)F (z,θ) 1

FIR z−nkB(z, θ) 1

BJ z−nk B(z,θ)F (z,θ)

C(z,θ)D(z,θ)

Assuming that the MacMillan degree2 n of the model G(z, θ) in M → ∞, the covariancecov(G(ejω , θN )) can be approximated:

cov(G(ejω , θN )) ≈ n

N

Φv(ω)

Φu(ω). (4-39)

N is the number of measurements, Φv(ω) and Φu(ω) are the power spectra of the disturbanceand input. Φu(ω) is defined as

∑+∞τ=−∞Ru(τ)ejwτ , with Ru(τ) ≡ E(u(t)u(t − τ)) (for the

definition of E see equation (4-26)). Equation (4-39) shows clearly that the covariance ofG(ejω, θN ) can be decreased by increasing the number of data points N and/or increasing thepower spectrum of the input signal φu at the frequencies ω with too high covariance.

4-6-4 Different Model Structures

M can have different shapes. There are 5 common model structures: Autoregressive modelwith exogenous inputs (ARX), Autoregressive moving average model with exogenous inputs(ARMAX), Output Error (OE), Finite Impulse Response (FIR), Box Jenkins (BJ). Thesemodels have different transfer functions G(z, θ) and H(z, θ). In Table 4-1 the different modelstructures are given.

In Table 4-1 the polynomials A(z, θ), B(z, θ), C(z, θ), D(z, θ) and F (z, θ) are polynomialswith respective degrees na, nb − 1, nc, nd and nf . nk stands for the number of samples timedelay. After choosing a model structure a model order has to be determined. In order toestimate the transfer functions in a proper way, the input signal must be sufficiently rich.Input signal u(t) is persistently exciting (and therefore sufficiently rich) of order n if theToeplitz matrix Rn is non-singular. The Toeplitz matrix is defined in equation (4-40). Theideal identification criterion of equation (4-27) has a unique solution if the input signal u(t) is

2The MacMillan degree of the dynamical G(z) is defined as the state dimension of any minimal realizationof G(z).



sufficiently exciting of order ≥ ng, where ng denotes the number of parameters in the functionG(z, θ).

Rn =

Ru(0) Ru(1) · · · Ru(n− 1)Ru(1) Ru(0) · · · Ru(n− 2)

.... . .

. . ....

Ru(n− 1) · · · Ru(1) Ru(0)

(4-40)

4-6-5 Prediction Error Identification Applied to Well Testing

The goal of this section is to apply the theory of PEI, described in the previous subsectionof this section, to the field of well testing, as an alternative identification method opposed tothe identification method introduced by [von Schroeter et al., 2004], described in section 4-5.

Another method to identify the system has to be determined. In order to do this three mainquestions have to be answered. These questions are related to the three main aspects ofsystem identification already stated in section 4-2: What data to use? Which model set canbe used best? What is the best identification criterion?

As data the measurements of the flow rate and pressure response at the top of the wellbore areused. The input is the flow rate q(t) and the output is the pressure response ∆p(t). In orderto determine which model set can be used best, some basic information about the system isneeded. This information is gathered by the preparatory experiments.

Preparatory experiments

As described in section 4-2 the goal of preparatory experiments is to gain some basic informa-tion about the system. This information enables the decision making about different modelsets to identify the system

Time [s]

Pro

duct

ion

rate

[BB

L/d

ay]

Skin factor: 2Permeability: 200mDarcyBoundaries: Infinity

Time [s]

Pre

ssure

[PSI]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5×105

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5×105

3750

3800

3850

3900

0

500

1000

1500

(a) A reservoir without boundaries shows time invari-ance (TI) behaviour.

Time [s]

Pro

duct

ion

rate

[BB

L/d

ay]


Time [s]

Pre

ssure

[PSI]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5×105

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5×105

3750

3800

3850

3900

0

500

1000

1500

(b) A reservoir without boundaries shows linear (L)behaviour

Figure 4-6: Test results show the LTI property of a reservoir without boundaries.



Figure 4-6 shows two simulations for a reservoir without boundaries (infinite size). Assumedis that the simulator represents the true reservoir in a sufficient way to get reliable results.Figure 4-6(a) shows the time invariance property of the reservoir as a system. Figure 4-6(b)shows that a reservoir without boundaries is linear when the flow rate is between 0 and 1000BBL/day (barrels per day).

Time [s]

Pro

duct

ion

rate

[BB

L/d

ay]

Skin factor: 2Permeability: 200mDarcyBoundaries: 300m

Time [s]

Pre

ssure

[PSI]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5×105

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5×105

3700

3750

3800

3850

3900

0

500

1000

1500

(a) A reservoir with closed boundaries shows the effectof a reservoir getting empty.

Time [s]

Pro

duct

ion

rate

[BB

L/d

ay]

Skin factor: 2Permeability: 200mDarcyBoundaries: 300m

Time [s]

Pre

ssure

[PSI]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5×105

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5×105

3750

3800

3850

3900

0

500

1000

1500

(b) A reservoir with closed boundaries shows nonlinearbehaviour.

Figure 4-7: Test results show that a reservoir with closed boundaries shows non-linear behaviour.

Figure 4-7 shows the same two simulations for a reservoir with a closed boundary (for detailsabout closed boundaries, see section 2-3). The boundary of the radial reservoir is at such adistance (300 meters) that the reservoir cannot be considered infinite. Figure 4-7(a) showsthat the response of the system can be considered to be time invariant, but does not reacha steady state situation when the flow rate is non zero. Figure 4-7(b) shows that a reservoirwithout boundary contains nonlinear dynamics when the flow rate is again between 0 and1000 BBL/day.

Because a transfer function is needed in order to estimate the properties (direct or indirect byusing a type curve) a parametric model has to be chosen. The reservoir properties are expectedto be related to very different dynamics, e.g. wellbore storage is expected to be related to fastdynamics but the boundaries are probably determined by relatively slow dynamics. This willprobably result in estimating small and large time constants at the same time, which makesa finding a good estimate for all dynamics difficult.

Within parametric models there are a five model structures to choose from, as already statedin section 4-6-4: ARX, ARMAX, OE, FIR and BJ. The use of simulated data causes the OEto perform best. When using real well testing data the model structure has to be selectedagain.

After choosing the model structure the model order has to be defined. The final choice forthe model order is made after evaluating the capability of predicting the pressure responsefor the various model orders. More details about the order selection are shown in Chapter 5.

When the model is estimated, the reservoir properties can be estimated. There are severalways to link the model parameters and the reservoir properties. The direct method is by far



the most appealing, since it removes the interpretation of the type curves from the currentwell testing practice, and therefore removes a cumbersome step in well test analysis (section3-3). However, since the reservoir properties cannot be expressed as a function of the modelparameters, type curve estimation is used in this research. Various representations of thesystem can be chosen. The pressure response to a shut in, the Horner plot and the latest andmost popular derivative LogLog type curve: the derivative of the shut in response with respectto time multiplied with time on a LogLog scale. All three representations of the estimatedshut in (step) response can be given. The results can be compared with the estimations madeusing the method of section 4-5 and the reference result generated with the simulator.InFigure 4-8 the implementation of the simulator in the research is visualized. In Appendix Dsome demands and alternatives for the simulator are given.

MoReS

System Identification

Toolbox

MoReS

Reference type curveType curve

estimated model

Reservoir model

Varying flowrate

Pressure response

Step input

Stepresponse

Estimated model

Evaluation

Figure 4-8: The evaluation method of the estimated type curve. The estimatedtype curve is compared with the expected type curve. The expectedtype curve is generated by applying a step function as input for thereservoir simulator MoReS.

4-6-6 Summary

In this section PEI is introduced in the field of well testing. At first, the general concept isexplained. After that, a detailed method description is given. In the previous subsection,


4-7 Summary 43

section 4-6-5, the theory of PEI is applied to well testing and a way to evaluate the resultsof the new method is presented. In this method, not only points of the derivative LogLogtype curve are estimated, but the impulse response is identified completely. Also uncertaintybounds of the estimated parameters can be calculated.

4-7 Summary

The high potential of identification theory in well test analysis is un-questioned [Bourdarot, 1998, von Schroeter et al., 2004, Onur et al., 2008,Whittle and Gringarten, 2008]. It may not only lead to removal of the need for shutin (see the reduction in oil production loss in Figure 1-5), but also more data can be takeninto account and therefore more precise estimations can be made (see equation (4-39)). Thelast ten years various methods are developed to perform identification in this field, oftenreferred to in literature as deconvolution techniques. These identification methods have notyet removed the shut in periods in practice. If used at all, this is only done to correct thepressure step response for the some variation in flow rate before the shut in. It has not yetremoved the use of shut in but only enhanced the shut in analysis.

Two methods of system identification are described for well testing. The first method (section4-5) is developed at Imperial College and is described in [von Schroeter et al., 2004]. Thisidentification method estimates a number of nodes of the derivative LogLog type curve (tg(t)represented in a LogLog domain). This domain is introduced in [Bourdet et al., 1983] and isthe domain commonly used by reservoir engineers. It takes measurement errors in the in- andoutput into account. The relative weight of the measurement errors in the in- and output inthe cost function has to be set manually. This method is implemented in the latest version(v4.10) of the well testing software package Saphir (see section 1-4). In section 4-6 a secondand new method is explained to identify the system using PEI. PEI is definitely much easier tohandle and more straight forward than the method introduced in [von Schroeter et al., 2004].PEI not only estimates a number of nodes in a specified domain, but it generates an completetransfer function of the system, without parameters that need to be set by hand, like thesmoothness factor in section 4-5.

In the next chapter results of both identification methods are compared with each other, butalso with the reference result (see Figure 4-8).


Chapter 5

Results

5-1 Introduction

In this chapter the results of two identification methods are presented. Both methods aredescribed and explained in Chapter 4.

First, as described in section 4-6, the model order for the method using Prediction ErrorIdentification (PEI) has to be determined. The result is presented in section 5-2.

In section 5-3 the Horner plot and several type curves are estimated by both methods, forseveral error levels. These curves are estimated using a varying flow rate without shut in.The permeability k and the skin factor S are estimated for an reservoir of infinite size, i.e.there are no boundaries present, making use of the Horner plots. To evaluate these reservoirproperty estimations, the same properties are also estimated after applying a complete shutin.

The ability to identify a closed boundary are evaluated in section 5-4. This is done for bothidentification methods.

In section 5-5 a summary of the results is given.

5-2 Model Order Selection

In the situation of an infinite reservoir, the reservoir shows linear and time invariant behaviour,see Figure 4-6. PEI can therefore be used to estimate the model. As already stated in Chapter4, Output Error (OE) is most likely to perform well for simulated data.

The next step is to determine the order of the model. In order to determine which modelperforms best, performance has to be defined. In this research, the level of performance isdefined as the percentage fit. A definition for the percentage fit is given by [Ljung, 1987]:

Percentage fit = 100 ·(

1 − ||ym − y||2||y − y||2

)

. (5-1)


46 Results

||.||2 stands for the ℓ2-norm:

||x||2 =√xTx =

√

x21 + x2

2 + ... (5-2)

ym is the predicted output by the model, y is the measured output, and y is the time averageof the measured output. Note that these are all vectors. For increasing order the models havebeen estimated and the resulting fit is shown in Figure 5-1(b). The input, Figure 5-1(a), isthe same input as used as input in [von Schroeter et al., 2004]. The simulated data contained4 · 104 data points. The simulated time is 2 · 105 seconds. The reservoir is of infinite size, theskinfactor S is 2 and the permeability k is 300 mDarcy.

Time [s]

Flo

wra

te[B

BL/d

ay]


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2×105

0

100

200

300

400

500

600

700

(a) flow rate (input)

Time [s]

Pre

ssure

[PSI]

MoReSOE110 83.10%OE220 94.53%OE330 98.10%OE440 99.29%OE550 99.60%OE660 99.68%OE770 97.71%OE880 99.54%

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2×105

0

10

20

30

40

50

60

(b) Pressure responses for different OE models

Figure 5-1: The output for different model order.

The fit percentages in Figure 5-1(b) show an upward trend in the fit for increasing modelorder. The seventh and eight order have a lower fit, this must be caused by numerical issuesestimating the model. The simulated data did not contain any measurement errors. However,since in reality there will be measurement errors, more model parameters (higher order) willresult in a higher uncertainty of the model, and a trade off must be made between number ofparameters and uncertainty. Until now, this is not integrated in the performance criterium asdescribed in equation (5-1). Higher order than the fourth order model only slightly improvesthe performance. Therefore the fourth order model is chosen.

5-3 Infinite Reservoir

For the following simulations a flow rate sequence is used as input without a shut in. Thissequence is shown in Figure 5-2. Compared with a complete shut in of the same length, thisis a reduction of the oil production loss with 75 percent. The same reservoir is used as before.

In Figure 5-3 various representations are given of the results using the two identification meth-ods. For the identification experiments, the number of data points is 6·104, and the simulatedtime is 3 · 105 seconds. Again, the simulated data does not contain simulated measurement


5-3 Infinite Reservoir 47

Time [s]

Pro

duct

ion

rate

[BB

L/d

ay]

Production rate

0 0.5 1 1.5 2 2.5 3×105

0

100

200

300

400

500

600

Figure 5-2: Flow rate sequence without uncertainty.

errors. In Figure 5-3(a) the simulated pressure response is given together with the estimatedpressure response of both methods. Both methods are capable of estimating the pressure re-sponse very well, and no significant differences can be found. In Figure 5-3(b) the estimationof the step response (for a shut in) is shown for both identification methods. The simulatedresult, which is the reference for evaluating the two estimations, is generated by giving thereservoir a shut in as input, and simulate the resulting pressure response. For more detailsabout the implementation of the simulator see Figure 4-8 in section 4-6-5. Again, no significantdifference can be found. Both methods, the one introduced by [von Schroeter et al., 2004] andPEI are capable of estimating the step response of the system equally well. Figure 5-3(c) showsthe Horner plot. As introduced in Chapter 3, this is the step response presented in a plot witha logarithmic time axis. Both methods show a minor deviation with the reference. Note thatin Figure 5-3(b) and Figure 5-3(c), the method introduced by [von Schroeter et al., 2004] onlygenerates points, whereas PEI generates a full expression of the curves. Figure 5-3(d) showsthe derivative LogLog representation, i.e. tg(t) on a LogLog scale. The expression generatedby PEI contains an oscillation with its mean. The estimation using the other identificationmethod also shows a significant difference with the reference signal. The ∆p curve in theLogLog domain, also Figure 5-3(d), is estimated equally well by both methods .

The situation without measurement errors in the flow rate and pressure response is not veryrealistic. A measure is needed for the level of uncertainty. In [von Schroeter et al., 2004] therelative standard deviation ηq and ηp are used for the level of uncertainty in the flowrate andpressure response. ηq and ηp are constant throughout the whole well test and are defined asin equation (5-3) and equation (5-4). The only unknown in equation (5-3) is the standarddeviation σq and can therefore be easily calculated. Note that since ||q||22 increases linearlywith the number of flow periods N a correction factor

√N is needed. This definition for

the uncertainty measure is dependent on the flow rate sequence. For the error level of thepressure response the same reasoning can be followed with M the number of pressure responsemeasurements [von Schroeter et al., 2004].

Error level flow rate = 100 · ηq = 100 ·[√

Nσq

||q||2

]

(5-3)


48 Results

Time [s]

∆P

[PSI]

MoReSOE440 98.81%von Schroeter

0 0.5 1 1.5 2 2.5 3×105

0

10

20

30

40

50

60

(a) Pressure response.

Time [s]

∆p

[PSI]

MoReSOE440von Schroeter

0 0.5 1 1.5 2 2.5 3×105

0

5

10

15

20

25

30

35

40

45

50

(b) Step response on a linear scale.

Time [s]

∆p

[PSI]


100 101 102 103 104 105 1060

5

10

15

20

25

30

35

40

45

50

(c) Step response on a semilog scale.

Log time [s]

Log

∆p

and

Log

tg(t

)

MoReS ∆pMoReS tg(t)OE440 ∆pOE440 tg(t)von Schroeter ∆pvon Schroeter tg(t)

Log time [s]

Log

∆p

and

Log

tg(t

)

100 101 102 103 104 105

100

101

(d) Step response on a LogLog scale.

Figure 5-3: Identification results for data without uncertainty.

Error level pressure response = 100 · ηp = 100 ·[√

Mσp

||p||2

]

(5-4)

In the results presented in this chapter, no measurement errors are added to the simulatedpressure response. The addition of the pressure response error level to the simulated dataas defined in equation (5-4) is not expected to have significant influence on the identificationresults, mainly because of the relatively high measurement frequency used: 0.2 Hz. Identifi-cation experiments including the measurement errors as defined in equation (5-4) showed nosignificant different result as the identification experiments without measurement errors inthe pressure response. Note that there is not much knowledge available to evaluate whetherthe error level as defined in equation (5-4) is in agreement with the error level of real welltest data.

In Figure 5-4 the same flow rate sequence is shown as in Figure 5-2 but also a sequence withan error level of 2 percent: ηq = 0.02. This last signal is used as input for identification,together with the simulated pressure response for the ‘clean’ flow rate sequence. The errorlevel on the flow rate is expected to have much more influence on the identified model, since



the flow rate q is only measured N times. As can be seen in Figure 5-4, the number of flowperiods is 19 in this case.

Time [s]

Pro

duct

ion

rate

[BB

L/d

ay]

Production rateProduction rate with uq = 2 percent

0 0.5 1 1.5 2 2.5 3×105

0

100

200

300

400

500

600

Figure 5-4: Flow rate sequence without measurement errors and a sequence with2 percent measurement errors.

In Figure 5-5 the results are presented of the two identification methods using the inputsequence of Figure 5-4 with an error level of 2 percent. Again the same reservoir of infinitesize is used (skin factor S is 2 and the permeability k is set to 300 mDarcy. The number ofdata points used for the identification is 6 ·104 and the simulated time is 3 ·105 seconds. Bothmethods manage to predict the pressure response without any significant difference comparedto the reference output. The pressure responses for a shut in, shown in Figure 5-5(b), showthat both identification methods estimate the pressure response very close to the simulatedpressure response using an shut in as input for the simulator. In the Horner plot, Figure 5-5(c),the result of method of [von Schroeter et al., 2004] slightly outperforms the result of themethod using acPEI. In the last representation, shown in Figure 5-5(d), it can be seen that theoscillations in the derivative curve are a little larger than the oscillations in the identificationusing the data without uncertainty. The estimated nodes of the other method do not differmuch, at least not significant, from the reference curve. This can be explained by the fact thatthe factor ν, as defined in equation (4-18), is set to 0.1. This implies that the measurementerror of the flow rate is weighted a factor 10 less in the estimation of the nodes than the errorin the pressure measurements. Since there is at the moment no measurement error in thepressure, the 2 percent error level in the rate does not influence the estimation significantly.The reason for the value of ν is the assumption that the error level of the rate measurementsis in practise 10 times as large as the error level of the pressure measurements. The value of0.1 for ν accounts for this difference.

When the error level of the rate sequence is increased to a value of ηq = 0.05, a possible flowrate sequence is shown in Figure 5-6.

When the sequence without measurement errors is the input of the same reservoir as used forthe earlier simulations, the simulated pressure response is shown in Figure 5-7(a), togetherwith the two estimated pressure responses based on identification using the data includingthe measurement errors. Both methods are capable of estimating the pressure response againwithout significant differences. The plot of the shut in pressure response, Figure 5-7(b),shows however a difference between the estimations of the two methods. At later times,


50 Results

Time [s]

∆P

[PSI]


0 0.5 1 1.5 2 2.5 3×105

0

10

20

30

40

50

60


Time [s]

∆p

[PSI]


0 0.5 1 1.5 2 2.5 3×105

0

5

10

15

20

25

30

35

40

45

50


Time [s]

∆p

[PSI]


100 101 102 103 104 105 1060

5

10

15

20

25

30

35

40

45

50


Log time [s]

Log

∆p

and

Log

tg(t

)


Log time [s]

Log

∆p

and

Log

tg(t

)

100 101 102 103 104 105

100

101


Figure 5-5: Identification results for input with 2 percent uncertainty and outputwithout uncertainty.

Time [s]

Pro

duct

ion

rate

[BB

L/d

ay]

Production rateProduction rate with uq = 5 percent

0 0.5 1 1.5 2 2.5 3×105

0

100

200

300

400

500

600

Figure 5-6: Flow rate sequence without measurement errors and a sequence with5 percent measurement errors.



the estimation of PEI underestimates the pressure response. The estimate of the methodintroduced by [von Schroeter et al., 2004] shows a result more close to the reference stepresponse. The Horner plot (section 3-3) of Figure 5-7(c) shows this difference as well. Italso shows an overestimation of the method by [von Schroeter et al., 2004] compared to thereference signal. In the last plot of Figure 5-5, Figure 5-7(d), tg(t) is plotted again on a LogLogscale. The underestimation of step response by the method using PEI (Figure 5-7(b)) resultsin a decrease of this curve, since the derivative of the estimation is lower than the referencesignal. The estimate of the method developed at Imperial College, is again not significantlyaffected by the measurement error in the flow rate sequence, for the same reason as arguedin the situation with an error level of 2 percent of the flow rate measurements.

Time [s]

∆P

[PSI]


0 0.5 1 1.5 2 2.5 3×105

0

10

20

30

40

50

60


Time [s]

∆p

[PSI]


0 0.5 1 1.5 2 2.5 3×105

0

5

10

15

20

25

30

35

40

45

50


Time [s]

∆p

[PSI]


100 101 102 103 104 105 1060

5

10

15

20

25

30

35

40

45

50


Log time [s]

Log

∆p

and

Log

tg(t

)


Log time [s]

Log

∆p

and

Log

tg(t

)

100 101 102 103 104 105

100

101


Figure 5-7: Identification results for input with 5 percent uncertainty and outputwithout uncertainty.

The results above give an indication of the uncertainty in the estimations of the variousrepresentations. These representations are, however, not the final goal of well testing. Thereservoir properties have to be estimated using these plots. As described in Chapter 3 thiscan be done directly, using the Horner plot, and indirectly, using type curves. In this researchit is done directly, despite the fact that using the comparison with type curves is preferred inrecent literature. The goal of this direct estimation is to show the sensitivity of the estimation


52 Results

of the reservoir properties to changes in the plots.

5-3-1 Estimation of Reservoir Properties

As described in section 3-3, the Horner plots can be used for direct estimation. Using equations(3-2) and (3-3) the permeability k and the skinfactor S can be estimated. For these estimationsthe same reservoir is used as before (infinite size, skin factor S is 2 and the permeability is300 mDarcy). The error level of the input is set to 2 percent and there is no uncertainty onthe output. The identification experiment is done 25 times with every time a different inputsignal for the identification using equation (5-3). The ranges of estimated values for the skinfactor S and the permeability k are given in Table 5-1. The true system is assumed to beequal to the settings of the simulator.

ηq ηp Range True

Skin factor S 0.02 0.00 1.8 < S < 2.3 2Permeability k 0.02 0.00 293 < k < 312 300 mDarcy

Table 5-1: Estimation of permeability and the skin factor after input shown inFigure 5-2.

Using modern well test analysis software, such as Saphir (introduced in section 1-4), to es-timate the reservoir properties as described in section 3-3, this range is expected to be evensmaller. Note that these estimations are done using an input sequence that reduces the lossof oil production with 75 percent with respect to the case of a shut in. In this latter case, theranges of estimated values for the skin factor S and the permeability k are given in Table 5-2.All values for S and k are estimated with, again, ηq = 0.02 and ηp = 0. During the shut in,there is no uncertainty of the flow rate, but before the shut in there is uncertainty of the flowrate. Note that in the case of varying flow rate (e.g. Figure 5-2), the flow rate is measuredN times, whereas in the case of the shut in, the flow rate is only measured once. Again, thetrue system is assumed to be equal to the settings of the simulator.

ηq ηp Range True

Skin factor S 0.02 0.00 1.8 < S < 2.2 2Permeability k 0.02 0.00 294 < k < 307 300 mDarcy

Table 5-2: Estimation of permeability and the skin factor after complete shut in.

5-4 Finite Reservoir

Until now all results are generated using a reservoir of infinite size. As can be seen fromFigure 4-6, this type of reservoir shows linear behaviour. The reservoir used to generate Fig-ure 4-7 however, shows nonlinear behaviour. This is caused by the closed boundary relativelyclose to the wellbore and the well located in the center of the reservoir. When a radial reser-voir of very large size is considered but with the well located not in the center, but near theboundary, the nonlinearities will be much smaller. This is caused by the fact that the amountof oil produced during the well test is small compared to the total amount of oil present in the


5-4 Finite Reservoir 53

reservoir. In this case again both identification methods are tested. At first, the error level ofboth the flow rate and the pressure response is zero. The input used is shown in Figure 5-2.The number of data points used for the identification is again 6 · 104 and the simulated timeis 3 · 105 seconds. Figure 5-8 shows the result.

Time [s]

∆P

[PSI]


0 0.5 1 1.5 2 2.5 3×105

0

10

20

30

40

50

60


Time [s]

∆p

[PSI]


0 0.5 1 1.5 2 2.5 3×105

0

10

20

30

40

50

60


Time [s]

∆p

[PSI]


100 101 102 103 104 105 1060

10

20

30

40

50

60


Log time [s]

Log

∆p

and

Log

tg(t

)


Log time [s]

Log

∆p

and

Log

tg(t

)

100 101 102 103 104 105

101


Figure 5-8: Identification results for data without uncertainty and the boundaryclose to the well.

Both methods estimate the pressure response in Figure 5-8(a) without a significant differencewith each other and the simulated reference response. The estimated step response is alsonot significant different and equal to the simulated shut in pressure response. Note the highlevel of similarity with the shut in pressure responses of the reservoir of infinite size shown inFigure 5-3(b). The influence of the boundary is not visible in these representations. When thestep response is shown with a semi log scale, the Horner plot in Figure 5-8(c), the differencebecomes visible by an increase of the pressure drop. The semi log straight line (described insection 3-3) is not lasting till the end of the test, as was the case in Figure 5-3(c). Instead,the slope of ∆p(t) increases after 2 · 103 seconds (compared with the semi log straight linein Figure 3-1). Physically this makes sense, by noting that the closed boundary causes anincreasing pressure drop because there is no oil flowing through this boundary to replace theoil produced by the well. In Figure 5-8(d) this effect is also visible in the ∆p(t) and tg(t) curve.


54 Results

The points generated by the method of [von Schroeter et al., 2004] follow the reference signalsignificantly better than the the curve estimated by the method using PEI. PEI however doesestimate a full expression for this curve, whereas the method by [von Schroeter et al., 2004]only estimates K nodes.

The oscillations of the derivative LogLog representations estimated by using PEI can beexplained by analyzing the fourth order OE model. In the time domain, the transfer functionof this model is given by:

gOE440(t) = constant +4∑

j=1

cjedjt, (5-5)

where the constant, cj and dj are estimated coefficients.

Equation (5-5) is visualized in Figure 5-9 by plotting tgj(t) = tcjedjt on a LogLog scale for

j = 1..4. The solid red line represents the sum of these. In this example, the coefficients cjand dj of a estimated model based on simulated data is used. The solid red line has thereforemany similarities with the derivative LogLog type curves.

time[s]

tgi(t)

Components of tg(t)

tg1(t) = tc1ed1t

tg2(t) = tc2ed2t

tg3(t) = tc3ed3t

tg4(t) = tc4ed4t

tg(t) = t∑

i gi(t)

10−2 100 102 104 10610−10

10−8

10−6

10−4

10−2

100

Figure 5-9: Examples of weighted basis functions for tg(t) of PEI

Based on this observation, one might expect that by increasing the order of the model, theestimations of the derivative LogLog plot become better. Indeed, sometimes the amplitude ofthe oscillations becomes less using an OE model of higher order, but often the estimation ofthe large number of coefficients in the higher order model generates numerical troubles. Notethat these coefficients are estimated for the linear domain. In that domain, as can be seen inFigure 5-1(b), the fit of the fourth order model is already nearly 100 percent.


5-5 Summary 55

5-5 Summary

In this chapter the results are presented of the evaluation of both the identification methodpresented in section 4-5 and the method introduced in section 4-6. The fourth order OEappeared to be the most suitable model to model the simulated data. The results of bothmethods, the method described in section 4-5 and the newly introduced method using PEI,described in section 4-6, are compared to each other and also to reference results for variouserror levels of the flow rate. For a low level (0 - 2 percent) of measurement errors in the flowrate, differences between the two methods are only visible in the derivative LogLog type curverepresentation. In this domain, the method described in section 4-5 outperforms the methodusing PEI. This is mainly caused by the manually set parameters in the first method. Bothmethods performed well in all other domains. For an error level of 5 percent in the flow rate,the differences between the results of both methods are also visible in the estimation of thestep response (linear domain) and the semi log domain.

Permeability and the skin factor are estimated using PEI in combination with the directinterpretation of the semi log straight line as introduced in section 3-3. These estimations arecompared with the estimations of the reservoir parameters after applying a complete shut in.The results are comparable. Important to notice is that for the estimations using the datawithout a shut in, the flow rate is measured N times, whereas the flowrate is only measuredonce in the case of a complete shut in.

Also the ability for closed boundary detection of both methods is evaluated (using an noncentered well). Both methods are able to detect the closed boundary. Important to notice isthat the level of nonlinearity of the system is dependent on the size of the reservoir and theduration of the well test.

Using these results the research questions stated in section 1-4 can be answered and conclu-sions can be drawn. This is done in Chapter 6.


56 Results


Chapter 6

Conclusions and Recommendations

6-1 Introduction

After the presented results in the previous chapter, conclusions can be drawn. Based onthe results, the five research questions stated in the introduction are answered. This isdone in the first section, section 6-2. This is the first research known to the author thatintroduces Prediction Error Identification (PEI) in the field of well testing. Many subjectscan be investigated in more detail. In section 6-3 a number of recommendations are given forfurther research.

6-2 Conclusions

In this chapter the conclusion of the research are presented by answering the five researchquestions stated in section 1-4.

Q1 ‘How do the introduced deconvolution techniques work?’

The working method of the ‘deconvolution’ method introduced by [von Schroeter et al., 2004]is described in section 4-5. Note that, according to the definition of deconvolution used in thisreport (see section 1-4), the method described is not a real deconvolution method. However,it is an identification method. K equally spaced nodes are estimated of the tg(t) curve in theLogLog domain (section 3-3), with time t and the impulse response g(t) of the system. Theflow rate is divided in N flow periods, visualized in Figure 4-2. Therefore, this method worksin a situation with a limited number of flow periods. A flow period is a period in which theflow rate is assumed to be constant. A Total Least Squares (TLS) method and regularizationis used to minimize the error, accounting for measurement errors in the flow rate and pressureresponse, see section 4-5-2. The relative weights of these errors are determined by a factorν. This factor has to be set manually. In the cost function (equation (4-18)) the curvature ofthe curve estimated by the nodes is punished. The weight λ is another factor that has to be


58 Conclusions and Recommendations

set manually. This method directly estimates in the domain preferred by reservoir engineersat the moment.

Q2 ‘How do the already developed deconvolution algorithms relate to systemidentification. Are the used methods justified and reliable in terms of systemidentification?’

The method introduced by [von Schroeter et al., 2004] can be considered as a system iden-tification method (section 4-5). The method is reliable in the very narrow field for whichit is designed: interpretation of the curve of tg(t) in the LogLog domain (section 3-3). Thejustification of the factors that have to be set manually can be questioned (section 4-5-3).Experiments are needed to validate the information they add to the estimation: informationabout the expected error levels in the data and the expected curvature of the estimated curve.

Q3 ‘Is it possible to remove the type curves from well test analysis, using systemidentification techniques?’

The system identification method introduced in [von Schroeter et al., 2004] cannot be usedto remove the type curve analysis, since the only goal of this method is to estimate themost recent type curve (section 4-5). Applying the method using PEI, the impulse responseg(t) of the system is estimated (section 4-6-1). The parameters of this model are not yetlinked directly to the reservoir properties (section 4-6-5). The estimated model is used to plotthe type curves, as shown in the Chapter 5. These type curves are linked to the reservoirproperties using the direct estimation method, described in section 3-3. Until now type curvesare needed to estimate the reservoir properties.

Q4 ‘Is system identification able to deliver reliable uncertainty estimations forthe estimations of reservoir properties?’

System identification as described in section 4-6, is able to deliver uncertainty estimations ofthe model estimated. This uncertainty can then be mapped onto every result created withthis model. In section 4-6-3 the equations for these uncertainty regions are presented, e.g.equation (4-38). Since the estimated model is able to present a type curve with uncertaintyregion, the estimations of the wellbore storage, skin and permeability have an uncertaintyregion as well. However, in this research the simulated results are not suited to generateuncertainty regions that are reliable for real measured data. Note that common systemidentification techniques expect no disturbance on the input. Therefore, the identification byPEI neglects this disturbance at the moment. This is one of the major challenges for furtherresearch.

At the moment uncertainty of the reservoir properties estimations can only be estimated byperforming Monte Carlo (MC) simulations. An important remark is that when using typecurves the estimated uncertainty in the reservoir property estimate is not only caused by theuncertainty in the estimated model but also by the uncertainty in the type curve interpretationmethod.

Q5 ‘Is it possible to reduce the costs of well testing using deconvolution and/orsystem identification?’

The use of the two system identification algorithms can reduce the costs of well testing,by decreasing the amount of production loss during the well test, see Figure 1-5. Bothidentification methods enable identification of the system using multiple flow rate periods, as


6-3 Recommendations 59

described in chapter 4. Using these multiple flow rates as alternative for a complete shut inreduces the production loss, assumed that the estimations of the reservoir properties using thesystem identification techniques are satisfactory. The results show that the estimation methodof the reservoir properties, skinfactor S and the permeability k, during a production sequenceas shown in e.g. Figure 5-1, are comparable to the estimations using a shut in. Important isthe sensitivity of both identification methods to measurement errors. The method introducedin [von Schroeter et al., 2004] has shown to give a satisfactory estimate of K nodes describingthe derivative LogLog type curve. The method using PEI shows good results for a numberof domains (see Figure 5-3 - Figure 5-8), but is less effective is describing this same curve.However, the method using PEI estimates a full version of the impulse response function(section 4-6), whereas the method by [von Schroeter et al., 2004] only estimates nodes of afunction of the impulse response (section 4-5). Considering direct estimation of the reservoirproperties, instead of evaluating estimated type curves, the method using PEI has muchhigher potential to be successful because of this full expression (section 1-4).

6-3 Recommendations

Since this research has an explorative character, a lot of recommendations can be made. Themain recommendations for further research are listed.

- To get a reliable estimation of the uncertainties using PEI, real well test data are needed.There is no evidence that simulated disturbances will have the same properties as thereal disturbances. Simulated disturbances are therefore not reliable at the moment.Testing the described methods on real well test data will determine the value of thedescribed methods for identification.

- In order to use PEI to estimate open boundaries, more research is needed. The factthat very small and large time constants are estimated at the same time, makes a goodestimation difficult. Separating fast dynamics from slow dynamics is therefore an optionthat can be investigated in the future. The well test should then be divided in two parts.One part is used to estimate the fast dynamics and the other part is used to estimatethe slow dynamics.

- Until now it is not possible to express the reservoir properties as a function of the esti-mated model parameters. The relation between the parameters in the estimated modeland the reservoir properties is unknown. With more research it might be possible to con-struct a model in which the parameters are directly related to the reservoir properties.Note that the total number of parameters is not (significant) reduced using the currentOutput Error (OE)-model. Type curves are redundant in that case, since estimationsof the reservoir properties can be made directly after estimating the reservoir model.Removing the type curves from well test analysis would be a major improvement.

- The contribution of the LogLog domain to well test analysis is unquestioned. However,until now, the use of this domain is strictly limited to the use of LogLog derivativetype curves. It would be interesting to see how the advantages of this domain could beincorporated in other well test analysis methods, i.e. without first estimating the typecurve.



- Using the identification technique according to [von Schroeter et al., 2004] or PEI re-moves the need for a shut in. Further research is needed to investigate the exact re-duction of costs that can be achieved. The required level of accuracy for the reservoirestimations will then play a major role. Also the capabilities of the well test equipmentare important to design the input signal. Concerning this input design, it is importantto notice the difference between theory and reality. On one hand you could argue thatthe practical possibilities of the well equipment are limiting factors for the input design,since input signals that cannot be implemented are useless in the field. On the otherhand there are feasible arguments for little restriction of the research to these limitationssince the well equipment will only be improved in case the return is worth the effort.Better well testing results or less oil production loss can be that return.

- In literature about system identification, a lot is written about how to deal withnoise-corrupted output. On the other hand, estimation of the parameters for lineardynamic systems where also the input is affected by uncertainty is recognized as amore difficult problem. Representations where errors or measurement uncertainties arepresent on both inputs and outputs are usually called Errors In Variables (EIV) models[Soderstrom, 2006]. The EIV problem is a violation of the assumption that the inputsignal is known. PEI theory has not yet a clear solution. Also it is unknown whether thechoice of the model structure influences the impact of the EIV-problem on the results.

Systemu0(t)

u(t)

++

++

y(t)y0(t)

u(t)

y(t)

(a) An error-in-variables problem.

Systemu0(t)u(t)

u(t)

++++ y(t)y0(t)

y(t)

(b) A false error-in-variables problem.

Figure 6-1: The basic setup for a dynamic error-in-variables (EIV) and a falseEIV problem.

Figure 6-1(a) shows the EIV problem. The undisturbed input is denoted by u0(t) andthe undisturbed output by y0(t). When the observations are corrupted by additivemeasurement uncertainties u(t) and y(t), the available, measured, signals in discretetime are given by:

u(t) = u0(t) + u(t)y(t) = y0(t) + y(t)

(6-1)

Figure 6-1(b) shows a situation that is often confused with the EIV problem. In Fig-ure 6-1(b), the input disturbance u(t) is added to u0(t) before the input reaches thesystem. In this situation the dynamics between u0(t) and y0(t) is the same as be-tween u(t) and y0(t). The available, measured, signals in discrete time are given by[Soderstrom, 2006]:

u(t) = u0(t) − u(t)y(t) = y0(t) + y(t)

(6-2)


6-3 Recommendations 61

More research on how to deal with the uncertainty in the flow rate would mean a majorimprovement of the well test analysis using PEI.

- In this research the identification is done in the time domain. However, there are alsopossibilities to identify the reservoir properties in the frequency domain. In Appendix Cthe sensitivity of the bode plot for a changing skin factor and permeability is illustrated.It might be investigated whether identification of the system is possible in the frequencydomain.


Bibliography

[Agarwal, 1970] Agarwal, R. G. (1970). An investigation of wellbore storage and skin effectin usteady liquid flow: I. analytical threatment. Society for Petroleum Engineers, pages279–290.

[Bourdarot, 1998] Bourdarot, G. (1998). Well Testing: Interpretation Methods. EditionsTechnip.

[Bourdet et al., 1983] Bourdet, D., Whittle, T., Douglas, A., and Pirard, Y. (1983). A newset of type curves simplifies well test analysis. World Oil, pages 95–106.

[Dake, 1978] Dake, L. P. (1978). Fundamentals of Reservoir Engineering. Elsevier.

[Dranchuk and Quon, 1967] Dranchuk, P. M. and Quon, D. (1967). Analysis fo the darcycontinuity equation. Producers Monthly, pages 25–28.

[Earlougher, 1977] Earlougher, R. C. (1977). Advance in Well Test Analysis. SPE of AIME.

[Grader and Horne, 1988] Grader, A. and Horne, R. N. (1988). Interference testing: Detect-ing an impermeable or compressible sub-region. SPE Formation Evaluation, pages 424–437.

[Gringarten, 2008] Gringarten, A. C. (2008). From straight lines to deconvolution: The evolu-tion of the state of the art in well test analysis. SPE Reservoir Evaluation and Engineering,11(1):41–62.

[Horne, 1995] Horne, R. N. (1995). Modern Well Test Analysis, a Computer-Aided Approach.Petroway, Inc., second edition.

[Horner, 1951] Horner, D. R. (1951). Pressure build-up in wells. Third World Petroleum

Congress, pages 503–523.

[Jansen, 2007] Jansen, J. D. (2007). Systems theory for reservoir management. Delft Univer-sity of Technology, version 3v edition.

[Lee et al., 2003] Lee, J., Rollins, J. B., and Spivey, J. P. (2003). Pressure Transient Testing.Society for Petroleum Engineers.


64 Bibliography

[Levitan et al., 2005] Levitan, M. M., Crawford, G. E., and Hordwick, A. (2005). Practi-cal considerations for pressure/rate deconvolution of well-test data. Journal of petroleum

technology, 57(2):71–72.

[Ljung, 1987] Ljung, L. (1987). System identification. Springer.

[Matthews and Russell, 1967] Matthews, D. G. and Russell, D. (1967). Pressure Buildup and

Flow Tests in Wells. Society for Petroleum Engineers.

[Onur et al., 2008] Onur, M., Cinar, M., IIK, D., Valko, P. P., Blasingame, T. A., and Hege-man, P. S. (2008). An investigation of recent deconvolution methods for well test dataanalysis. SPE.

[Raghavan, 1993] Raghavan, R. (1993). Well Test Analysis. Prentice Hall.

[Schlumberger, 2008] Schlumberger (2008). Origin of natural oil and gas.http://www.seed.slb.com.

[Simpson and Weiner, 2008] Simpson, J. and Weiner, E. (2008). The Oxford English Dictio-

nary. Oxford University Press.

[Soderstrom, 2006] Soderstrom, T. (2006). Error-in-variables methods in system identifica-tion. Automatica, pages 939–958.

[van den Akker and Mudde, 2007] van den Akker, H. E. A. and Mudde, R. F. (2007). Fysis-

che Transportverschijnselen. VSSD.

[van Everdingen and Hurst, 1949] van Everdingen, A. F. and Hurst, W. (1949). The appli-cation of Laplace transformation to fluid problems in reservoir. AIME Petroleum Transac-

tions, 189:305–324.

[von Schroeter et al., 2001] von Schroeter, T., Hollaender, F., and Gringarten, A. (2001).Deconvolution of well test data as a nonlinear total least squares problem. Society for

Petroleum Engineers.

[von Schroeter et al., 2004] von Schroeter, T., Hollaender, F., and Gringarten, A. (2004).Deconvolution of well test data as a nonlinear total least squares problem. Society for

Petroleum Engineers.

[Whittle and Gringarten, 2008] Whittle, T. and Gringarten, A. (2008). The determinationof minimum tested volume from the deconvolution of well test pressure transients. SPE.

[Zandvliet, 2008] Zandvliet, M. J. (2008). Model-based Lifecycle Optimization of Well Lo-

cations and Production Settings in Petroleum Reservoirs. PhD thesis, Delft University ofTechnology.


Appendix A

The Diffusion Equation and Solutions

A-1 Introduction

Based on the three laws introduced in Chapter 2 the diffusion equation, equation (2-7), canbe derived. In the next section of this appendix, this derivation is given in more detail. Basedon equation (2-7), and the conditions described in section 2-3-2, solutions of the diffusionequation can be derived. The details of the derivations are given in the third and last section.

A-2 Derivation of the Diffusion Equation

In this section the derivation of equation (2-7) will be given in more detail.

Combining equation (2-1) with the mass balance gives:

1

r

∂

∂r

(kr

µρ∂p

∂r

)

= φ∂ρ

∂t(A-1)

The time derivative of the density appearing on the right hand site of (A-1) can be expressedin terms of a time derivative of the pressure by using the basic thermodynamic definition ofisothermal compressibility:

cρ =1

ρ

(∂ρ

∂p

)

T

(A-2)

From (A-2) it can be deduced that

cρρ∂p

∂t=∂ρ

∂t(A-3)

Combining (A-1) and (A-3), it results in

1

r

∂

∂r

(kρ

µr∂p

∂r

)

= φctρ∂p

∂t(A-4)


66 The Diffusion Equation and Solutions

Note that the compressibility cρ from (A-3) is replaced by ct because according to (2-3), ct isequal to co when cp, the compressibility of the porous medium, is zero.

When the source term is included , (A-4) looks like

1

r

∂

∂r

(kρ

µr∂p

∂r

)

+ ρq∗ = φctρ∂p

∂t(A-5)

q∗ represents the flow rate q per unit volume ([q∗] = s−1)[Jansen, 2007].

(A-4) is a nonlinear equation because of the implicit pressure dependence of the density ρ,compressibility c and viscosity µ. Therefore, there are no simple solutions of (A-4) withoutlinearizing this equation. [Bourdarot, 1998, Dake, 1978]

When considering radial and vertical flow, the diffusion equation is

∂2p(r, t)

∂r2+

1

r

∂p(r, t)

∂r+∂2p(r, t)

∂z2=φctµ

k

∂p(r, t)

∂t. (A-6)

[Horne, 1995]

The same equation, but now in cartesian coordinates, including a source term, looks like

−∇.[αρ

µ~K∇p

]

+ αρφct∂p

∂t− αρq∗ = 0, (A-7)

in which α(~x) is a geometric factor. h is the reservoir height. Gravity is again assumed to benegligible [Jansen, 2007].

When equation (A-7) is rewriten in scalar 2D form, assuming isotropic permeability, smallcompressibility of porous medium and fluid, uniform reservoir thickness and absence of gravityforces the result is

−hµ

∂

∂x

(

k∂p

∂x

)

− h

µ

∂

∂y

(

k∂p

∂y

)

+ hφct∂p

∂t− hq∗ = 0. (A-8)

Note that k still can vary in space [Jansen, 2007]. Also it still includes the geometrical factorh introduced in (A-7).

In order to be able to generate simple solutions, liberalization of equation (A-4) is applied.Using the chainrule for differentiation equation (A-4) can be written as

1

r

[∂

∂r

(k

µ

)

ρr∂p

∂r+k

µ

∂ρ

∂rr∂p

∂r+kρ

µ

∂p

∂r+kρ

µr∂2p

∂r2

]

= φctρ∂p

∂t. (A-9)

In the same way as equation (A-2) is rewritten to equation (A-3), equation (A-2) can berewritten to

cρ∂p

∂r=∂ρ

∂r(A-10)


A-2 Derivation of the Diffusion Equation 67

When this equation is substituted into equation (A-9) the result is

1

r

[

∂

∂r

(k

µ

)

ρr∂p

∂r+k

µcρr

(∂p

∂r

)2

+kρ

µ

∂p

∂r+kρ

µr∂2p

∂r2

]

= φctρ∂p

∂t. (A-11)

If the viscosity, µ, and the permeability, k, are assumed to be practically independent ofradius r and may be regarded as a constant for liquid flow, the resulting equation is

1

r

[

k

µcρr

(∂p

∂r

)2

+kρ

µ

∂p

∂r+kρ

µr∂2p

∂r2

]

= φctρ∂p

∂t. (A-12)

Another assumption that can be made is that the pressure gradient ∂p∂r

is small and therefore,

terms of the order(

∂p∂r

)2can be neglected.

Considering this last assumption, equation (A-12) reduces to

∂2p

∂r2+

1

r

∂p

∂r+q∗µ

k=φµctk

∂p

∂t. (A-13)

In this equation a source term is included. Equation (A-13) can be more conveniently ex-pressed as

1

r

∂

∂r

(

r∂p(r, t)

∂r

)

+q∗µ

k=φµctk

∂p(r, t)

∂t(A-14)

The basic equation has been linearized, assuming that the compressibility is constant. Thissimple linearization by deletion must be treated with caution and can only be applied whenthe product cp≪ 1 [Dranchuk and Quon, 1967, Dake, 1978].

In this research focus is on reservoir models without heterogeneities but with so-called homo-geneous properties. In order to reduce the complexity of the model, assumptions are oftenmade. Properties like porosity, compressibility and permeability are considered to have thesame value everywhere in the reservoir. This eliminates the angular dependency of the radialdiffusion equation. Oil reservoirs always contain both hydrocarbons and water. Hydrocarbonsconsists of many chemical components which, theoretically, should each be considered individ-ually in the modeling process [Zandvliet, 2008]. Due to the computational complexity, thesevarious components are often simplified to three phases: oil, gas and water. In this researchoil is considered to be the only phase present: only one-phase-flow is considered. Gravityeffects will be neglected, just like capillary pressures. These are common made simplificationsin this field. Different boundary conditions can be present for reservoirs.

There are more assumptions commonly made regarding reservoir properties. The main as-sumptions are:

- Porosity, permeability, viscosity and compressibility are constant

- Fluid compressibility is small.

- Pressure gradients in the reservoir are small.

- Thermal effects are negligible.

- The porous medium is completely saturated with the fluid (oil in our case).



A-3 Solutions of the Diffusion Equation

Transient flow During the initial transient flow period, it has been found that the constantterminal rate solution of the radial diffusivity equation can be approximated by the line sourcesolution which assumes that in comparison to the apparently infinite reservoir the wellboreradius is negligible en the wellbore itself can be treated as a line in stead of a cylinder (or acylinder with radius zero). Therefore the boundary conditions can be restated as

- p = pi at t = 0 ∀r.Before producing, the pressure everywhere within the drainagevolume is equal to the initial equilibrium pressure pi.

- p = pi at r = ∞ ∀t. The pressure at the outer, infinite boundary is not affected by thepressure disturbance at the wellbore and vice versa.

- limr→0 r∂p∂r

= qµ2πkh

for t > 0. The line source inner boundary condition.

Furthermore, the formation is still assumed to be homogeneous and isotropic, and drained bya fully penetrating well to ensure radial flow. The fluid itself must have a constant viscosityand a small and constant compressibility.

First equation (A-4) is rewriten as

1

r

∂

∂r

(

r∂p

∂r

)

= K∂p

∂t(A-15)

Where K = φµck

is substituted. Then s = K r2

4tis substituted. It follows that

∂s

∂t= −K r2

4t2(A-16)

∂s

∂r= K

r

2t. (A-17)

Rewriting equation (A-15) it results in

1

r

∂

∂s

∂s

∂r

(

r∂p

∂s

∂s

∂r

)

= K∂p

∂s

∂s

∂t. (A-18)

Inserting equation (A-16) and equation (A-17) gives:

1

rKr

2t

d

ds

(

rKr

2t

dp

ds

)

= −(

Kr

2t

2)dp

ds. (A-19)

Dividing both sides through K r2t

generates

1

r

d

ds

(S

2

dp

ds

)

= −(

Kr

2t

) dp

ds(A-20)


A-3 Solutions of the Diffusion Equation 69

Multiplying both sides with r and rewriting gives

d

ds

(

sdp

ds

)

= −sdpds

(A-21)

When p′ = dpds

is introduced, it can be written as

dsp′

ds= −sp′ (A-22)

sdp′

ds+ p′

ds

ds= −sp′ (A-23)

sdp′

ds+ p′ = −sp′ (A-24)

sdp′

ds= −(s+ 1)p′ (A-25)

dp′

p′= −s+ 1

sds (A-26)

Now integrating both sides

ln p′ = − ln s− s+C (A-27)

With C a constant.

It can be seen that

p′ = C2e−s

s(A-28)

When the so called line source boundary condition is now introduced (which assumes thatin comparison to the apparently infinite reservoir the wellbore radius is negligible and thewellbore itself can be treated as a line), it can be assumed that at r = ∞ the pressure p isconstant in time at pi) limr→∞r

∂p∂r

= qµ2πkh

. (following from Darcy’s law : q = 2πrhkµ

∂p∂r

). Theequation that follows is

2sp′ =qµ

2πkh(A-29)

since

r∂p

∂r= r

dp

ds

∂s

∂r(A-30)

Combining this with equation (A-28) gives the value for C2 = qµ4πkh

.

When equation (A-28) is integrated to get p in stead of p′, we have to do this between thelimits. On the right side these limits are t = 0, (s → inf) and the current value of t, for whichs = x. On the left side these limits are the initial pressure pi and the current pressure p.



This results in

p(r, t) = pi −qµ

4πkh

[

Ei(φµcr2

4kt)

]

+ q, (A-31)

where

Ei(x) =

∫ ∞

x

e−s

sds (A-32)

The integration limits are t = 0 (s = ∞) and the current value for t, for which s = x[Dake, 1978].

Assuming x < 0.01, this can be approximated by

p(r, t) − pi = − qµ

4πkh

[

ln4kt

γφµctr2+ 2S

]

(A-33)

Semi Steady State flow The solution technique for semi steady flow is set out in somedetail since the method is a perfectly general one which can be applied for a variety of radialflow problems.

Substituting the constant of (2-8) into (A-14) gives

1

r

∂

∂r

(

r∂p

∂r

)

= − qµ

πr2ekh(A-34)

Integrating this function once to radius r gives

r∂p

∂r= − qµr2

2πr2ekh+ C1 (A-35)

Using the outer no-flow boundary (∂p∂r

= 0) we can calculate C1 as C1 = qµ2πkh

, which trans-forms (A-35) into

∂p

∂r=

qµ

2πkh

(1

r− r

re2

)

(A-36)

Which, when integrated over the interval [rw r], results in

p(r) − pwf =qµ

2πkh

(

lnr

rw− r2

2r2e+rw

2

2re2+ S

)

(A-37)

p(r) is the pressure at a distance r from the center of the wellbore, and pwf is the pressure at

the wellbore radius rw. In (A-37) rw2

2re2 can be considered to be negligible [Dake, 1978]. The

difference between p(r) and pwf is independent of time t, since ∂p∂t

is constant ∀r, t.


A-3 Solutions of the Diffusion Equation 71

Steady State flow Using the same mathematical steps, we can derive the steady statesolution. Starting with (A-14), and using ∂p

∂t= 0

1

r

∂

∂r

(

r∂p

∂r

)

= 0 (A-38)

Integrating (A-38) once to radius r, results in

r∂p

∂r= C2 (A-39)

with C2 a constant. Now note that when p = pe = constant at r = re, then ∂p∂t

= 0. Hence,it follows that C2 = qµ

2πkh.

Integrating once more to r, in the same way as the semi steady state (over the interval [rw r])results in equation (A-40). The solution is independent of time t:

p(r) − pwf =qµ

2πkh

(

lnr

rw+ S

)

. (A-40)


Appendix B

Errors in SPE 77688

During the research some mistakes are found in the paper presenting the deconvolutionmethod [von Schroeter et al., 2004] described in this report.

These mistakes are probably typing mistakes.

• In equation (28), a minus sign is forgotten in the second term of the array. In order tobe consistent with equation (25) it should be

ν(x′′) =

p√νq

−√λ(Dx′′ − k)

• In the text after equation (A-6), the term eαk is missing in the second term. The correctversion is:

∂Cij

∂zk= eα1δk,1

∫ τ1

−∞θj(ti − eτ )eτβkdτ + eαk

∫ ln(T )

τ1

θj(ti − eτ )ψk(τ)eτβkdτ

• The last error is also a missing minus sign. In equation (B-6b), the second term in thearray should be

− τi+1 − τi−1

(τi+1 − τi)(τi − τi−1)if j = i


74 Errors in SPE 77688


Appendix C

Reservoir Parameters and Frequency

Domain

In literature about well testing a lot is written about identifying the reservoir parametersin the time domain. Also a lot is written about the challenges that this way of identi-fying brings along. Until now, no article describes a way of identifying the reservoir pa-rameters in the frequency domain (deconvolution sometimes does use the frequency domain[von Schroeter et al., 2004]).The use of the frequency domain has proven its value in a lot ofother applications of system identification. In order to evaluate the frequency domain thebode magnitude and phase plot of the identified model might be able to help. It might beworth the effort to try to identify the reservoir parameters in this way, since if it is successful,the type curves, together with their inconvenient mapping might become unnecessary.

The length of the well test is mainly determined by the reservoirs boundaries. When a well isproducing for a while, the information about the boundaries is expected to be in the measureddata. Frequency analysis might be able to get this information out.

C-1 Reservoir Parameters

In this section the influence of the four physical reservoir properties from section 1-2-4 on thefrequency content of the pressure response is shown.

Since MoReS is not generating pressure responses with a constant sampling rate, looking atthe bode plots of a fitted Ouput-Error model is preferred above a Fast Fourier Transform ofthe pressure response generated by MoReS.

C-1-1 Skin and permeability

In order to see the influence of the value of the skin and permeability in the frequencycontent of the estimated model, the simulator generated the response for varying values of


76 Reservoir Parameters and Frequency Domain

these parameters. The responses are generated with the same input sequence. Results areshown in Figure C-1.


C-1 Reservoir Parameters 77

Input MoReS

Time [days]

Flo

wra

te[B

BL/d

ay]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

100

200

300

400

500

600

(a) flow rate

Output MoReS

Time [days]

∆p

[psi

]

skin 0 OE330skin 2 OE440skin 3 OE440skin 4 OE440

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

(b) Pressure response for a varying skin factor.

Bode plot of estimated OE model

Frequency (Hz)

Mag

nitude

(dB

)

Frequency (Hz)

Phas

e(d

eg)

skin 0 OE330skin 2 OE440skin 3 OE440skin 4 OE440

10−7 10−6 10−5 10−4 10−3 10−2 10−1

10−7 10−6 10−5 10−4 10−3 10−2 10−1

120

140

160

180

0

0.1

0.2

0.3

0.4

(c) Bode plot of estimated OE model for a varyingskin factor.

Output MoReS

Time [days]

∆p

[psi

]

perm 10 OE330perm 50 OE440perm 100 OE440perm 200 OE440

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

(d) Pressure response for a varying permeability.

Bode plot of estimated OE model

Frequency (Hz)

Mag

nitude

(dB

)

Frequency (Hz)

Phas

e(d

eg)

perm 10 OE330perm 50 OE440perm 100 OE440perm 200 OE440

10−7 10−6 10−5 10−4 10−3 10−2 10−1

10−7 10−6 10−5 10−4 10−3 10−2 10−1

0

50

100

150

200

0

1

2

3

4

(e) Bode plot of estimated OE model for a varyingpermeability.

Figure C-1: For a fixed input (a), and a varying skin factors and permeability,the output is simulated (b and d). Also the bode plots are generated(c and e).MSc. Thesis I.T.J. Kuiper

78 Reservoir Parameters and Frequency Domain


Appendix D

Well Test Simulation: MoReS

D-1 Why Using a Simulator

In order to find a solution that will work in practice, a test facility is needed. It is needed tofor several reasons. The first reason is that it enables the possibility to see what the influenceof changing parameters is on the measured signal. The influence of the varying values forthe reservoir properties on the pressure response can be easily shown. Also, interpretationmethods can be checked by comparing the resulting estimates with the reference value in theinput file of the simulator. Of course theory is an important part of the research, but thesimulator enables us to theory with practise.

D-2 Demands for Simulator

There are several demands for a simulator. The first requirement is that the simulator isa good representation of real reservoirs. The generated output must correspond with themeasured output from a reservoir with the same properties as the ones set in the simulator.A second requirement is that the reservoir must be able to simulate all the reservoirs thatneed to be tested. Besides these two main requirements it is of course an advantage wheneverybody that is going to see the results trusts the used simulator. Also, the easy of usecan play a minor role in choosing the simulator. In this research Shell’s inhouse simulatorDynamo - MoReS is used as simutalor.

D-3 Alternatives for Simulator

Besides Dynamo - MoReS one other reservoir simulator is available. It is the Matlab basedSimSim, developed at Delft University of Technology (DUT). Chosen is to use MoReS, sinceSimSim isn’t yet able to simulate well bore storage. Another minor reason to chose for MoReSwas that results would be more easily accepted within Shell, since they have more experience


80 Well Test Simulation: MoReS

with MoReS than with SimSim. A disadvantage of MoReS is the input language, which isdifferent than the more common Matlab input. However, it is not a very difficult language.


Appendix E

Software

To complete this research, several software packages are used. In alphabetic order:

• Dynamo, MoReS. Shell International E & P Version 2008.1 build 80623.

• Ecrin, Saphir. Kappa Eng. Version 4.10.02.

• Matlab. The MathWorks Inc. Version 7.2.0.232 R2006a.

• Maple. Waterloo Maple Inc. Version 11.01.


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