TWO-PHASE FLOW PRESSURE TRANSIENT ANALYSIS OF …

TWO-PHASE FLOW PRESSURE TRANSIENT ANALYSIS OF CARBONATE

RESERVOIRS WITH HIGH PERMEABILITY LENS INTERSECTED BY THE

WELLBORE

Tatiana Lipovetsky

Dissertação de Mestrado apresentada ao

Programa de Pós-graduação em Engenharia

Civil, COPPE, da Universidade Federal do

Rio de Janeiro, como parte dos requisitos

necessários à obtenção do título de Mestre

em Engenharia Civil.

Orientadores: Paulo Couto.

José Luis Drummond Alves.

Rio de Janeiro

Dezembro de 2013



WELLBORE

Tatiana Lipovetsky

DISSERTAÇÃO SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO

LUIZ COIMBRA DE PÓS-GRADUAÇÃO E PESQUISA DE ENGENHARIA

(COPPE) DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE

DOS REQUISITOS NECESSÁRIOS PARA A OBTENÇÃO DO GRAU DE MESTRE

EM CIÊNCIAS EM ENGENHARIA CIVIL.

Examinada por:

________________________________________________ Prof. Paulo Couto, Dr. Eng.

________________________________________________

Prof. José Luis Drummond Alves, Dr.

________________________________________________

Prof. Patrick William Michael Corbett, Dr.

________________________________________________

Prof. Luca Roberto Augusto Moriconi, Dr.

RIO DE JANEIRO, RJ - BRASIL

DEZEMBRO DE 2013

iii

Lipovetsky, Tatiana

Two-Phase Flow Pressure Transient Analysis Of Carbonate

Reservoirs With High Permeability Lens Intersected By The

Wellbore / Tatiana Lipovetsky. – Rio de Janeiro: UFRJ/COPPE,

2013.

XVIII, 138 p.: il.; 29,7 cm.

Orientadores: Paulo Couto

José Luis Drummond Alves

Dissertação (mestrado) – UFRJ/ COPPE/ Programa de

Engenharia Civil, 2013.

Referências Bibliográficas: p. 94-96.

1. Simulação de Fluxo Multifásico. 2. Reservatórios

Carbonáticos do Pré-Sal. 3. Análise de Teste de Poço. I. Couto,

Paulo et al. II. Universidade Federal do Rio de Janeiro, COPPE,

Programa de Engenharia Civil. III. Titulo.

iv

ACKNOWLEDGEMENTS

First of all, I’d like to thank and dedicate this work to my parents, grandparents

and sister. They gave me all kinds of support, so I could accomplish this mission I gave

myself, never questioning why. God and Guardian Angel, thank you so much for

putting me in such loving and caring family. Else, I’d like to thank the amazing friends I

made in Rio de Janeiro and those I left back home, that never let me down.

Also, I’d like to thank COPPE/UFRJ and staff for the provided apprenticeship.

The NIDF and LAMCE laboratories for all the opportunities and friends I’ve made!

Petrobras for funding my scholarship. Schlumberger and its team for believing in me,

supporting and preparing me with their training courses and software programmes. BG

Group members for being always present and concerned about us all.

To my professors and colleagues, I appreciate all you have done. Paulo Couto,

Patrick Corbett and José Luis Alves, I have no words to describe how thankful I am.

You believed and trusted me, gave me opportunity, pushed me, taught me more than

you can ever possibly imagine. You were more than professors: role models. Professors

Luca Moriconi and Fábio Ramos, I sincerely feel grateful for all the help, advices and

friendship in and outside school. My NIDF friends: Marcelo Marsili, Luis Carrión,

Ronny Santana, Rodrigo Pereira, Ricardo Fernandes and Daniel Rodrigues – thank you

for the great times and for our “brainstorms”!

v

Resumo da Dissertação apresentada à COPPE/UFRJ como parte dos requisitos

necessários para a obtenção do grau de Mestre em Ciências (M.Sc.)

ANÁLISE DE TRANSIENTE DE PRESSÃO DE FLUXO BIFÁSICO DE

RESERVATÓRIOS CARBONÁTICOS COM LENTE DE ALTA

PERMEABILIDADE INTERSECTADA PELO POÇO

Tatiana Lipovetsky

Dezembro / 2013

Orientadores: Paulo Couto


Programa: Engenharia Civil

Carbonatos têm se tornado objeto de estudo de grande importância desde a

descoberta da bacia do Pré-Sal no Brasil. Os reservatórios carbonáticos são

caracterizados por apresentarem porosidade primária e secundária. Um tipo de

carbonato encontrado no Brasil é o de “coquinas”, que é formado por conchas

depositadas, que após a dissolução dão origem à porosidade secundária. Seu

comportamento de pressão de fluido durante a produção de petróleo ainda não é

totalmente compreendido. O objetivo deste trabalho é entender um tipo de reservatório

de carbonato, no qual o poço intersecta lentes de alta permeabilidade e extensões

limitadas.

A Análise de Teste de Poços, uma ferramenta poderosa para descrição de geologia

e fluxo, então, é aplicada aos reservatórios de carbonatos, que por sua vez são

modelados através de análises estatísticas da formação carbonática do Morro de Chaves.

Quatro diferentes reservatórios são construídos a partir dos resultados estatísticos. Para

que sejam simulados, um fluxo bi-fásico é incorporado em um código computacional

desenvolvido usando o Método dos Volumes Finitos aplicado ao Método IMPES

(Pressão Implícita, Saturação Explícita). O código mostra, entre outros, a pressão do

óleo no fundo do poço com o tempo. Através de dados de pressão, curvas são

construídas para que seu comportamento seja analisado: um teste de poço é feito.

vi

Abstract of Dissertation presented to COPPE/UFRJ as a partial fulfillment of the

requirements for the degree of Master of Science (M.Sc.)



WELLBORE

Tatiana Lipovetsky

December / 2013

Advisors: Paulo Couto


Department: Civil Engineering

Carbonates have become a very important object of study since the discovery of

the Pre-Salt basin in Brazil. Carbonate reservoirs are characterised by primary and

secondary porosities. A type of carbonate found in Brazil is the “coquinas”, which is

formed by deposited shells, that after dissolved form a secondary porosity. Their

pressure behaviour when producing the petroleum field is still not completely

understood. The objective of this work is to understand a kind of carbonate reservoir,

where a well intersects lenses of high permeability and limited extensions.

The Well Test Analysis, a powerful tool for geological and flow description, then,

is applied to the carbonate reservoirs, modelled from statistical analyses of the Morro de

Chaves carbonate formation samples. Four different reservoir scenarios are built from

the statistical results. To simulate them, a two-phase flow is incorporated into a code

developed using the Finite Volume Method applied to the IMPES (Implicit Pressure,

Explicit Saturation) Method. The code shows, among others, the oil pressure at the well

bottom with time. From the pressure data, curves are built so its behaviour can be

analysed: a well test analysis is done.

vii

INDEX

1. INTRODUCTION .......................................................................................................... 1

1.1 MOTIVATION .................................................................................................................... 1

1.2 OBJECTIVES ...................................................................................................................... 3

2. CARBONATE RESERVOIRS, WELL TEST ANALYSIS AND FIN ITE VOLUME

METHOD BIBLIOGRAPHICAL REVIEW ....................................................................................... 6

2.1 USEFUL PETROPHYSICAL CONCEPTS ................................................................................ 6

2.1.1 Porosity and Permeability Relationships .................................................................... 7

2.1.2 Capillary Pressure .................................................................................................... 10

2.2 CARBONATE RESERVOIRS............................................................................................... 11

2.2.1 Lenses ........................................................................................................................ 13

2.3 RESERVOIR STATISTICAL CHARACTERISATION .............................................................. 13

2.3.1 Arithmetic Average ................................................................................................... 14

2.3.2 Geometric Average ................................................................................................... 14

2.3.3 Harmonic Average .................................................................................................... 15

2.3.4 Comparing the arithmetic, geometric and harmonic averages ................................. 15

2.3.5 Well tests and average permeabilities ....................................................................... 16

2.3.6 Standard deviation .................................................................................................... 17

2.3.7 Coefficient of variation ............................................................................................. 18

2.4 WELL TEST ANALYSIS .................................................................................................... 19

2.4.1 The effect of reservoir heterogeneities on well responses ......................................... 20

2.4.2 Cross-flow reservoirs ................................................................................................ 21

2.4.3 Layered reservoirs with or without crossflow ........................................................... 22

2.4.4 Previously studied Well Test extreme responses in Cross-Flow Reservoirs ............. 23

2.4.5 Wellbore storage effect ............................................................................................. 28

2.4.6 Multiphase flow reservoirs ........................................................................................ 30

2.5 RESERVOIR SIMULATION AND FINITE VOLUME METHOD ............................................... 31

2.5.1 The Reservoir Simulator ........................................................................................... 32

2.5.2 Numerical Solution Methods versus Analytical Solution Methods............................ 33

2.5.3 IMPES Method .......................................................................................................... 33

3. THE DEVELOPED COMPUTER CODE FOR MULTIPHASE FLOW. .................. 35

3.1 THE LACUSTRINE CARBONATE FORMATION OF MORRO DE CHAVES .............................. 35

3.1.1 The Morro de Chaves Formation Reservoir and its initial conditions ...................... 36

3.2 THE MULTIPHASE FLOW SIMULATOR DEVELOPED IN THIS RESEARCH ............................ 42

3.2.1 The Discretised Hydraulic Diffusivity Equation for Oil and Water Reservoir ......... 43

viii

3.2.2 The Model Generated in Wolfram Mathematica ....................................................... 47

3.3 THE MODELLED SCENARIOS ............................................................................................ 55

3.3.1 Scenario 1 ................................................................................................................. 55

3.3.2 Scenario 2 ................................................................................................................. 59

3.3.3 Scenario 3 ................................................................................................................. 63

3.3.4 Scenario 4 ................................................................................................................. 67

3.3.5 Scenarios 1 to 4 and Homogeneous Case of 26mD: Mathematica (FVM) Simulation

Pressure Diffusivity Graphical Results ............................................................................................. 71

3.4 THE SCHLUMBERGER ECLIPSE 100 SIMULATION ............................................................ 73

3.5 REMARKS ON THE FINITE VOLUME METHOD CODE DEVELOPED IN THIS WORK ............. 74

4. WELL TEST ANALYSIS: DATA INTERPRETATION AND THE PARTIAL

PERFORATION HUMP EFFECT .................................................................................................... 76

4.1 THE WELL TEST ANALYSIS ............................................................................................ 76

4.2 SCENARIOS COMPARISON ............................................................................................... 77

4.2.1 Group 1 Analysis ....................................................................................................... 78

4.2.2 Group 2 Analysis ....................................................................................................... 78

4.2.3 Groups 1 and 2 Comparison ..................................................................................... 79

4.3 THE WELL TEST CURVES INTERPRETATION ................................................................... 79

4.3.1 The Partial Perforation Hump Effect ........................................................................ 79

4.3.2 Geochoke ................................................................................................................... 84

4.3.3 Radial Flow ............................................................................................................... 85

4.3.4 Sealed Boundary ....................................................................................................... 85

4.4 THE WELL TEST ANALYSIS COMPARISON ...................................................................... 86

5. CONCLUSION ............................................................................................................. 92

BIBLIOGRAPHY ............................................................................................................ 94

7. APPENDIX ................................................................................................................... 97

7.1 IMPES METHOD – FORMULAE DEVELOPMENT FOR OIL AND WATER DIFFUSIVITY

EQUATION 97

7.1.1 Oil phase equation discretisation .............................................................................. 99

7.1.2 Water phase equation discretisation ....................................................................... 101

7.1.3 The IMPES method applied to two-phase flow of oil and water ............................. 104

7.2 AUXILIARY SCENARIOS GRAPHS .................................................................................. 106

7.2.1 Scenario 1 ............................................................................................................... 106

7.2.2 Scenario 2 ............................................................................................................... 109

7.2.3 Scenario 3 ............................................................................................................... 111

7.2.4 Scenario 4 ............................................................................................................... 113

ix

7.3 EXTRA TESTS MADE FROM SCENARIO 1 ........................................................................ 115

7.4 SCHLUMBERGER ECLIPSE 100 CODE FOR SCENARIO 1 .................................................. 118

8. ANNEXES .................................................................................................................. 133

8.1 FLOW REGIMES ............................................................................................................. 133

8.1.1 Steady State ............................................................................................................. 134

8.1.2 Transient State ........................................................................................................ 134

8.1.3 Closed reservoir: pseudo steady state regime ......................................................... 135

8.2 WELL TEST INTERPRETATION ROUTINE ....................................................................... 136

x

FIGURES

Figure 1-1 – Typical Well Test Analysis responses [2] (FEKETE website, 2012). Δp,DER is the pressure

derivative, measured downhole at the wellbore; BDF is Boundary Dominated Flow. ________________ 2

Figure 1-2 – Example of a core from a Carbonate Rock [5] showing patchy nature of oil-bearing porosity

in black. White limestone matrix is clear in some tighter intervals. (from LARTER in CORBETT 2013) _ 3

Figure 1-3 – Example of Reservoir with high permeability Lenses [6]. In this case, k means permeability.

(SAGAWA et al., 2000) _________________________________________________________________ 4

Figure 2-1 – This plot shows a wide range of matrix permeability for a narrow range of matrix porosity.

These data are referred to Morro de Chaves. (CORBETT and BORGHI, 2013) [7] _________________ 7

Figure 2-2 – Lorenz Plot for a braided fluvial reservoir. The best permeability units (HU1) contain 70%

of the kh (transmissivity) but only 15% of the �h (storage). Braided fluvial reservoirs contain double matrix porosity – part of the matrix is transmissivity-dominated (in this case, only 15% of the pore volume) and part is storage-dominated (85% of the pore volume). (CORBETT et al., 2005) [9] __ 10

Figure 2-3 – Examples of capillary pressure curves. The red curves represent well sorted grains (higher

permeability, easier hydrocarbon rock invasion), and the black curves represent poorly sorted grains

(lower permeability, more difficult hydrocarbon rock invasion). Image from the Kansas Geological

Survey website [10]. __________________________________________________________________ 11

Figure 2-4 – Alternative estimators for well test permeabilities depend on the geometry of the lenses at

the bed-scale. Well testing (in the middle time region) is essentially a bedform scale measurement. Well

tests give a measure of an effective property over the volume investigated at any point in the test.

Considering the nature and scale of the layering in the volume of investigation of a well test is necessary.

This schematic relates to sandstones but the same geometries might be expected in carbonates [5]. The

top schematic can be related to dual permeability, and the middle and bottom ones to double porosity.

(CORBETT, 2013) ____________________________________________________________________ 17

Figure 2-5 – Hump effect example (CORBETT, 2005)________________________________________ 24

Figure 2-6 – “Cake” reservoir example. k1 ≠ k2. ____________________________________________ 25

Figure 2-7 – The effect of different parameter on the well-test response of a double permeability

reservoir. (HAMDI, 2012) ______________________________________________________________ 25

Figure 2-8 – A simplified two-layer reservoir representing the double permeability model. The thin high

permeable layer can have a variable lateral and vertical extension. (HAMDI, 2012) _______________ 26

Figure 2-9 – A set of drawdown derivative response curves of a two layer reservoir model (referent to

Figure 2-8) where the lateral extension of the thin high permeable layer (i.e. 10 ft) varies from 25 ft to

6400 ft. (HAMDI, 2012)________________________________________________________________ 27

Figure 2-10 - Definition sketch of the pressure profile away from the wellbore showing skin as an

increase or decrease in pressure in the immediate wellbore region. As the reservoir pressure drops from

P1 to P2 in the expected, radial uniform case, the pressure profile drops as shown. If the pressure

measured at the wellbore is higher than expected from this profile then this defines a negative pressure

xi

drop and an increase in production. If the pressure measured in the wellbore is much lower than this

gives a positive skin and results in reduced production. (CORBETT, 2012) _______________________ 28

Figure 2-11 – Pressure drawdown, build up and wellbore storage. Area 1 represents the pure wellbore

storage effect, Area 2 represents the afterflow storage effect. (BOURDET, 2002) __________________ 29

Figure 3-1 – Bivalve Coquinas from Morro de Chaves Formation. The Picture on the left shows the

coquinas fabric: presence of shells. The picture on the right shows a coquinas thin section

photomicrograph: porosity is in blue. Note that the black outlines represent dissolved shells, or moldic

porosity (secondary porosity). (CORBETT and BORGHI, 2013) _______________________________ 36

Figure 3-2 – Morro de Chaves Global Hydraulic Elements: Poro-perm distribution (CORBETT and

BORGHI, 2013). If there were a fracture, it would be represented by data in the upper redish-left region

(low porosity, high permeability values). The lens data is indicated in the orange band, and corresponds

to 33% of the inflow to the well. _________________________________________________________ 37

Figure 3-3 – Morro de Chaves Lorenz Plot. (CORBETT and BORGHI, 2013) Box shares 33% of

transmissivity (due to only 3 plugs). ______________________________________________________ 38

Figure 3-4 – Schematic figure of a 2-D radial reservoir, with well placed on the left (represented in blue

volumes). When rotating the image around the left boundary, the well takes the centre of the 3-D

reservoir. ___________________________________________________________________________ 43

Figure 3-5 – Comparison of oil saturation (So) versus time, for different time steps. (SAVIOLI and

BIDNER, 2005) ______________________________________________________________________ 49

Figure 3-6– Formation Volume Factor (m³/m³ std) behaviour of Oil (blue) and Water (red) with pressure

(Pa). _______________________________________________________________________________ 51

Figure 3-7 – Oil (blue) and water (red) viscosities (Pa.s) with pressure (Pa). _____________________ 52

Figure 3-8 – Water (red curve) and oil (blue curve) relative permeabilities versus saturation curves,

extracted from the developed code in Wolfram Mathematica. __________________________________ 54

Figure 3-9 – Capillary pressure versus water saturation curve, extracted from the developed code in

Wolfram Mathematica. The 0.15 water saturation is the connate, or immobile, water saturation. The

straight line at y=0 (capillary pressure) is merely due to mathematical boundaries. This value doesn’t

exist, since the water saturation has a minimum value equal to 0.15. ____________________________ 54

Figure 3-10 – Scenario 1 Representation: � (porosity) and k (permeability) distribution. ________ 56

Figure 3-11 – Scenario 1: Pressure distribution in the radial reservoir at the time of three years: the

closer to the well bore, the smaller the pressure values. ______________________________________ 57

Figure 3-12 – Scenario 1: Pressure distribution along the reservoir radius, at the time of 3 years, and

depth of the high permeable lens, which intersects the wellbore in its middle. _____________________ 58


depth equal to the bottom of the well. _____________________________________________________ 58

Figure 3-14 – Scenario 2 Representation: � (porosity) and k (permeability) distribution. ___________ 60


closer to the well bore, the smaller the pressure values. Note that the pressure range in this plot has

higher values than the pressure range in the Scenario 1 plot shown in Figure 3-11. ________________ 61

xii





Figure 3-18 – Scenario 3 Representation: � (porosity) and k (permeability) distribution. ________ 64



higher values than the pressure ranges in the Scenarios 1 and 2, shown in Figure 3-11 and Figure 3-15.

___________________________________________________________________________________ 65





Figure 3-22 – Scenario 4 Representation: � (porosity) and k (permeability) distribution. ___________ 68



higher values than the pressure ranges in the Scenarios 1, 2 and 3, shown in Figure 3-11, Figure 3-15

and Figure 3-19. _____________________________________________________________________ 68





Figure 3-26 – Wolfram Mathematica Code Simulation (FVM): Pressure Diffusivity in Image Shots for

five different cases. Pressure diffusion starts mainly by the lens. The well is localized at the left of each of

the twenty snap shots, but is not represented in the images. ___________________________________ 72

Figure 3-27 – Eclipse 100 Simulation: Pressure Diffusivity Graphical Results. Pressure diffusion starts

mainly by the lens. The software considers hydrostatic pressure. Pressure range in kgf/cm². Pressure

range goes from 182.24Kgf/cm² in dark blue to 204.78Kgf/cm² in vibrant red. ____________________ 74

Figure 4-1 – Pressure (ΔP) and Derivative Pressure (P’) versus time (Δt) for all simulated Scenarios. The

pressure and time data were obtained from the Wolfram Mathematica code developed in this work. The

lower set of curves refers to the pressure derivative plots; the upper set of curves refers to the pressure

drawdown plots. ______________________________________________________________________ 77




drawdown plots. In the zoomed box: the early time well bore effect. ____________________________ 80

Figure 4-3 – Enhanced Pseudo-Fracture Channel Skin Test, Pseudo-Fracture Channel Skin Test and

Partial Perforation Skin Test: Pressure and Pressure Derivative versus time. _____________________ 82

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Figure 4-4– Partial Perforation Skin Test: Pressure distribution at 0.1 hour of simulation. Pressure

range: from 204 kgf/cm² (red) to 203 kgf/cm² (blue). _________________________________________ 83

Figure 4-5– Partial Perforation Skin Test: Pressure distribution at 1 hour of simulation. Pressure range:

from 204 kgf/cm² (red) to 201 kgf/cm² (blue). _______________________________________________ 83

Figure 4-6– Partial Perforation Skin Test: Pressure distribution at 10 hours of simulation. Pressure


Figure 4-7– Partial Perforation Skin Test: Pressure distribution at 100 hours of simulation. Pressure


Figure 4-8– Partial Perforation Skin Test: Pressure distribution at the end of the simulation. Pressure





drawdown plots. In the zoomed box: the hump effect, or geochoke. _____________________________ 84

Figure 4-10 – Pressure (ΔP) and Derivative Pressure (P’) versus time (Δt) for all simulated Scenarios.

The pressure and time data were obtained from the Wolfram Mathematica code developed in this work.

The lower set of curves refers to the pressure derivative plots; the upper set of curves refers to the

pressure drawdown plots. In the zoomed box: the radial flow behaviour. ________________________ 85

Figure 4-11 – Pressure (ΔP) and Derivative Pressure (P’) versus time (Δt) for all simulated Scenarios.

The pressure and time data were obtained from the Wolfram Mathematica code developed in this work.

The lower set of curves refers to the pressure derivative plots; the upper set of curves refers to the

pressure drawdown plots. In the zoomed box: the sealed boundary pressure drop behaviour. ________ 86

Figure 4-12 – Well Test Analysis for the Finite Volume Method code case, developed in Wolfram

Mathematica for this work. _____________________________________________________________ 87

Figure 4-13 – Well Test Analysis for the Eclipse 100 case. The early time phenomena show a possible

numerical storage. ____________________________________________________________________ 87

Figure 4-14 – Well Test curves for the Wolfram Mathematica code, developed with the Finite Volume

Method, during the middle time. _________________________________________________________ 88

Figure 4-15 – Well Test curves for the Schlumberger Eclipse 100 simulation, developed with the Finite

Volume Method, during the middle time. __________________________________________________ 89

Figure 7-1 – Scenario 1: Oil flow rate variation, at standard conditions, with time. _______________ 106

Figure 7-2 – Scenario 1: Water flow rate variation, at standard conditions, with time. ____________ 107

Figure 7-3 – Scenario 1: Pressure draw down at the bottom of the well, versus elapsed time: due to the

constant flow rate at the surface, and lack of external inflow, the pressure at the bottom of the well falls in

order to keep the production at a constant rate. BHP stands for Bottom Hole Pressure. ____________ 108

Figure 7-4 – Scenario 1: ΔP versus elapsed time: the graph confirms the one fromFigure 7-3, showing

that the pressure at the bottom of well bore decreases with time. ΔP = (Initial pressure) – (Pressure at

time). ______________________________________________________________________________ 108


xiv

Figure 7-6 – Scenario 2: Water flow rate variation, at standard conditions, with time. ____________ 109




Figure 7-8 – Scenario 2: ΔP versus elapsed time: the graph confirms the one from Figure 7-7, showing


time). ______________________________________________________________________________ 110


Figure 7-10 – Scenario 3: Water flow rate variation, at standard conditions, with time. ___________ 111






time). ______________________________________________________________________________ 112

Figure 7-13 – Scenario 4: Oil flow rate variation, at standard conditions, with time. ______________ 113

Figure 7-14 – Scenario 4: Water flow rate variation, at standard conditions, with time. ___________ 113






time). ______________________________________________________________________________ 114

Figure 7-17 – Scenario 1 variations: refined mesh case and no wellbore storage case. ____________ 116

Figure 7-18 – Pressure draw down evolution for the Wolfram Mathematica FVM code. The cases

considered here are of no wellbore storage and consideration to wellbore storage effect, both run for

Scenario 1. _________________________________________________________________________ 117

Figure 8-1 – Pressure profiles of a circular closed reservoir. t1: the boundaries are not reached, infinite

reservoir behaviour; t2: boundaries reached, end of infinite acting; t3 and t4: pseudo steady state regime,

the pressure profile drops. (BOURDET, 2002) ____________________________________________ 135

Figure 8-2 – Closed system drawdown and buildup responses. Linear scale. (BOURDET, 2002) ____ 136

xv

TABLES

Table 2-1 – Hydraulic Unit lower boundaries (shown as FZI values) for 10 Global Hydraulic Elements

(CORBETT, 2013) ........................................................................................................................................ 9

Table 2-2 – Types of Average and their use. .............................................................................................. 15

Table 2-3 – Most appropriate permeability average, according to reservoir characteristics [5]. ............ 16

Table 2-4 – Coefficient of variation (CV) and its relation to reservoir heterogeneity [5] ......................... 18

Table 3-1 – Transmissivity, Porosity and Permeability values for Lens and Reservoir, through the

assessment of the Lorenz Plot. .................................................................................................................... 38

Table 3-2 – Porosity-permeability values (CORBETT and BORGHI, 2013). The shaded values

correspond to the orange band, in Table 3-1. ............................................................................................ 39

Table 3-3 – Statistical treatment of porosity and permeability data .......................................................... 40

Table 3-4 – Initial Reservoir and Fluids Properties................................................................................... 41

Table 3-5 – Scenario 1: Porosity and Permeability values ........................................................................ 56

Table 3-6 – Scenario 1: Initial and Final values (at 3 representative years) simulated with the Finite

Volume Method, in Wolfram Mathematica ................................................................................................. 59

Table 3-7 – Scenario 1: Initial and Final values (at 3.5 representative years) simulated with the Finite


Table 3-8 – Scenario 2: Porosity and Permeability values ........................................................................ 60



Table 3-10 – Scenario 3: Porosity and Permeability values ...................................................................... 63



Table 3-12 – Scenario 4: Porosity and Permeability values ...................................................................... 67


Volume Method, in Wolfram Mathematica. ................................................................................................ 70

Table 4-1 – Scenarios Groups: Scenarios are grouped together if they have the same matrix reservoir

properties. K stands for permeability. ........................................................................................................ 78

Table 4-2 – Enhanced Pseudo-Fracture Channel Skin Test: Porosity and Permeability values ............... 80

Table 4-3 – Pseudo-Fracture Channel Skin Test: Porosity and Permeability values ................................ 81

Table 4-4 – Partial Perforation Skin Test: Porosity and Permeability values ........................................... 81

Table 4-5 – Scenarios 1 to 4 Well Test Behaviour, Expectations and Results from Eclipse 100 and

Mathematica code. This table summarizes the derivative plots. k stands for permeability. ....................... 90

Table 8-1 – Summary of important usual log-log responses, extracted from BOURDET (2002), Geochoke

image extracted from CORBETT et al. (2005) ......................................................................................... 137

xvi

LIST OF SYMBOLS

- cowP : oil-water capillary pressure;

- rwk : water relative permeability.

- wP : water pressure;

- B. : oil FVF;

- B./ : initial oil FVF;

- B0 : water FVF;

- B0/ : initial water FVF;

- c. : oil compressibility factor;

- c0 : water compressibility factor;

- P/ : initial or static reservoir pressure;

- T234 : reservoir temperature;

- 5676683 : bubble point pressure;

- 5/ : initial reservoir pressure,

- 509(:) : wellbore pressure during the drawdown,

- ;234 : volume of the fluid at reservoir conditions;

- ;4<= : volume of the fluid at standard conditions;

- ;0388 : wellbore volume.

- >. : oil compressibility;

- ?@AB,AB : oil density;

- C0388 : wellbore radius;

- : oil density;

- : oil pressure;

- : oil flow rate per volume unit in standard conditions;

- : Oil Formation Volume Factor;

- : oil saturation;

- : oil-relative permeability;

- : permeability tensor;

- API degree: American Petroleum Institute degree;

oρ

oP

,o stdq′′′ɺ

oB

oS

rok

k̂

xvii

- C :wellbore storage coefficient;

- cm: centimetre

- Ct: total compressibility

- CV: coefficient of variation;

- e, w, n, s sub-indexes: related to the volume boundaries at east (e), west (w),

north (n), south (s) of the centre of the volume.

- E, W, N, S sub-indexes: related to the volumes at east (E), west (W), north

(N), south (S) of the central volume (C).

- FDM: Finite Difference Method;

- ft: feet

- FVF: Formation Volume Factor;

- FVM: Finite Volume Method;

- FZI:Flow Zone Indicator;

- g: gravity;

- rok : oil relative permeability;

- wρ : water density;

- ,w stdq′′′ɺ : water flow rate per volume unit in standard conditions;

- wB : water formation volume factor;

- wS : water saturation;

- K : the permeability

- Kgf: kilogram-force

- kh: average flow capacity

- m: metre

- N: number of data in sample;

- o and w sub-indexes: property of oil (o) or water (w).

- P: Pressure;

- P’: Pressure derivative

- Pa: Pascal

- q: flow rate;

- SD: the standard deviation;

- std: standard

- Upper index “0” : related to the value of the property at the previous iteration.

- WTA: Well Test Analysis;

xviii

- z: vertical distance to datum;

- Δr : radial distance between volumes centres or boundaries, depending on the

sub-index;

- Δt : time step;

- 5: pressure.

- 5 : oil pressure;

- C : radius;

- : : time.

- PF.0 : capillary pressure;

- P. : oil pressure;

- P0 : water pressure;

- G0 : water saturation;

- G0F.H : connate water saturation;

- K2. : oil relative permeability;

- K20 : water relative permeability.

- mD: miliDarcy;

- Log: logarithm;

- KPa: kilopascal;

- PFC: Pseudo-Fracture Channel;

- EPFC: Enhanced Pseudo-Fracture Channel;

- PP: Partial Perforation;

- h: hour;

- BHP: bottom hole pressure;

- μ. : oil viscosity;

- μ.K : dead oil viscosity;

- μ0 : water viscosity;

- : medium porosity;

- λ: interlayer cross flow coefficient;

- L : storativity ratio.

φ

1

1. Introduction

Petroleum Reservoir Engineering makes use of scientific principles to the

drainage problems that arise during the production and development of a reservoir,

aiming a high economic recovery. Integrating geology, applied mathematics, physics

and chemistry it is possible to describe and even predict the behaviour of the fluids

present within the reservoir rock.

When a reservoir is modelled, the conduct of reservoir simulation studies to

determine optimal development plans for oil and gas reservoirs. As part of the

appraisal, well tests, which are going to be described below, are also performed and

integrated to other reservoir studies, before a reservoir is modelled.

In this project, well test modelling is being done in advance of well testing.

1.1 Motivation

Well testing has become one of the most powerful tools for predicting complex

reservoir characteristics. Various kinds of Well Test Analyses (WTA) can be executed,

depending on one’s purpose, at stages of construction, completion and production of the

well. During a well test it is possible to infer reservoir properties from its responses, or

pressure behaviour. Therefore, a well test analysis can be synonymous with pressure

transient analysis.

The primary objective of pressure transient analysis and well testing is to assess

the well (e.g., productivity) and formation properties, by measuring flow rate, downhole

or at surface, and pressure, downhole at the well. This information is elementary for

knowing the in situ dynamic reservoir properties [1].

For well test analysis, a mathematical model is used to relate pressure response to

flow rate. Then, by assuming that the modelled reservoir input is the same as that in the

field, one can infer that the model parameters and the reservoir parameters are the same,

as long as the pressure output is the same as the measured reservoir pressure output.

2

Despite the difficulties and ambiguities, it is possible to minimize the dangers of

reservoir modelling by careful specification of the well test. A combination of typical

collection of well test responses is show in Figure 1-1.

Figure 1-1 – Typical Well Test Analysis responses [2] (FEKETE website, 2012). Δp,DER is the pressure derivative, measured downhole at the wellbore; BDF is

Boundary Dominated Flow.

Usually, the objectives of a well test, which range from simple identification of

produced fluids and determination of reservoir deliverability to the characterization of

complex reservoir features, fall into three major categories [3]:

- Reservoir evaluation;

- Reservoir management; and

- Reservoir description.

The description of a reservoir’s geology is very important when it comes to

evaluating the formation and its fluids. Data presented by the SCHLUMBERGER

website in 2013 [4] point that about 60% of world’s oil reserves and 40% of world’s gas

reserves are held in carbonate reservoirs, of which 70% are of Middle East’s oil and

90% are of Middle East’s gas reserves. The Pre-Salt Region in Brazil is also held in a

carbonate formation. Hence, the carbonate reservoir was chosen to be object of analysis

here. An example of a carbonate rock is show in Figure 1-2. It is possible to see that,

3

depending on the location, the rock is more or less heterogeneous, varying its

properties, which is a great challenge to Petroleum Engineering and Geology.

Figure 1-2 – Example of a core from a Carbonate Rock [5] showing patchy nature of oil-bearing porosity in black. White limestone matrix is clear in some tighter intervals.

(from LARTER in CORBETT 2013)

From Figure 1-2, it is possible to see the limestone core shows oil-bearing

porosity (in black) and also shows non-oil-bearing porosity segments (white limestone).

1.2 Objectives

The high permeability lenses reservoirs consist of concentrated deposits of large

grains that form vertically and laterally limited lenses in a low permeability matrix. An

example is show in Figure 1-3. This example is based on a sandstone reservoir but is

4

being extended to carbonates where high heterogeneity is also present in the carbonate

matrix. This work does not consider the role of fractures.

Figure 1-3 – Example of Reservoir with high permeability Lenses [6]. In this case, k means permeability. (SAGAWA et al., 2000)

A method of verifying the reservoir’s performance is through measuring its

pressure and flow rate at the well at a certain flow condition. During a certain period of

test, the formation pressure is monitored in time. The analysis of these changes may

provide information on the size and geometry of the formation, as well as its capacity to

produce fluids.

The main goal of this work is to describe, through computational analyses, the

behaviour of pressure in a model that represents lenticular reservoirs, which might

portray a carbonate reservoir, intersected by a production well, with multiphase flow

composed of oil and water, for a given flow rate. Thereunto, a computational

programming software was developed in Wolfram Mathematica and, afterwards, two

commercial software programmes were run: Schlumberger ECLIPSE 100, which is a

black-oil simulator, and FEKETE Well Test.

The code developed in Wolfram Mathematica consists of the Finite Volume

Method, applied to the IMPES (Implicit pressure, explicit saturation) method, that

calculates implicitly the oil pressure, and explicitly the saturations of oil and water, for

each time step, for a given constant flow rate.

The Schlumberger ECLIPSE 100 is a black-oil simulator, used here as a

benchmark for the results acquired from Mathematica. Being a black-oil simulator

5

means that a fully-implicit finite difference method, three-phase, 3D simulation is

assumed and no compositional changes are observed.

The FEKETE software was, then, used to compute the well testing type-curves,

which are used to describe and characterise a petroleum reservoir.

6

2. Carbonate Reservoirs, Well

Test Analysis and Finite Volume

Method Bibliographical Review

In this chapter, the most relevant aspects of petrophysics, carbonate reservoirs, its

lenses, well test analysis (WTA) and WTA applied to carbonate and bounded reservoirs,

and the Finite Volume Method will be reviewed and presented.

2.1 Useful Petrophysical Concepts

Petrophysics, by definition, is the study of physical rock properties and their

interaction with fluids. When it comes to carbonate reservoir characterisation, the most

important and essential petrophysical property is capillary pressure, which is a unique

measurement that gives insight to the relationships between pore size and pore throat.

Pore size distributions are usually obtained from capillary pressure. Porosity in

carbonates is extremely challenging as there are so many pore types which can have

complex primary or secondary process, so that the same porosity can have very different

permeabilities, and this is far more of a challenge in carbonates than in clastics.

Capillary pressure is linked through petrophysical rock typing to porosity and

permeability relationships. Rock typing, in this particular petrophysical use of word, is

critical to carbonate reservoir petrophysics [5].

Here, the most relevant petrophysics concepts related to carbonate reservoirs will

be presented.

7

2.1.1 Porosity and Permeability Relationships

Relationships between porosity and permeability can be extremely complex, but

necessary for geoengineering predictions. This complexity happens considerably in

carbonate reservoirs, hence it is important to understand that, within the same reservoir,

it is possible to find repetitively one same value of matrix porosity with many

corresponding values of matrix permeability, as shown in Figure 2-1.

Figure 2-1 – This plot shows a wide range of matrix permeability for a narrow range of matrix porosity. These data are referred to Morro de Chaves. (CORBETT and

BORGHI, 2013) [7]

The Morro de Chaves Formation outcrop, described by CORBETT and BORGHI

in 2013 and depicted in Figure 2-1, shows lacustrine carbonate petrophysical data,

indicating a wide range of rock types, which specifies a very heterogeneous formation.

2.1.1.1 Global Hydraulic Elements (GHE)

Global Hydraulic Elements is an alternative and adaptation of the traditional Rock

Typing Method, used to improve the description of a formation. In this new approach,

8

developed by CORBETT and POTTER (2004) [8], the petrophysical elements have

their boundaries specified prior to the core analysis, which allows a rapid and

systematic approach for varying data sets in different wells.

For a given porosity, the permeability can be calculated as:

k = N(OPQ) R ∅1 − ∅U0.0314 XY

2.1

Where:

- � is the porosity;

- K is the permeability (mD);

- FZI is the Flow Zone Indicator, which defines the petrotypes boundaries, has

its values chosen with consideration to:

o The range of porosity and permeability in typical reservoirs;

o The acceptable variability – from an engineering viewpoint – within a

hydraulic element for its porosity-permeability data pairs;

o The need for discretization of the porosity-permeability space to

eliminate less important variations, such as “noise” in data.

Therefore, the definition of the FZI (petrotype boundaries) values is done in order

to fragment a wide range of possible combinations of porosity-permeability data into a

manageable number of Hydraulic Elements. By fixing those boundaries, they can be

applied globally and, thereunto, become Global Hydraulic Units.

Table 2-1 shows ten GHE. For each one of them, there is an FZI bounding value:

9

Table 2-1 – Hydraulic Unit lower boundaries (shown as FZI values) for 10 Global Hydraulic Elements (CORBETT, 2013)

FZI GHE

45 10

24 9

12 8

6 7

3 6

1.5 5

0.75 4

0.375 3

0.1875 2

0.0938 1

2.1.1.2 Lorenz Plot

The Lorenz plot is based on a plot of porosity (�), or porosity-thickness (�h)

product, versus permeability-thickness (kh) product of data obtained from reservoir core

samples. The points are plotted in order of decreasing values of k/�. In an ideal scenario of uniform rock properties, the points fall on a diagonal from the upper right to the lower left corner of the plot, which suggests that the porosity (or porosity-thickness product) is a linear function of permeability-thickness. With the increasing of reservoir heterogeneity, the plotted points are shifted further away from the diagonal. The area between the diagonal and the deviated points is the basis of the Lorenz coefficient: ̀ @Cabc >@adde>eab:= fCag ab>h@ia? jk lh@: l@eb:i gb? ?egm@bghfCag ab>h@ia? jk ?egm@bgh gb? j@::@n Cemℎ: >@CbaC @d :ℎa lh@:

2.2

With increasing heterogeneity, the Lorenz coefficient gets closer to unity, while

with increasing homogeneity, it gets closer to zero.

An example of heterogeneous medium is shown in a Lorenz plot, in Figure 2-2.

10

Figure 2-2 – Lorenz Plot for a braided fluvial reservoir. The best permeability units (HU1) contain 70% of the kh (transmissivity) but only 15% of the �h (storage). Braided fluvial reservoirs contain double matrix porosity – part of the matrix is transmissivity-dominated (in this case, only 15% of the pore volume) and part is storage-dominated (85% of the pore volume). (CORBETT et al., 2005) [9]

2.1.2 Capillary Pressure

The capillary pressure phenomenon occurs in porous media which have two or

more immiscible fluids.

The shape of the capillary pressure curve is controlled by textural properties of the

sediment, such as pore distribution, grain size and sorting. The distribution of capillary

curves in a reservoir will control the distribution of hydrocarbons in place. A matrix

formed of well sorted grains, which form large pores, can be an easier path for

hydrocarbons to flow, as in opposition to poorly sorted grains that form smaller pores.

Selecting an effective capillary pressure curve in a reservoir with many rock types is a

geoengineering challenge as it will depend on the distribution of rock types [5]. Rock

typing is a link between sedimentological description and fluid flow: it classifies rocks

according to their porosities and permeabilities by similar pore throat and sizes. For

reservoir modelling, it’s sometimes necessary to choose one capillary curve to represent

the whole reservoir volume. Figure 2-3 shows the relation between water saturation,

11

capillary pressure (or free water height) and permeability. For poorly sorted rocks, the

permeability is lower and the capillary pressure is higher, and for well sorted rocks, the

permeability is higher and the capillary pressure is lower.

Figure 2-3 – Examples of capillary pressure curves. The red curves represent well sorted grains (higher permeability, easier hydrocarbon rock invasion), and the black curves represent poorly sorted grains (lower permeability, more difficult hydrocarbon rock

invasion). Image from the Kansas Geological Survey website [10].

2.2 Carbonate Reservoirs Carbonate is a type of chemical/biochemical sedimentary rock, originated by

precipitation of minerals from water through various chemical or biochemical

processes. Chemical/biochemical sedimentary rocks can be distinguished from

siliclastic sedimentary rocks by its chemistry, mineralogy and texture. Carbonaceous

sedimentary rocks, such as coals and oil shales, make up a further special group of rocks

that contain abundant non-skeletal organic matter in addition to various amounts of

siliclastic or chemical (e.g. carbonate) constituents. The latter (carbonaceous) are not to

be confused with carbonates.

The carbonate rocks are the most abundant kind of chemical/biochemical

sedimentary rock, and can be divided on the basis of mineralogy into limestones

(composed of calcite and aragonite, that are different crystal forms of CaCO3) and

dolostones (composed of dolomite, (CaMg(CO3)2). Limestones are mainly composed of

calcite minerals, while dolostones are mainly composed of dolomite minerals, and it´s

12

estimated that both act as reservoir rocks for more than one-third of the world’s

petroleum reserves.

Carbonates vary from chalks through sandstones to dolomites. They can contain

evaporites such as disseminated anhydrite, clay minerals, and electronically conducting

materials such as pyrite. Their porosity is laterally and vertically localized, within a

layer, but, on the other hand, the carbonate rocks pores may be much larger than the

sandstone pores, for example, causing the rock to have high permeability.

In carbonate reservoirs, the primary pore system comprises interparticle porosity

that coexists with a highly variable secondary system of dissolution voids and/or

fractures. As a consequence, carbonate reservoirs are markedly heterogeneous from

pore to reservoir scales, and this variability poses significant challenges to data

acquisition, petrophysical evaluation and reservoir description [11]. The primary

porosity, which results from the original rock deposition, is accumulations of shells and

reefs, and the oolitic carbonates. When shells are the dominating carbonate grains, these

rocks can be called “coquinas”. Still, there´s the clastic dolomites and carbonates, which

are results from the accumulation of grains of older carbonate rocks (in this case, the

carbonate porosity would be almost the same as the sandstone porosity). Whatsoever,

due to the calcite or dolomite solutions deposition and to recrystallization, the original

porosity is largely reduced. Carbonate rocks porosity is almost always secondary and

due to processes of dissolution, dolomitization and fracturing. The most important

process of these is dissolution (or solution), where dolomite is “washed” by the

subterranean water, resulting in cavities with dimensions that vary from tiny to cave-

sized pores.

As in comparison, clastic rocks are predominantly sandstones that comprise

quartz and other mineral components, which are transported from elsewhere and

modified through weathering, lithification, and diagenetic processes to form a clastic

reservoir. Sandstones can show a wide range of reservoir quality through variations in

mineralogy made from carbonate grains (shells), grain size distribution and sorting,

texture and degree of induration. Carbonate rocks, on the other hand, can be formed in-

situ through the growth of calcitic organisms and precipitation, with subsequent

evolution being governed by compaction, cementation, dolomitization, dissolution, and

other diagenetic processes. This way, carbonates can show a very wide range of

reservoir quality through pore-size distribution, pore connectivity, fracturing, and the

degree of dolomitization. Therefore, one can say that sandstone reservoir quality is

13

governed by texture, whereas carbonate reservoir quality is governed more by pore

character. The relative brittleness and solubility of carbonate rocks give rise to

secondary porosity in the form of natural fractures and dissolution fissures, respectively.

Where these fractures are open, their occurrence enhances drainage locally/regionally.

Where they are closed or sealed, they act as flow inhibitors and can even

compartmentalize reservoir units [11]. This work doesn´t consider fractures.

2.2.1 Lenses

Differences in grain size, permeability, porosity, grain shape, sorting, packing,

orientation, diagenesis, among other grain characteristics, can cause to reservoirs great

heterogeneities. Where there is a coarse grain concentration in channel deposits,

vertically and horizontally limited high permeability lenses are formed, surrounded by a

low permeability matrix, which can often occur in fluvial environments, but can also

occur in other environments [12]. In carbonates, the presence of “lenses” is not so clear

but matrix heterogeneity exists in many forms – including lenses (CORBETT, personal

communication).

2.3 Reservoir Statistical Characterisation Reservoir Statistical Characterisation has been the subject of study of many

scientists. CORBETT (2013) presented in his “Petrophysics and Characterisation of

Carbonate Reservoirs” [5] material the Reservoir Statistical Characterisation, used as

base for this chapter.

Beforehand, it is important to clearly understand some basic important

terminology:

- Population: properties values of a reservoir unit for which is required to be

inferred.

- Population parameter: used to analyse the population data. Examples are mean

and standard deviation.

- Sample: set of measurements.

- Mean: estimated by an appropriate descriptive statistic from a sample. An

average is a kind of mean. In this text, mean is the population parameter for

central tendency.

14

- Average: is the estimator of the population mean calculated by a sample

statistic (i.g., arithmetic average).

In the petroleum industry, the available samples are constantly very small and not

necessarily representative. “It is common to infer parameters for an entire reservoir

(order 108-1016 m³) from a few cores (10-102 m³), from which limited samples are taken

(10-2-10-3 m³).”(CORBETT, 2013)

Quantity variables have numerical quantity that can either be discrete (e.g., fossil

specimens, number of channels) or continuous (e.g., permeability, porosity), which is

the type of interest in this work.

Effective properties are the equivalent properties of an equivalent homogeneous

medium, and permeability averages are often used to estimate the effective

permeability. Notice that while permeability is an intensive variable generally non-

additive, porosity is additive.

2.3.1 Arithmetic Average

The arithmetic average can be obtained by adding the quantities and diving by the

number of data in the sample:

Kpq2 = 1r s t/u

/vw 2.3

Where:

- N is the number of data in sample;

- K is the permeability

The arithmetic average is equally sensitive to all values, and it can be biased (bias

is a systematic error in the estimator) because of many reasons [5], not discussed here.

2.3.2 Geometric Average

Used to determine the Nth root of the product of a number (N) of data:

Kpx3.K = yz t/u

/vw {w/u

2.4

Or:

15

Kpx3.K = a}wu ∑ �� 2.5

Very low values (≤ 0.001 mD) should be used instead of zeroes.

2.3.3 Harmonic Average

The harmonic average for N data is:

Kp�q2 = r �s 1t/u

/vw ��w

2.6

Very low values (≤ 0.001 mD) should be used instead of zeroes.

Since the inverse of permeability can be considered as resistance to flow, the

harmonic average is permeability that corresponds to the arithmetic average resistance

to flow.

2.3.4 Comparing the arithmetic, geometric and harmonic averages

Considering these averages as a function of the sample heterogeneity, commonly

observed in permeability datasets, the differences are always: Kp�q2 ≤ Kpx3.K ≤ Kpq2

These averages are proper for the following conditions, presented Table 2-2.

Table 2-2 – Types of Average and their use.

Average Use

Kp�q2

Bed series, single phase flow (i.e., vertical

flow in a horizontally layered, bounded

system) Kpx3.K Single phase flow in a random, 2-D field

Kpq2

Bed parallel, single phase flow (i.e.

horizontal flow in a horizontally layerd,

bounded system)

Under other flow conditions, such as two phase flow, extreme care is needed to

select the appropriate average. For example, see Table 2-3.

16

Table 2-3 – Most appropriate permeability average, according to reservoir characteristics [5].

Situation Recommendation

Layered system at low dips (< 20 degrees) Vertical flow: use Kp�q2

Horizontal flow: use Kpq2

Layered system at steeply dipping beds (>

70 degrees)

Vertical flow and horizontal flow along

the beds: use Kpq2

For horizontal permeability across the

beds: use Kp�q2

Random systems Use Kpx3.K for vertical and horizontal

permeability

2.3.5 Well tests and average permeabilities

Usually, well tests are dominated by the properties of beds in clastic reservoirs.

Well test permeability is most closely associated with average bed permeability. Hence,

the proper average will depend of the bed geometry, as shown in Figure 2-4, which

indicates the best estimators for well test permeabilities depending on the geometry of

the lenses at the bed-scale.

17

Figure 2-4 – Alternative estimators for well test permeabilities depend on the geometry of the lenses at the bed-scale. Well testing (in the middle time region) is essentially a

bedform scale measurement. Well tests give a measure of an effective property over the volume investigated at any point in the test. Considering the nature and scale of the layering in the volume of investigation of a well test is necessary. This schematic

relates to sandstones but the same geometries might be expected in carbonates [5]. The top schematic can be related to dual permeability, and the middle and bottom ones to

double porosity. (CORBETT, 2013)

2.3.6 Standard deviation

A deviation is the distance from the mean. Therefore, the mean deviation is the

average deviation for a sample. The variance is the average squared deviation.

-k

ar

-k

geom

-k

har

1-5ft

5-10ft

10-50ft

18

The standard deviation can be calculated as:

SD = ys �t/ − t��Yr − 1u/vw {B.�

2.7

Where:

- SD is the standard deviation;

- N is the number of data;

- �∑ (�� )�u�wu/vw � is the sample variance;

- K is the permeability (in miliDarcy, or mD).

2.3.7 Coefficient of variation

The coefficient of variation (CV) is an absolute measure of dispersion, very used

in reservoir characterisation for definition of level of heterogeneity.

CV = G�Kpq2 2.8

According to the CV value, the reservoir can be classified according to its

heterogeneity, as seen in Table 2-4.

Table 2-4 – Coefficient of variation (CV) and its relation to reservoir heterogeneity [5]

CV value Reservoir heterogeneity

classification Type of distribution

0.0 < CV < 0.5 Homogeneous Normal (Gaussian)

0.5 < CV < 1.0 Heterogeneous Skewed

1.0 < CV Very heterogeneous Increasingly skewed

As known, it is important to remember that carbonate reservoirs can have

different heterogeneity classifications, so extra care must be taken when choosing the

statistical analysis methods.

19

2.4 Well Test Analysis Well Test Analysis (WTA) consists of temporary changes in production rates,

causing transient pressure responses, usually measured at the bottom of the well while

the flow rate is measured at the surface (standard conditions). Before opening the well,

the initial pressure is the static reservoir pressure, which is constant and uniform. When

the well is opened to [13]flow, the drawdown pressure is measured, and when the well

is shut, the build-up pressure can be measured. The well response is generally

monitored during a relative short period of time compared to the life of the reservoir.

The observed pressure data are usually plotted in a log-log graph, which consists of two

separate curves: ΔP (pressure) and ΔP’ (pressure derivative), and they are defined, for a

drawdown test, as:

∆P = 5/ − 509(:) 2.9

∆P� = d(∆P)dln(t) 2.10

Where:

- 5/ is the initial reservoir pressure,

- 509(:) is the wellbore pressure during the drawdown,

- : is time.

WTA provides information on reservoir and well, which, associated with

geological and geophysical data, are used to build a reservoir model for prediction of

the field behaviour and fluid recovery to different scenarios. For well evaluation, tests

don’t need to be run for more than a couple of days, while for reservoir evaluation, the

tests might last up to several months [13]. The reservoir information obtained from

pressure transient measurements are essential since they reflect its dynamic behaviour

[1]. Testing a particular kind of permeability heterogeneity, like stratification, can

provide information on individual and/or group of layers, as the impact of pressure and

saturation distribution under multi-phase flow conditions. This in-situ information can

give estimates of pressure, saturation and fluid mobility in the layers, not to mention the

movement of possible fluid banks [14].

The well-test interpretation is an inverse problem, with non-unique solutions. Its

input is the flow rate, and the output is the wellbore bottom pressure. With the input and

the output, it’s possible to characterise the unknown system (reservoir + well). The non-

20

uniqueness of these solutions can be reduced by adding to the problem analysis

additional (or external) information, such as geological and geophysical data.

In this chapter, some Well Test concepts will be shown as an introduction, with

the intention of acquainting what the WTA is about and how it can be done. Later, the

WTA will be better explained while the tests are being run and presented. The WTA in

this work will be done after using the FEKETE F.A.S.T. Well TestTM software, where

pressure and its derivative curves will be automatically computed. The input used to

FEKETE will be extracted from the computer simulation code, developed in Wolfram

Mathematica software especially for this work.

The next sections in this chapter are mainly theoretical, intended to show

examples of what the WTA can be used for. Plots are also presented, to depict the

reservoir scenarios.

2.4.1 The effect of reservoir heterogeneities on well responses

The reservoir heterogeneity has its pressure data deviated from the homogeneous

behaviour. These deviations may take only a few minutes to become evident, but can

also take up to several days. In this study, it is assumed that the well is affected only by

the wellbore storage and skin. Primarily, the assumed reservoir acts as infinite, then the

outer boundary effects are reached, or “seen”, by the diffusion pressure.

It is possible to combine basic homogeneous solutions to construct heterogeneous

ones: describing double porosity models (restricted or unrestricted interporosity flow),

double permeability models and composite systems (radial or linear interfaces). Also,

the extension of the basic solution to a larger number of elementary behaviours can be

considered, such as multi-porosity systems for changing matrix block sizes, multi-layer

systems and multi-composite formations. For more details about each particular

behaviour, see BOURDET’s Well Test Analysis: The use of advanced interpretation

models, 2002, or items 8.1 to 8.2 [13].

2.4.1.1 Previous outstanding researches on how layers in multi-layered reservoirs

work

According to LEFKOVITS et al. [15], for a two-layer single phase case, the

average reservoir flow capacity (kh) is equal to the arithmetic average of the layer’s

21

reservoir flow capacities. The individual layer contribution to total flow (qi/qt) in the

transient period depends on the fractional flow capacity (kihi/∑kihi).

Then, EHLIG-ECONOMIDES [16] used simulation models to study individual

layer-properties in a multi-layered reservoir. She used flow-rate transient curve analysis

so she could understand the flow mechanism within the reservoir. To do so, she built

two models: a commingled reservoir (no cross-flow between layers) with two zones,

each divided into several cross-flow layers, and another commingled one with five

zones. She concluded that, during the early time, the pressure behaviour of the first

model (two zones) acts like if it were a commingled reservoir. The zonal properties

were determined using commingled reservoir type curves, while the individual layer

properties in each zone were calculated from cross-flow type curves. Since there is no

cross-flow between adjacent layers, the multi-layer system can be described as two

separate reservoirs with varying flow rates, but with total fixed production.

2.4.2 Cross-flow reservoirs

Cross-flow reservoirs are those in which flow takes place between units within the

reservoir [17]. For that kind of reservoir (“cake” reservoir model), a V-shape pressure

derivative response is usually observed. HORNE [3] stated that “If the layers are in

hydraulic communication within the reservoir (as well as through the wellbore) the

layers are said to experience cross-flow”. According to HAMDI (2012) [17], cross-flow

reservoirs regularly are detected through three different transient responses extrema: the

geoskin (CORBETT et al., 1996) [18] and the geochoke (CORBETT et al., 2005) [19].

The extrema of interest will be later presented, in Section 2.4.4.

When cross-flow happens in macro scale, it means that it occurs between

reservoirs zones or layers. When the reservoir is commingled, there is no hydraulic

communication between the layers or zones, which only communicate with the

wellbore. In this second case, there is no cross-flow in macro-scale. Studies made by

CHAUDHRY (2004) [20] show that there is a growing interest in converting

commingled reservoirs into the cross-flow reservoirs, through the use of hydraulic

fracturing techniques. This is due to economic preferences in the cross-flow reservoirs

compared to the commingled ones: the cross-flow reservoirs have shorter operating life,

higher ultimate recovery and a reduced perforating and completing cost and requires

less engineering time for interpretation.

22

The macro cross-flow occurs after some time of production: the high permeability

layer depletes faster than the low permeability layer, provoking a vertical pressure

difference between those layers. The flow, then, occurs from the low to the high

permeability layer, and can happen during the pseudo-steady state or a transient state

[17].

Previous studies have shown that when testing all reservoirs layers concomitantly,

the pressure derivative response usually is very similar to that of a single layer reservoir

[3]. Conversely, for layers with high contrasts in thickness, skin factor and permeability,

the response can resemble that of a double porosity reservoir, showing a V-shape

signature [17].

2.4.3 Layered reservoirs with or without crossflow

This is an extension of the double porosity models. The double permeability

behaviour is observed in stratified reservoirs, where the permeability of the different

layers is participating to the response (this scenario corresponds to when, in a fissured

reservoir, the matrix blocks are connected to the fissures).

The double permeability model, proposed by BOURDET in 1985 [13], considers

that:

- The well, intercepting two homogeneous layers, is affected by wellbore

storage. At each layer, a skin defines the communication between the well and

the formation.

- The initial pressure is the same in the two layers;

- After some production, a difference of pressure is established between the two

layers and a cross flow takes place in the reservoir.

From now on, the subscripts 1 and 2 will refer, respectively, to layer 1 and 2.

Layer 1 has higher permeability than layer 2.

The total permeability-thickness product is:

tℎ�.<q8 = twℎw + tYℎY 2.11

The reservoir total storativity is:

(�><ℎ)�.<q8 = (�><ℎ)w + (�><ℎ)Y 2.12

The kh contrast between the two layers is expressed by the mobility ratio �:

� = �� = �� ¡¢ 2.13

23

Contrast of storage between the layers is expressed by the storativity ratio L,

which defines the contribution of layer 1 (high permeability layer) to the total

storativity:

L = (£F �)�(£F �)�� (£F �)� = (£F �)�(£F �)�� ¡¢ 2.14

The reservoir cross flow is defined by the interlayer cross flow coefficient λ: the

smaller the λ, the more difficult the communication between the layers. λ=0 corresponds

to two layers without cross flows (commingled system):

¤ = C0Ytℎ�.<q822 ℎ′t¦� ℎwt¦w ℎYt¦Y

2.15

λ is a function of the vertical permeability t¦� in the low permeability “wall” of

thickness ℎ′ between the layers and of vertical permeability in the two layers t¦w and t¦Y, C0 is the well radius. λ is equivalent to the effective interporosity flow parameter

λeff defined by MOENCH (1984) [21], and if there’s no skin at the interface and

pressure gradients are negligible in the high permeability layer, λ is equivalent to the

interlayer flow parameter of the transient double permeability model of CHEN et al.

(1990) [22].

The total, or equivalent, properties presented in this section can be used to build

equivalent reservoir models.

2.4.4 Previously studied Well Test extreme responses in Cross-Flow

Reservoirs

Certain reservoir kinds have extreme variations in their properties, at various

scales within its layers. Flow happens laterally within the layers with varying degrees of

vertical communication. This type of cross-flow (micro-cross-flow) manifests as either

negative geoskin, a geochoke (hump effect) or a ramp effect (cross-flow not between,

but within layers). V-shape, as explained before, is a characteristic signature of a

layered reservoir. These effects (or signatures) will be here better explained, except for

the Ramp effect (HAMDI, 2012) [17], which is not a macro cross-flow response.

2.4.4.1 The hump effect

The well test, when representing the period of transient regime production,

possibly shows the layered reservoir as a homogeneous reservoir, with average

properties, and sometimes the difference between permeabilities is not detected. For a

24

reservoir with cross-flow, when this difference between permeabilities is perceived, the

hump effect (or geochoke) is shown in the pressure derivative plot, usually during the

middle time of the WTA. Normally, this effect gives rise to a negative skin, which, for

being provoked by the geology of the reservoir (its heterogeneities) and not for by-the-

well attributes, was named by CORBETT [18], in 1996, a Geoskin.

The hump effect signature can be seen in reservoirs with high permeability lenses

intersecting the wellbore, laterally and vertically limited, embedded in a low

permeability matrix. An example can be seen in Figure 2-5.

Figure 2-5 – Hump effect example (CORBETT, 2005)

2.4.4.2 The V-shape signature

The V-shape signature also happens in cross-flow reservoirs, but unlike the hump

effect (geochoke), the V-shape signature happens in the “cake” kind reservoir. The V-

shape signature becomes more evident as the lateral extension of the high permeable

layer equals the lateral extension of the low permeable layer. An example of a “cake”

kind reservoir can be seen in Figure 2-6, where the well is intersected by two layers of

equal or unequal length and different permeabilities:

Elapsed Time (hours)

Del

ta P

/Del

ta Q

(ps

i/ST

B/d

ay)

25

Figure 2-6 – “Cake” reservoir example. k1 ≠ k2.

According to HAMDI (2012) [17], the smaller the capacity contrast (ω), or the

larger the transmissivity contranst (κ), the deeper the valley amplitude in a derivative

pressure versus time log-log plot. However, the capacity contrast or the transmissivity

are not always dependent, as for carbonates. In sandstone reservoirs, for example, there

are usually relations between permeability and porosity, meaning that a high permeable

layer usually has a higher porosity. Hence, the thickness of the layers can be considered

as the controlling parameter of the V-shape signature appearance on the derivative

curve. This is shown in Figure 2-7.

Figure 2-7 – The effect of different parameter on the well-test response of a double permeability reservoir. (HAMDI, 2012)

The interlayer cross-flow coefficient (λ), on the other hand, affects the time at

which the valley appears on the derivative plot.

26

2.4.4.3 The Geochoke (hump effect) and the V-shape signature

The Geochoke (hump effect) occurs when there is a delay in the provision of flow

from the low permeable (matrix) to the high permeable layer (lens) – the high

permeable layer depletes faster than the low permeable layer, therefore, the geochoke,

which is a probable cause of a negative geoskin.

The V-shape signature happens due to a vertical difference in pressure between

the “cake” reservoir layers.

HAMDI (2012) also presented a set of drawdown derivative responses of a two-

layer system, where the lateral extension of the thin high permeable layer varies from 25

feet to 6400 feet, shown below in Figure 2-8.

Figure 2-8 – A simplified two-layer reservoir representing the double permeability model. The thin high permeable layer can have a variable lateral and vertical extension.

(HAMDI, 2012)

He also showed that the geoskin effect at the shorter length changes to a classical

double permeability response with a V-shape signature in the larger lengths. The small

hump after the early linear flow is due to the double permeability and will convert to a

perfect valley in larger patch lengths. This hump can disappear as the layer thickness

contrast is reduced. This is depicted in Figure 2-9.

27

Figure 2-9 – A set of drawdown derivative response curves of a two layer reservoir model (referent to Figure 2-8) where the lateral extension of the thin high permeable

layer (i.e. 10 ft) varies from 25 ft to 6400 ft. (HAMDI, 2012)

2.4.4.4 The Geoskin

The skin effect is a decrease (negative skin) or increase (positive skin) in the

pressure drop predicted from Darcy’s law, using the permeability times thickness (kh)

value predicted from a well test (buildup or drawdown well test). This decrease or

increase is assumed to be caused by the skin.

A damage near the wellbore can cause a positive skin, because of the near

wellbore permeability, that is lower than the formation permeability (knear well bore <

kformation), meaning that the well productivity is lower than the one predicted with no

skin near the wellbore. If the opposite happens (knear well bore > kformation), the skin is

negative, meaning that the well productivity is greater than the one predicted with no

skin near the wellbore. In cases where the skin effect is null, the well productivity is the

original one. Figure 2-10 shows the positive and negative skin effects.

28

Figure 2-10 - Definition sketch of the pressure profile away from the wellbore showing skin as an increase or decrease in pressure in the immediate wellbore region. As the

reservoir pressure drops from P1 to P2 in the expected, radial uniform case, the pressure profile drops as shown. If the pressure measured at the wellbore is higher than expected

from this profile then this defines a negative pressure drop and an increase in production. If the pressure measured in the wellbore is much lower than this gives a

positive skin and results in reduced production. (CORBETT, 2012)

A negative skin (or geoskin) can, for instance, indicate the presence of high

permeability lenses of limited extent (less than 10% of the depth of investigation, or

radius of investigation, for example) intersecting the well. Since this response is usually

associated with fractures, they have been named “Pseudo-fracture channels (PFC)”

(CORBETT et al., 1996 [18]) but the response is caused by lenses, not fractures.

This means that a half slope derivative in the early time response is usual [17].

CORBETT et al. (1996) performed sensitivity analyses and concluded that the geoskin

level is sensitive to radius, thickness and permeability contrast of the PFC’s and the

background matrix, and insensitive to number or location of PFC’s intersecting the

wellbore. Else, they showed that the geoskin can be used along with core analysis and

production logs to predict the extent of the PFC’s.

2.4.5 Wellbore storage effect

When the well is opened, the production at surface is initially due to the

expansion of the fluid stored in the wellbore, and the reservoir contribution is initially

negligible. This is the pure wellbore storage effect, and it can last from a few seconds to

a few minutes, until the reservoir production really starts and the sand face rate

29

increases until it becomes the same as the surface rate. When this condition is reached,

the wellbore storage has no effect anymore on the bottom hole pressure response, so the

data describes the reservoir behaviour and it can be used for a transient analysis [13].

Figure 2-11 shows the wellbore storage for drawdown and build-up. Drawdown is the

given name to the period after which the well is opened, representing the continuously

dropping pressure, while build-up period is the one after the well is shut, representing

the continuously increasing pressure.

Figure 2-11 – Pressure drawdown, build up and wellbore storage. Area 1 represents the pure wellbore storage effect, Area 2 represents the afterflow storage effect.

(BOURDET, 2002)

The wellbore storage can be calculated as:

© = >.;0388 ª5ª: «2v2¬�¢¢ 2.16

Where:

- C is the wellbore storage coefficient;

- >. is the oil compressibility;

- 5 is the oil pressure;

- C is the radius;

- C0388 is the wellbore radius;

- : is time; and

- ;0388 is the wellbore volume.

Drawdown Buildup

Wellbore storage (Area 1 = Area 2)

Area 1

Area 2

30

2.4.6 Multiphase flow reservoirs

So far, the well test interpretation methods are designed for wells producing a

single-phase fluid. Although the reservoir can be considered as a single-phase fluid

reservoir, at surface, most wells produce more than one kind of fluid.

A two-phase flow condition is found in oil and water reservoirs, which is the case

studied in this research.

When more than one fluid phase is flowing in a porous system, each phase

reduces the ability of the other phase to flow, and the effective permeability for each

phase is less than the permeability for a single phase. The relative permeability is a

function of saturation, which changes in space and time.

The Perrine Method (PERRINE, 1956) [23] is the simplest and one of the most

used methods proposed for analysing tests in multiphase reservoirs. The Perrine method

considers the oil wells method of analysis, and an equivalent liquid flow rate. The

method is a modified single phase approach, that an equivalent liquid of constant

properties is defined as the sum of the three phases: oil, water and gas (PERRINE,

1956; MARTIN, 1959) [23], [24]. The analysis yields the effective mobility of this

equivalent fluid, but it does not give the absolute reservoir permeability directly.

The Perrine Method hypothesis are:

- The three phases are uniformly distributed in the reservoir;

- The saturations are constant and independent of pressure;

- The capillary pressures are neglected: the pressure is the same in the different

phases.

This method becomes less reliable, for gas reservoirs, as the gas saturation

increases (WELLER, 1996) [25]. Furthermore, neglecting the changes of saturation

around the wellbore may cause the skin to be over-estimated and the effective

permeability is frequently underestimated (AL-KHALIFAH, 1987; RAGHAVAN;

1989) [26], [27].

Unlike the Perrine Method, that is still being largely used, the capillary pressure

will not be neglected in this work, meaning that the different fluids will have different

pressures at different times, and the saturations will be dependent of pressure, also

varying with time. This way, a more reliable result can be presented. The IMPES

(Implicit Pressure, Explicit Saturation) method will be used, and will be presented on

section 2.5.

31

2.5 Reservoir Simulation and Finite Volume Method Well test analytical solutions of the diffusivity equation with various boundary

conditions are mainly derived for the constant flow rate cases, and are used for the

idealized reservoir characterisation based on the estimation of reservoir parameters.

Analytical solutions tend to be very complex and computationally costly or may not be

applicable on complex geology and reservoir. On the other hand, numerical techniques

are proven to be accurate against analytical simulations: they can be used to tackle the

non-linearity (e.g. non Darcy flow, multi-phase flow, and non-consolidated formation),

complex well situations (e.g. multi-segment and slanted well) and combined complex

reservoir heterogeneities (e.g. multi-layered, multi-facies, highly faulted and fractured)

(HAMDI, 2012). Therefore, along with associated boundary conditions, the numerical

techniques are implemented to solve well-test diffusivity equation.

The main objective of this study is to characterise a given geological model,

through a numerical simulator, looking at well-test responses to different geological

phenomena. According to the geology of the reservoir, trends in the well-test responses

can be set as a family of pressure diagnostic plots, or the so called “Geotype curves”, as

explained by CORBETT et al., (2013) [5].

The reason why a carbonate reservoir has been chosen in this work is that it can

be a great challenge to be modelled, concerning numerical and gridding issues, due to

its pore systems, which are complex through depositional and diagenetic processes, and

fractures are often present.

When well tests analyses are made, analytical models are recommended to do, for

example, a pressure-transient analysis. The analytical models represent exact solutions

to simplified problems, meaning that the physics of the problem is preserved. This way

the analytical methods are often used to determine how the reservoir parameters affect

its performance.

The equations derived during the formulation process, if solved analytically,

would give the pressure, saturation, and production rates as continuous functions of

time and location. Because of the non-linear nature of these equations, analytical

techniques cannot be used and solutions must be obtained with numerical methods.

This means that reservoir problems, which are more complicated than an

analytical model could stand, demand simplifications, which will not properly represent

32

the reservoir and the perforated well. Numerical models are, therefore, more acceptable

to this kind of problem, since they yield approximate solutions to exact problems.

Usually, commercial software programmes, such as Eclipse and other similar

ones, use the Finite Difference Method (FDM), but as an alternative, in this work, the

chosen numerical model is the Finite Volume Method (FVM). The main reason for this

change is that while the FDM discretizes the reservoir medium in spaced points, the

FVM uses volume discretization, “embracing”, or comprehending, the whole reservoir

volume (no spaces between volumes, unlike “points”), comprehending its entirety, with

no importance to its geometry or boundaries. Another name to the FVM is the Volume

Control Method, resembling that within each discretised volume, properties don’t

change. Theoretically, the FVM brings the problem representation closer to reality,

being able to simulate a heterogeneous and anisotropic reservoir, with non-regular

boundaries.

2.5.1 The Reservoir Simulator

When a petroleum reservoir is discovered, the information available of the field is

scanty at best, and can be disjointed too, because bits of information emanate from

different parts of the field. Therefore, it is necessary to integrate these pieces of

information as accurately as possible, in order to construct a picture of the system. As

field development evolves, more information is acquired, and the reservoir description

becomes closer to reality, and the better the reservoir is represented by the simulation

code.

A robust formulation computer code can be used dynamically with laboratory data

and field observations, as it can be used to shed light on the validity of laboratory

experimental approaches.

Reservoir simulation is based on the principle that: (Mass in) – (Mass out) =

Accumulated Mass; or (Flow in) – (Flow out) = Accumulated Flow Rate.

The best way to simulate a reservoir is to use discrete solutions, because it

represents a continuous system through the finite differences method or the finite

volume method, taking into account the variations of fluid/rock properties in space and

time, by analysing the reservoir blocks and advancing temporally in discrete steps.

Reservoir simulation is used to quantify and interpret physical phenomena with

the ability to extend these to project the reservoir’s future performance. To do this, the

33

model used is numerical, and not analytical, due to the complexity and non-linearity of

the variables involved. The simulator divides the reservoir into a number of discrete

units, up to three dimensions, and models the progression of reservoir and fluid

properties through space and time in a series of discrete steps. It is important to take

caution, because the model being simulated differs from the reservoir itself. This

happens due to data uncertainty, averaging, inferred of extrapolated information, among

others. In this case, both the simulation code developed for this work and the

Schlumberger software Eclipse 100 are black-oil simulators, which means that they may

be unsuited to modelling certain processes, such as highly-compressible fluids.

2.5.2 Numerical Solution Methods versus Analytical Solution Methods

Analytical solution techniques provide exact solutions, when they can be found,

that are continuous throughout the system. The problem with the analytical solutions is

that they are very limited, falling short when dealing with non-uniformity of

permeability and porosity, varying formation thickness, varying fluids properties, and

other conditions that bring non-linearity to the problem. Therefore, finding analytical

solutions for petroleum reservoirs generally means that it will be necessary to simplify

the problem, so it can be plausible to be handled analytically. This means that an exact

solution, provided by the analytical techniques, is associated with approximate

problems (e.g., a classical well test analysis model).

Numerical solutions, on the other hand, tackle non-linearity bringing approximate

solutions to exact problems.

2.5.3 IMPES Method

IMPES (Implicit Pressure, Explicit Saturation) is a method that uncouples the

partial differential non-linear equations system, evaluating pressure and saturation

coefficients at the previous time step.

The idea of the IMPES Method consists of solving a partial differential coupled

system for up to three-phase flow in a porous medium, by separating the computation of

pressure from that of saturation. Basically, the IMPES method allows to implicitly

calculate the pressure in the porous medium, and then, through using the obtained

pressure values, to explicitly calculate the saturations of the two fluids in the porous

medium. The saturation can be explicitly calculated as it takes longer than pressure to

34

change as the reservoir is depleted. Therefore, The IMPES method applied to the Finite

Volume Method is a quick and accurate alternative to the analytical method.

35

3. The developed computer code

for multiphase flow.

3.1 The Lacustrine Carbonate Formation of Morro de Chaves

According to CORBETT and BORGHI (2013) [7], lacustrine carbonates have

been less studied in terms of reservoir characterisation, as they are not historically

common as major carbonate reservoirs. Nonetheless, the Pre-Salt of the South Atlantic

Margin in Brazil discoveries have set the lacustrine carbonates as a major target. The

Morro de Chaves Formation is localised in the Sergipe-Alagoas Basin, Northeast of

Brazil, and is basically made up of coquina deposits with varying amount of clastic

input.

The lacustrine carbonates are formed in lakes sensitive to saline, pH, water level

and sediment supply, which results in a wide range of primary fabric types and

associated porosity types, likely formed by the interaction of either microbial and/or

chemical processes. Lacustrine grainstones can be formed from spherulites or coquina.

Stromatolites and stromatolite bioherms can form cavities and laminated pore systems.

These and other processes can potentially influence porosity and permeability. The rock

properties in lacustrine carbonates are, still according to CORBETT and BORGHI

(2013), not so different from those in other carbonate systems, but just less well

understood and documented. Lacustrine carbonates can be very heterogeneous, with

layering on the scale of centimetres and laterally varying from metres to decimetres.

They also stated that in reservoirs with high-permeability, high-porosity rock-types, the

fluid production into the well is likely to be uneven. This can be used to identify the

flowing clusters determined by the complex porosity system and to distinguish them

36

from fracture-contribution, what highlights the importance of well-testing in this type of

reservoir.

3.1.1 The Morro de Chaves Formation Reservoir and its initial

conditions

The Morro de Chaves Formation’s characterisation and geology quantification in

terms of properties involve the measurement of parameters such as porosity and

permeability. Alongside fluid properties, initial reservoir conditions and production

data, the porosity and permeability parameters will be used as basis for the reservoir

simulation scenarios. A study (provided by BG do Brasil) made in situ shows the

coquinas formation, which is made by shells and presents primary and secondary

porosities, as seen in Figure 3-1.

Figure 3-1 – Bivalve Coquinas from Morro de Chaves Formation. The Picture on the left shows the coquinas fabric: presence of shells. The picture on the right shows a

coquinas thin section photomicrograph: porosity is in blue. Note that the black outlines represent dissolved shells, or moldic porosity (secondary porosity). (CORBETT and

BORGHI, 2013)

The Morro de Chaves Formation is a very heterogeneous carbonate (permeability

CV = 1.43), as can be seen in the Lorenz Plot (Figure 3-3) and in the GHE poro-perm

chart (Figure 3-2) characterising it as a worthy object of study for this work.

37

Figure 3-2 – Morro de Chaves Global Hydraulic Elements: Poro-perm distribution (CORBETT and BORGHI, 2013). If there were a fracture, it would be represented by data in the upper redish-left region (low porosity, high permeability values). The lens data is indicated in the orange band, and corresponds to 33% of the inflow to the well.

From Figure 3-2 (Global Hydraulic Elements) it is noticeable that the data are

mainly situated in the yellow zones, meaning that, regardless the porosity value, which

varies from 0.05 to almost 0.25, the permeability varies from slightly less than 10-1 mD

to 103 mD. Another conclusion to this chart is that more than one petrotype is observed.

The Formation’s Lorenz Plot shows that 80% of the fluid inflow to the wellbore

comes from 40% of the plug. This is depicted in Figure 3-3 and, again, indicates a very

heterogeneous medium, which can be proved as the Lorenz Plot curve that is distant

from the diagonal line. Orange band, in Figure 3-2, defines the best quality reservoir in

this case. Permeability is not extremely larger than seen in other bands so it is not a

fracture. A possible fracture could appear in the circled zone in Figure 3-2(small

porosity, great permeability). The lens, in this case, is represented by the data plotted in

the orange band, and is responsible for 33% of the inflow to the well. This can be seen

in the Lorenz Plot (Figure 3-3), where the lens data are within the box.

Fractures Lenses

38

Figure 3-3 – Morro de Chaves Lorenz Plot. (CORBETT and BORGHI, 2013) Box shares 33% of transmissivity (due to only 3 plugs).

This Lorenz Plot from Morro de Chaves doesn´t show extreme heterogeneity, as

Figure 2-2. By summing the porosity and permeability values, the lens (three points in

the black box) shows total summed porosity of 18.8%, and arithmetic average

permeability of 533 mD. The other points are assumed to be outer reservoir data, which

have 14.9% of porosity (summed values), and 21.14 mD of permeability (geometric

average). This is shown in Table 3-1, and can be verified in Table 3-2. Thereunto, one

can conclude that 33% of the entire reservoir’s transmissivity comes from the high

permeable lens, as can be seen in Table 3-1.

Table 3-1 – Transmissivity, Porosity and Permeability values for Lens and Reservoir, through the assessment of the Lorenz Plot.

Property Lens Reservoir

Porosity (arithmetic average) 18.8% 14.9%

Permeability (geometric average) 533 mD 21.14 mD

Transmissivity 33% 67%

The porosity-permeability (�-k) sample data extracted from CORBETT and

BORGHI`s 2013 work have the following values:

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,2 0,4 0,6 0,8 1

Tra

nsm

issi

vit

y (

kh

)

Storativity (phih)

Lorenz Plot

Lenses

39

Table 3-2 – Porosity-permeability values (CORBETT and BORGHI, 2013). The shaded values correspond to the orange band, in Table 3-1.

�� (decimal)(decimal)(decimal)(decimal) K (mD) �� (decimal)(decimal)(decimal)(decimal)

K (mD) �� (decimal)(decimal)(decimal)(decimal) K (mD) �� (decimal)(decimal)(decimal)(decimal)

K (mD)

0,113 12,9 0,164 16,6 0,164 481 0,206 106

0,125 14,7 0,158 21,5 0,128 2,49 0,184 13,6

0,120 13,0 0,137 5,05 0,147 199 0,173 36,7

0,141 36,6 0,145 30,4 0,153 48,0 0,195 41,2

0,108 15,8 0,163 73,9 0,187 148 0,177 12,1

0,161 37,3 0,175 103 0,192 148 0,059 0,062

0,145 9,09 0,164 57,9 0,193 238 0,050 0,061

0,157 93,9 0,211 179 0,200 742 0,065 0,208

0,136 5,55 0,168 86,7 0,193 258 0,058 0,052

0,144 41,0 0,049 0,210 0,205 241 0,180 72,1

0,145 4,90 0,159 224 0,201 375 0,165 63,9

0,151 46,3 0,120 6,70 0,194 281 0,161 147

According to SAGAWA et al. [6], having one lens across the wellbore being

modelled is equivalent to having more lenses. Therefore, only one lens intersecting the

wellbore is considered in this work. If necessary, an equivalent layer can be calculated

from the permeability and other properties of all layers (lens, in this case).

3.1.1.1 The Scenarios

From the observations made above, one can conclude that the presence of a lens is

significant when it comes to transmissivity. Hence, sensitivity scenarios will be run,

considering the following average estimatives (Table 3-3). For the given porosity and

permeability values for the lens, presented in Table 3-1, a different permeability value

for the rest of the reservoir will be tested, with the geometric porosity average.

40

Table 3-3 – Statistical treatment of porosity and permeability data

Statistical Analysis Method Porosity (Decimal) Permeability (mD)

Arithmetic Average 0.15 100

Geometric Average 0.14 26

Harmonic Average 0.13 0.76

Standard Deviation 0.04 143

Coefficient of Variation 0.28 1.43

Number of samples 48 48

Tolerance (%) 7.94 41

Nzero (20% Tolerance) 7.57 206

Nzero (50% Tolerance) 1.21 33

Minimum 0.05 0.1

Maximum 0.21 742

The permeability values presented in Table 3-3 are used as the horizontal

permeability. The vertical permeability is assumed to be 10% of the horizontal

permeability.

To simulate the multiphase flow in the reservoir, initial reservoir and fluids

properties are required. In Table 3-4, the initial reservoir and fluids properties are

shown. Those data were based in averages from general literature:

41

Table 3-4 – Initial Reservoir and Fluids Properties

Property Brazilian Field Units International Units

System

Initial Pressure 203.9432 kgf/cm² 2 x107 Pa

Bubble Pressure 81.57728 kgf/cm² 8 x106 Pa

Minimum Wellbore

Pressure

81.57728 kgf/cm² 8 x106 Pa

Permeability Varies according to scenario and locality.

Horizontal Permeability Taken from from the statistical analysis.

Vertical Permeability Is equal to 10% of the horizontal permeability.

Porosity Varies according to scenario and locality.

Oil density 0.875 0.875

Water density 1 1

Oil API 30.2143 30.2143

Oil compressibility at

reservoir conditions

1.47x10-10 /( kgf/cm²)

1.5 x10-9 /Pa

Water compressibility at

reservoir conditions

4.31x10-10 /( kgf/cm²)

4 x10-10 /Pa

Formation compressibility

at reservoir conditions

3.92x10-10 /( kgf/cm²)

4.4 x10-10 /Pa

Initial Water Formation

Volume Factor

1.03 m³/m³ std 1.03 m³/m³ std

Initial Oil Formation

Volume Factor

1.103 m³/m³ std 1.103 m³/m³ std

Initial water saturation 0.2 0.2

Connate water saturation 0.15 0.15

Water viscosity 1.02 x10-8 (kgf/cm²).s 10-3Pa.s

Formation thickness 50m 50m

Drainage radius 800m 800m

Well head producing flow

rate

150 m³/day 150m³/day

Well radius 0.0889 m 0.0889 m

42

The wellbore storage effect was considered in all simulations developed in the

Wolfram Mathematica code.

3.2 The Multiphase Flow Simulator developed in this research The conceptual model developed for this study is based on actual carbonate

reservoirs, where a low permeable matrix is interbedded with thin high permeable

limited lenses, based on a work presented in 2000 by SAGAWA et al. [6].

A generic model was, then, built in Wolfram Mathematica®, as follows:

The reservoir and its well are represented in a cylindrical axisymmetric-geometry,

which is ideal for describing single well-problems. The wellbore and the reservoir

boundary are concentric cylinders of same height. In radial problems, the principle flow

directions are radial (r), axial (z) and tangential (θ). A typical two-dimensional (r, z)

representation is interesting for axisymmetric single-well problems, where gravity and

layering effects are significant. The r,z plane can be taken at any θ location without

changing the problem due to its axis-symmetric nature.

For the problem being dealt with here, a radial-geometry reservoir consisting of oil

and water is considered. The well is placed in its centre and the reservoir boundaries are

considered to be of no-flow type. It is important to notice that the most often used boundary

conditions can be grouped into two main categories: the Dirichlet and Neumann type. In

Dirichlet type boundary condition, the values of the dependent variables are specified at the

boundaries, and in the Neumann type boundary condition, the gradients of the unknowns are

specified in the boundaries. Depending on what is known at the wellbore, it is possible to

choose between the Dirichlet and the Neumann boundary type conditions. It is chosen the

Dirichlet when the wellbore pressure is known, and the Neumann condition is chosen when the

wellbore flow rate is known. In this case, the Neumann condition is considered, because the

outer boundaries are sealed, allowing no flow in or out the reservoir, meaning that the pressure

gradient across the external boundaries is zero.

The mathematical model here used is the numerical model and the developed reservoir-

simulator can be classified as Black Oil, which is used when recovery processes are intensive to

compositional changes in the reservoir fluids. The wellbore is intersected by a high permeability

lens, which was described earlier in this text.

A schematic drawing is presented in Figure 3-4:

43

Figure 3-4 – Schematic figure of a 2-D radial reservoir, with well placed on the left (represented in blue volumes). When rotating the image around the left boundary, the

well takes the centre of the 3-D reservoir.

3.2.1 The Discretised Hydraulic Diffusivity Equation for Oil and

Water Reservoir

The equations that govern the laminar flow of oil and water in a porous medium

are, respectively:

,

ˆo ro

o o o stdo o o

S kP g z q

t B B

φ ρµ

∂ ′′′= ∇ ⋅ ∇ − ∇ + ∂

kɺ 3.1

,

ˆw rw

w w w stdw w w

S kP g z q

t B B

φ ρµ

∂ ′′′= ∇ ⋅ ∇ − ∇ + ∂

kɺ 3.2

1o wS S+ = 3.3

( )cow w o wP S P P= − 3.4

Where:

- is the medium porosity; φ

44

- is the oil saturation;

- is the Oil Formation Volume Factor;

- is the permeability tensor;

- is the oil-relative permeability;

- is the oil viscosity;

- is the oil pressure;

- is the oil density;

- g is gravity;

- z is the vertical distance to datum;

- is oil flow rate per volume unit in standard conditions;

- wS is water saturation;

- wB is water formation volume factor;

- wµ is water viscosity;

- wP is water pressure;

- wρ is water density;

- ,w stdq′′′ɺ is water flow rate per volume unit in standard conditions;

- cowP is oil-water capillary pressure;

- rok is the oil relative permeability;

- rwk is the water relative permeability.

Those equations are written in the two-dimensional (r,z) radial coordinates

system. By integrating these equations in space and time, then using the traditional

Taylor series expansion method for the derivatives, and discretising according to the

finite volume method, the oil and water equations are ready to be implemented into the

IMPES (Implicit Pressure, Explicit Saturation) routine. The discretised equations are:

For the oil phase:

C

stdowwCowooCop

CoCowowoeoeososonono

V

qSSCPPC

PPPPP

∆−−+−

=⋅−⋅+⋅+⋅+⋅

,0,

0,

,,,,,,,,,,

)()(ɺ

τττττ

3.5

oS

oB

k̂

rok

oµ

oP

oρ

,o stdq′′′ɺ

45

Where:

,

2 1 1

2r ro

o eC E C C o o e

k k

r r r r Bτ

µ

= + ∆ + ∆ ∆

,

2 1 1

2r ro

o wC W C C o o w

k k

r r r r Bτ

µ

= − ∆ + ∆ ∆

,

2 1 z roo n

C N C o o n

k k

z z z Bτ

µ

= ∆ + ∆ ∆

,

2 1 z roo s

C S C o o s

k k

z z z Bτ

µ

= ∆ + ∆ ∆

woeosonoCo ,,,,, τττττ +++=

Coooo

wCop PBBPt

SC

∂∂+

∂∂

∆−= φφ 1

)1

(1 0

,

CoCow Bt

C

∆−= 11

, φ

3.6

For the water phase:

Ccow

C

stdowwCowooCop

CoCowowoeoeososonono

DV

qSSCPPC

PPPPP

,,0

,0

,

,,,,,,,,,,

)()( +∆

−−+−

=⋅−⋅+⋅+⋅+⋅ɺ

τττττ

3.7

Where:

,

2 1 1

2r rw

w eC E C C w w e

k k

r r r r Bτ

µ

= + ∆ + ∆ ∆

,

2 1 1

2r rw

w wC W C C w w w

k k

r r r r Bτ

µ

= − ∆ + ∆ ∆

,

2 1 z rww n

C N C w w n

k k

z z z Bτ

µ

= ∆ + ∆ ∆

,

2 1 z rww s

C S C w w s

k k

z z z Bτ

µ

= ∆ + ∆ ∆

wwewswnwCw ,,,,, τττττ +++=

3.8

46

Cowwo

wCwp PBBPt

SC

∂∂+

∂∂

∆= φφ 1

)1

(0

,

CwCww Bt

C

∆= 11

, φ

wcow

ww

ecow

ew

scow

sw

ncow

nw

Ccow

Cw

Ccow PPPPPD ⋅+⋅+⋅+⋅+⋅−= τττττ

Where:

- e, w, n, s sub-indexes are related to the volume

boundaries at east (e), west (w), north (n), south (s) of

the centre of the volume.

- E, W, N, S sub-indexes are related to the volumes at east

(E), west (W), north (N), south (S) of the central volume

(C).

- Δr is the radial distance between volumes centres or

boundaries, depending on the sub-index;

- Δt is the time step;

- Upper index “0” is related to the value of the property at

the previous iteration.

- o and w sub-indexes are property of oil (o) or water (w).

Now, the water and oil equations are coupled, characterising the IMPES Method:

the reservoir’s dynamic fluids behaviour can be evaluated with respect to time, while it

is depleted, as pressure and saturation are evaluated.

First, the oil pressure (P) is implicitly calculated as follows:

][][][ 1 BAP ⋅= − 3.9

Where:

−= e

wCww

Cowe

oe

C

CA ττ

−= w

wCww

Coww

ow

C

CA ττ

3.10

47

−= n

wCww

Cown

on

C

CA ττ

−= s

wCww

Cows

os

C

CA ττ

( )

+−+= C

wpCwC

ww

CowC

opCo

C CC

CCA ττ

∆−

∆+

−

−=

V

q

V

q

C

CD

C

CPCC

C

CB stdostdw

Cww

CowC

cowCww

Cow

oCop

CwpC

ww

Cow

c,,0ɺɺ

Then, water saturation distribution can be explicitly obtained:

0

,0

1

( )

n n s s e e w ww o w o w o w o

w w C wstdC C C C Cww w o wp o o cow

P P P PS S q

C P C P P DV

τ τ τ τ

τ

⋅ + ⋅ + ⋅ + ⋅ = + − ⋅ − − − + ∆

ɺ

3.11

Regarding the well, the total flow rate (,w stdqɺ + ,o stdqɺ ), for a drawdown test must be

kept constant, and, since the analysis being made is a well test, the well bore storage

effect must be considered in the programmed code. That is necessary for plotting and

analysing the pressure behaviour in the early production time.

For the complete development of the IMPES method coefficients, see item 7.1.

3.2.2 The Model Generated in Wolfram Mathematica

3.2.2.1 Gridding

The gridding was developed with the intention of better refinement near the well

bore, and coarser refinement far from it. This is due to computational costs: a finer

refinement near the well is indispensable for more precise pressure and saturation

results. The more refined the gridding, the longer the computer takes to calculate the

reservoir’s properties. Since the objective here is a well test analysis, farther from the

well the grid can be coarser, to reduce computation costs such as time and use of

machine capacity. So, a logarithmic grid was assigned to 50 cells in the radial direction

to the model developed in this work. The axial direction has 11 cells of equal

dimensions.

48

3.2.2.2 Time Steps

According to SAVIOLI and BIDNER (2008) [28], the smaller the time step (Δt),

the smaller the pressure and saturation oscillation. They have showed that small time

steps, such as a 20 second interval, lead to better results in the beginning of the

simulation, than higher time step values, as shown in Figure 3-5. After a certain period

of simulation, smaller and greater time steps tend to the same response.

For this work, the time interval starts at 20 seconds, and has an increase of 10%

after each iteration. The time step increase is done until the pressure convergence error

is smaller than 10%. If the error is greater than 10%, the time step is decreased by a

factor of 50%, until the convergence is reached.

The time interval was built according to the well test industry practice.

49

Figure 3-5 – Comparison of oil saturation (So) versus time, for different time steps. (SAVIOLI and BIDNER, 2005)

50

3.2.2.3 Fluid properties

The fluid properties are calculated from international system of units formulae.

3.2.2.3.1 Formation Volume Factors (FVF)

The oil and water formation volume factors (FVF or B) were assigned to the

model through equations which show the FVF variation as the reservoir is depleted. The

FVF relates the fluids volume at reservoir conditions to its volume at standard

conditions:

FVF = ;234;4<= 3.12

For the oil, the FVF is given by:

B.(P) = B./ + c.B./(P/ − 5) 3.13

For the water phase, the FVF is given by:

B0(P) = B0/ + c0B0/(P/ − 5) 3.14

Where:

- FVF is Formation Volume Factor;

- ;234 is the volume of the fluid at reservoir conditions;

- ;4<= is the volume of the fluid at standard conditions;

- B. is the oil FVF;

- B0 is the water FVF;

- B./ is the initial oil FVF;

- B0/ is the initial water FVF;

- c. is the oil compressibility factor;

- c0 is the water compressibility factor;

- P/ is the initial or static reservoir pressure;

- 5 is pressure.

The representative graph of FVF against pressure is shown in Figure 3-6.

51

Figure 3-6– Formation Volume Factor (m³/m³ std) behaviour of Oil (blue) and Water (red) with pressure (Pa).

3.2.2.3.2 Viscosity (µ)

To calculate the oil’s viscosity behaviour with pressure, first it’s necessary to

know its API degree, given by the formula:

API = 141.5?@AB,AB − 131.5 3.15

The API for the oil in this work is 30.2143, characterising a light oil.

Then, the reservoir viscosity variation with pressure is calculated, considering the

reservoir temperature constant, from the following:

μ.K = ®0.32 + 1.8¯10°API±.�² ³ R 360(1.8T234 + 32) + 200UwB�µ.¶·¸¹.··º»¼�

3.16

μ.(5) = �μ.K + 0.001(5 − 5676683)000145(0.024μ.Kw.A+ 0.038μ.KB.�A)�¯10�²

3.17

The water viscosity is considered constant with pressure (see 3.2.2.3.4):

8.0µ106 1.0µ107 1.2µ107 1.4µ107 1.6µ107 1.8µ107 2.0µ107

1.04

1.06

1.08

1.10

1.12

Pressure

For

mat

ion

Vol

ume

Fac

torH

Bo,

BwL

52

μ0(5) = 10�² 5g. i 3.18

Where:

- μ.K is the dead oil viscosity;

- ?@AB,AB is the oil density;

- API is American Petroleum Institute inverted scale that defines the “lightness”

of the oil;

- T234 is the reservoir temperature;

- 5676683 is the bubble point pressure;

- 5 is pressure;

- μ. is oil viscosity;

- μ0 is water viscosity.

The representative graph of viscosity against pressure is shown in Figure 3-7.

Figure 3-7 – Oil (blue) and water (red) viscosities (Pa.s) with pressure (Pa).

3.2.2.3.3 Relative permeability and Capillary pressure curves

As the reservoir is depleted, the water saturation increases, and the oil saturation

decreases, as a function of pressure. The lower the oil saturation (non-wetting fluid), the

0 5.0µ106 1.0µ107 1.5µ107 2.0µ1070.000

0.001

0.002

0.003

0.004

Pressure

Vis

cosi

tyHm

o ,mw

L

53

harder it is for it to flow within the rock pores, hence its relative permeability curve

shows lower values as the production increases. The opposite happens to the wetting

fluid, which in this case is the water, as the reservoir is depleted. Figure 3-8 shows this

situation, and was generated in the developed computational code according to the

following equations:

K2. = y1 − RG0 − G0F.H1 − G0/ U{² 3.19

K20 = RG0 − G0F.H1 − G0/ U² 3.20

The capillary pressure (PF.0) is a function of saturation, and is equal to the

difference of pressure between the oil and water phases:

PF.0 = 0.1�G0 − G0F.H1 − G0/ �w.² 100000 3.21

PF.0 = P. − P0 3.22

Where:

- PF.0 is the capillary pressure;

- P. is the oil pressure;

- P0 is the water pressure;

- G0 is the water saturation;

- G0F.H is the connate water saturation;

- K2. is the oil relative permeability;

- K20 is the water relative permeability.

54

Figure 3-8 – Water (red curve) and oil (blue curve) relative permeabilities versus saturation curves, extracted from the developed code in Wolfram Mathematica.

The capillary pressure curve is:

Figure 3-9 – Capillary pressure versus water saturation curve, extracted from the developed code in Wolfram Mathematica. The 0.15 water saturation is the connate, or

immobile, water saturation. The straight line at y=0 (capillary pressure) is merely due to mathematical boundaries. This value doesn’t exist, since the water saturation has a

minimum value equal to 0.15.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Saturation

Rel

ativ

eP

erm

eabi

lity

0.0 0.2 0.4 0.6 0.8 1.00

50000

100000

150000

Saturation

Cap

illar

yP

ress

ure

55

3.2.2.3.4 Compressibility Factors and Water viscosity

According to LING and SHEN (2011) [29], for a simulation occurring above the

bubble point pressure, a 40% error in the oil viscosity value produces a 4% error in the

recoverable oil volume. Other factors, such as oil (co) and water compressibility (cw)

can be kept constant along the production simulated, without carrying significant errors.

The OOIP (Original Oil in Place) has a much higher influence on the production than

the change in compressibility. The most important factor for production simulation and

reserve estimation are porosity, permeability and pressure effects, while the least

important are those that can be kept constant, such as viscosity (here, only water

viscosity was kept constant) and oil and water compressibility factors.

3.3 The modelled scenarios The scenarios will be first simulated using the developed code for this work, in

Wolfram Mathematica. Then, the results will be validated using the Schlumberger

Eclipse 100 software.

3.3.1 Scenario 1

As explained in 3.1.1, the first scenario has its high permeable lens values of

permeability and porosity extracted from the Lorenz Plot. The properties of the outer

reservoir were chosen from the statistical analysis: to minimize contrasts in the

properties, the permeability value was chosen as the highest, so this case could be a

benchmark. The near well bore reservoir properties were deliberately chosen so the lens

intersecting the wellbore would be of much higher permeability.

The Scenario 1 has the schematic presented in Figure 3-10. The lens and the near

the wellbore area have radius equal to 10% of the total drainage radius. Table 3-5

summarizes the reservoir properties represented by Scenario 1. The permeability of the

outer reservoir was chosen to be equal to the one from the arithmetic average.

56

Figure 3-10 – Scenario 1 Representation: � (porosity) and k (permeability) distribution.

Table 3-5 – Scenario 1: Porosity and Permeability values

Lens Near the well matrix Reservoir Matrix

Porosity 18.8% 10% 14%

Permeability 533 mD 10 mD 100 mD (arithmetic)

In order to explain the mechanism of the simulation of a sealed reservoir,

producing at a constant total flow rate at the well head, the results of this first scenario

will also be used to describe the physics of this kind of reservoir’s behaviour. Therefore,

the next scenarios will have shorter explanation due to their similar behaviour.

3.3.1.1 Finite Volume Method in Wolfram Mathematica Simulation

The Scenario 1 case is three year long-simulation, at a constant total flow rate at

the well head of 150m³/day. The following graphs were extracted from the

computational code, developed in Wolfram Mathematica, especially for this work.

The most important graphs will be presented in this chapter, and other auxiliary

graphs can be found in the Appendix.

It is important to mention that the sum of the water and the oil flow rates is always

constant, equal to 150m³/day, which characterises a drawdown.

As the reservoir is depleted, a continuous pressure drop occurs from the well bore

toward the sealed edge, as shown in Figure 3-11. The pressure near the well starts

dropping at the moment that the well is opened to production, which happens due to the

fluids expansion and reduction of the porous volume. With production, the pressure

57

drop is transmitted throughout the reservoir, as it constantly reaches for its lowest

energy state. It’s through the reservoir’s transmissivity that the pressure drop is

transferred from one point to another, until it reaches the outer boundaries. Due to the

sealed boundaries, the regime achieved is the pseudo-steady state, or the depletion

regime.

Figure 3-11 – Scenario 1: Pressure distribution in the radial reservoir at the time of three years: the closer to the well bore, the smaller the pressure values.

A characteristic of the pseudo-steady regime is that, at the external radius point,

the pressure against elapsed time curve has null slope, representing the sealed

boundaries, as expected by the Darcy’s law equation, for the case with no external

inflow/outflow. To prove that, two graphs of pressure versus radius were extracted from

the model: the first data are relative to mid-height of the reservoir, therefore

comprehends the high permeable lens. The second graph comprehends the bottom of the

reservoir. These graphs are, respectively, Figure 3-12 and Figure 3-13, and clearly show

that =½=2¾2v2�¿ = 0.

58

Figure 3-12 – Scenario 1: Pressure distribution along the reservoir radius, at the time of 3 years, and depth of the high permeable lens, which intersects the wellbore in its

middle.

Figure 3-13 – Scenario 1: Pressure distribution along the reservoir radius, at the time of 3 years, and depth equal to the bottom of the well.

It is clear, from Figure 3-12 (Pressure at the bottom of the well versus radius), the

change in the pressure curve behaviour, before and after the lens limit. The slope of the

pressure versus radius curve changes drastically from one reservoir zone to another.

0 200 400 600 800

1.00 µ107

1.05 µ107

1.10 µ107

1.15 µ107

Radial distance to the well HmL

Pre

ssur

eHP

aL

0 200 400 600 800

1.00 µ107

1.05 µ107

1.10 µ107

1.15 µ107


Pre

ssur

eHP

aL

10% of radius

10% of radius

59

By the end of the three years of simulation, the pressure and water saturation data

are compared to their initial values, as presented in Table 3-6.

Table 3-6 – Scenario 1: Initial and Final values (at 3 representative years) simulated with the Finite Volume Method, in Wolfram Mathematica

Property Initial value Final value

Well Bottom Hole Pressure 2.0 x 107 Pa 9.51912 x 106 Pa

Reservoir water saturation

aside the well bottom 0.2 0.207197

Reservoir oil pressure aside

the well bottom 2.0 x 107 Pa 9.94447 x 106 Pa

An additional analysis was exceptionally made to Scenario 1: as in a time limit-

analysis, this case was run until the last pressure drop, before the minimum pressure at

the bottom of the well bore was achieved in the next time step, which happened at 3.5

years of production. Pressure and water saturation data are presented in Table 3-7.

Table 3-7 – Scenario 1: Initial and Final values (at 3.5 representative years) simulated with the Finite Volume Method, in Wolfram Mathematica







3.3.2 Scenario 2

The scenario 2 is a sensibility test of the first scenario: it was built by multiplying

the lens permeability of the first scenario, which used the Lorenz Plot values, by a factor

of two, as shown in Figure 3-14.

60

Figure 3-14 – Scenario 2 Representation: � (porosity) and k (permeability)

distribution.

Table 3-8 describes Scenario 2.



Porosity 18.8% 10% 14%



Unlike the first scenario, here the pressure across the reservoir and at the bottom

of the well draws down slower as in comparison to Scenario 1. The model was run, until

a maximum simulation time of three years, when the pressure throughout the reservoir

and at the bottom of the well was still far from being reached, as presented in Figure

3-15.

61

Figure 3-15 – Scenario 2: Pressure distribution in the radial reservoir at the time of three years: the closer to the well bore, the smaller the pressure values. Note that the pressure range in this plot has higher values than the pressure range in the Scenario 1 plot shown

in Figure 3-11.

The high permeable lens intersecting the wellbore, this time with a value twice

higher than that from Scenario 1, causes the fluids to flow through the rock of the sealed

reservoir more easily than in 3.3.1.1, characterising a better transmissivity. Thereunto,

the pressure doesn’t have to fall as much to keep the constant flow rate at surface.

One more time, the pressure behaviour on the reservoir radius is plotted in two

different depths: at the depth of the high permeable lens and at the bottom of the

reservoir. These graphs are shown in Figure 3-16 and Figure 3-17 and the same

observation regarding the slope of the curves before and after the lens extent limit can

be made.

62


middle.


By the end of the three representative years of simulation, the following results

were achieved:

0 200 400 600 800

1.36 µ107

1.38 µ107

1.40 µ107

1.42 µ107

1.44 µ107

1.46 µ107


Pre

ssur

eHP

aL

0 200 400 600 800

1.36 µ107

1.38 µ107

1.40 µ107

1.42 µ107

1.44 µ107

1.46 µ107


Pre

ssur

eHP

aL

10% of radius

10% of radius

63








3.3.3 Scenario 3

The third scenario is built from the first one, only that the reservoir matrix

permeability is equal to 26% of the one from Scenario 1. The permeability in Scenario 3

is that obtained from the geometric average, as shown in Table 3-10 and Figure 3-18.

The lens and the near well bore reservoir properties are kept equal to the first scenario,

in order to make of it another sensibility analysis case – this time, the Lorenz Plot lens

permeability was kept, but the outer reservoir permeability was reduced.



Porosity 18.8% 10% 14%

Permeability 533 mD 10 mD 26 mD (geometric)

64



The pressure across the reservoir and at the bottom of the well takes much longer

to reach the minimum well bore pressure. The model was run, until a maximum

simulation time of three years, and yet the minimum wellbore pressure was not reached

at the bottom of the well, nor at the reservoir. A plausible explanation is that due to the

great difference in permeability between the outer reservoir and the lens, the pressure

doesn’t have to drop as much to keep the fluids flowing at a constant total rate. This

permeability contrast causes the fluid to flow through the rock into the wellbore more

easily, so the pressure, for a constant total flow rate at the well head, shows a slower

pressure drop, as in comparison to Scenario 1. Later, the effects of this great difference

between permeabilities will be discussed.

The pressure distribution by the end of the simulation of the third scenario is

depicted in Figure 3-19.

65

Figure 3-19 – Scenario 3: Pressure distribution in the radial reservoir at the time of three years: the closer to the well bore, the smaller the pressure values. Note that the pressure range in this plot has higher values than the pressure ranges in the Scenarios 1 and 2,

shown in Figure 3-11 and Figure 3-15.

The oil and water flow rates were measured, with time. The total flow rate at the

well head was kept constant to 150m³/day.

The pressure behaviour on the reservoir radius is plotted in two different depths:

at the depth of the high permeable lens and at the bottom of the reservoir. These graphs

are shown in Figure 3-20 and Figure 3-21 and the same observation regarding the slope

of the curves before and after the lens extent limit can be made.

66


middle.



were achieved:

0 200 400 600 800

1.35 µ107

1.40 µ107

1.45 µ107

1.50 µ107

1.55 µ107


Pre

ssur

eHP

aL

0 200 400 600 800

1.35 µ107

1.40 µ107

1.45 µ107

1.50 µ107

1.55 µ107


Pre

ssur

eHP

aL

10% of radius

10% of radius

67








3.3.4 Scenario 4

The fourth scenario is built from the second one, only that the reservoir matrix

permeability is equal to that obtained from the geometric average. The lens permeability

if that from the Lorenz Plot multiplied by two. When comparing Scenario 4 to Scenario

3, the outer reservoir permeability decreases from 26% to 2,4% of the permeability of

the lens. This represents a very high contrast, and, as the other Scenarios presented

earlier, can be found in any carbonate reservoir. Figure 3-22 and Table 3-12 represent

the fourth scenario.



Porosity 18.8% 10% 14%

Permeability 1066 mD 10 mD 26 mD (geometric)

68



Because of the high permeability contrast between the lens and the outer

reservoir, the pressure drop along the reservoir, at the end of three simulated years, is

even smaller than in any of the Scenarios 1, 2 and 3. This can be seen in the pressure

versus radius and height plot, at the end of the simulation, in Figure 3-23.

Figure 3-23 – Scenario 4: Pressure distribution in the radial reservoir at the time of three years: the closer to the well bore, the smaller the pressure values. Note that the pressure range in this plot has higher values than the pressure ranges in the Scenarios 1, 2 and 3,

shown in Figure 3-11, Figure 3-15 and Figure 3-19.

69

The oil and water flow rates were measured, with time. The total flow rate at the

well head was kept constant at 150m³/day.

The pressure behaviour on the reservoir radius is plotted in two different depths:

at the depth of the high permeability lens and at the bottom of the reservoir. These

graphs are shown in Figure 3-24 and Figure 3-25 and the same observation regarding

the slope of the curves before and after the lens extent limit can be made.


middle.

0 200 400 600 800

1.62 µ107

1.64 µ107

1.66 µ107

1.68 µ107

1.70 µ107

1.72 µ107

1.74 µ107


Pre

ssur

eHP

aL

10% of radius

70



were achieved:

Table 3-13 – Scenario 4: Initial and Final values (at 3 representative years) simulated with the Finite Volume Method, in Wolfram Mathematica.







0 200 400 600 800

1.62 µ107

1.64 µ107

1.66 µ107

1.68 µ107

1.70 µ107

1.72 µ107

1.74 µ107


Pre

ssur

eHP

aL

10% of radius

71

3.3.5 Scenarios 1 to 4 and Homogeneous Case of 26mD: Mathematica

(FVM) Simulation Pressure Diffusivity Graphical Results

Here, the Scenarios 1 to 4 and the homogeneous case of 26mD are shown

graphically. Snapshots are printed while the Wolfram Mathematica FVM code is run for

each case.

72

Figure 3-26 – Wolfram Mathematica Code Simulation (FVM): Pressure Diffusivity in Image Shots for five different cases. Pressure diffusion starts mainly by the lens. The

well is localized at the left of each of the twenty snap shots, but is not represented in the images.

73

Extra tests were run with the Wolfram Mathematica code for the first scenario.

The extra tests consist of:

- Refining the mesh originally divided in 50 radial and 11 axial divisions, to 100

radial and 22 axial divisions;

- Not considering the wellbore storage effect.

These tests can be found in the Appendix, in Section 7.3.

3.4 The Schlumberger Eclipse 100 Simulation The Schlumberger Eclipse 100 software simulates Black-Oil reservoir models. It

is a production simulator, therefore, doesn’t consider early time effects, such as the

wellbore effects.

The reservoir is simulated through the use of the Finite Difference Method on the

oil and water diffusivity equations. A three-dimensional model was built for each

scenario presented above.

Unlike the model developed in this work, the boundaries are not of the no-flow

type. Eclipse doesn’t allow null permeability values; therefore a low permeability value

of 0.01mD was entered in the contour cells, outside the volume of interest. An aquifer

was entered at the depth of 1500m. The bottom of the well was at the depth of 1350m.

The total rate at the well head was input as the oil total rate, equal to 15m³/day.

It was observed that, for a well test simulation, not every value of flow rate was

accepted by the software. This can be seen as a limitation of the software to simulate a

draw down test. For example, at higher values, the rate can’t be kept constant, and drops

as production enhances.

Else, the Eclipse 100 Black Oil simulator considers hydrostatic difference in the

initial reservoir pressure, meaning that the static reservoir pressure varies with depth.

As a qualitative benchmark, the software shows the diffusion of pressure across

the reservoir, as it is depleted. This is shown below, where the three-dimensional

reservoir was built with two angular (theta) cells, each representing 180 degrees. To see

the inner distribution of pressure across the reservoir radius and height, only one radial

cell is shown, so its front view (axial and radial coordinates) can be captured in images.

Two homogeneous reservoir cases were generated in Eclipse. The properties used

are those of the outer reservoir matrix of Scenario 1 and Scenario 3. This last study was

74

done to show that the Eclipse simulation works properly. The images presented below

were extracted from the 26mD Homogeneous Case.

Figure 3-27 – Eclipse 100 Simulation: Pressure Diffusivity Graphical Results. Pressure diffusion starts mainly by the lens. The software considers hydrostatic pressure.

Pressure range in kgf/cm². Pressure range goes from 182.24Kgf/cm² in dark blue to 204.78Kgf/cm² in vibrant red.

The Eclipse 100 code developed for Scenario 1 can be found in the Appendix,

Section 7.4.

3.5 Remarks on the Finite Volume Method Code Developed in this work

Figure 7-3, Figure 7-7, Figure 7-11 and Figure 7-15 confirm the behavior

expected from a radial reservoir with sealed external boundaries, producing at constant

75

rate. Those are the typical type curve response from a draw down well test analysis

made in a reservoir like the one presented in this work.

The higher the permeability contrast between the lens that intersect the wellbore

and the outer reservoir, the longer the pressure takes to fall down and achieve the

minimum well bore pressure. This is due to the easiness of the fluids to flow through the

high permeable lens. The high permeability also increases the fluids trasmissivity: the

higher the transmissivity, the less the pressure at the well bore has to drop and therefore

reservoir pressure has to drop less to keep the total flow rate at the well head constant.

From the graphs plotted in Figure 3-12, Figure 3-13, Figure 3-16, Figure 3-17,

Figure 3-20, Figure 3-21, Figure 3-24 and Figure 3-25, it is possible to see a difference

in the pressure curve behaviour before and after the end of the radial lens limit,

equivalent to 10% of the total reservoir limit. The graph is more slanted within the lens

limit, and less slanted after the lens limit, which would be expected even for a

homogeneous and isotropic reservoir. The difference here is that slope changes with

different rates before and after the lens limit. Also, it is possible to see that the

difference in pressure behaviour is more evident when measured at the bottom of the

well, than at another depth, such as in the mid depth.

The Schlumberger Eclipse 100 simulator uses the finite difference method (FDM)

to approach the hydraulic diffusivity equation in a porous medium. Unlike the finite

volume method (or the control volume method), the FDM needs boundaries of regular

geometry. Being a software designed for production analysis, it doesn´t consider the

wellbore storage in the early time period of test (right after opening the well period),

characterising it a good benchmark for the well test type curves for middle and late

times. At early times, the Eclipse 100 simulator may report numerical issues, such as

numerical wellbore storage. Nonetheless, the results presented both from Eclipse 100

and the code developed in this work validate each other, proving that the finite volume

method can be expanded to more general cases, with non-regular boundaries in a

heterogeneous and anisotropic medium.

In other words, both simulations (Eclipse and the one developed for this work in

Wolfram Mathematica) have similar middle and late time results and show that,

initially, the pressure diffusivity is due especially to the high permeable lens, until the

reservoir starts to respond with a tendency to its relative homogeneous case. This can be

better seen when the pressure and its derivative are plotted against time, which will be

presented in the next chapter.

76

4. Well Test Analysis: Data

Interpretation and the Partial

Perforation Hump Effect

4.1 The Well Test Analysis After the scenarios were developed and run in the Wolfram Mathematica and

Eclipse 100 software programmes, as presented in the third chapter, the pressure at the

bottom of the well bore and the constant well head fluid rate with time were entered to

The Fekete F.A.S.T. Well Test software, along the reservoir properties. The pressure

and its derivative curves against time were plotted through Fekete and transferred to the

Excel format. These data are presented in Figure 4-1.

77

Figure 4-1 – Pressure (ΔP) and Derivative Pressure (P’) versus time (Δt) for all simulated Scenarios. The pressure and time data were obtained from the Wolfram

Mathematica code developed in this work. The lower set of curves refers to the pressure derivative plots; the upper set of curves refers to the pressure drawdown plots.

4.2 Scenarios Comparison Before the well test analysis is done, a short comparison between the scenarios is

made here, from their known parameters and curves presented on Figure 4-1.

From the Figure 4-1 Log-Log Plot, it is possible to see that Scenarios 1 (green)

and 2 (blue) are almost totally superposed by, respectively, Scenarios 3 (red) and 4

(orange) during the early and middle time. According to their permeability properties,

it’s possible to divide them in two different groups:

78

Table 4-1 – Scenarios Groups: Scenarios are grouped together if they have the same matrix reservoir properties. K stands for permeability.

Group Scenarios Unchanged Properties Changed

Properties

Group 1 Scenario1 Kmatrix,1 = Kmatrix,2

Knear wellbore,1 = Knear wellbore,2

K lens,1 = 0.5 Klens,2 Scenario 2

Group 2 Scenario 3 Kmatrix,3 = Kmatrix,4

Knear wellbore,4 = Knear wellbore,4 K lens,3 = 0.5 Klens,4

Scenario 4

4.2.1 Group 1 Analysis

Scenarios 1 and 2 (green and blue) differ continuously from each other by an

approximately constant offset (or gap). The pressure difference between Scenarios 1 and

2 has an average value equal to 1494.81 KPa.

Notice that the pressure and its derivative follow, for both scenarios, almost the

very same drawing, separated by an offset.

Since the first scenario has a lower permeability value near the well bore than the

second one, its curves are positioned at a higher position on the plot in the early time,

meaning that, to maintain the constant flow rate at the well head, it is necessary to drop

the pressure at the bottom of the well bore to lower values than in the second scenario.

This means that, the higher the permeability of the lens, the less the pressure needs to

drop to maintain the constant total flow rate at surface.

4.2.2 Group 2 Analysis

Scenarios 3 and 4 (red and orange) differ continuously from each other by an

approximately constant offset. The pressure difference between Scenarios 1 and 2 has

an average value equal to 1341.57 KPa.

Notice that the pressure and its derivative follow, for both scenarios, almost the

very same curve, but with different values separated by this offset.

Since the third scenario has a lower permeability value than the fourth one near

the well bore, its curves are positioned at a higher position on the early time plot,

meaning that, to maintain the constant flow rate at the well head, it is necessary to drop

the pressure at the bottom of the well bore to lower values than in the fourth scenario.

79

This means that, the highest the permeability of the lens, the less the pressure needs to

drop to maintain the constant total flow rate at surface.

4.2.3 Groups 1 and 2 Comparison

Comparing Group 1 and Group 2, it’s easy to see that due to the lower matrix

permeability value of Group 2, obtained from the geometric average (26 mD), the

pressure has to have a slightly higher drop than Group 1, which has the matrix

permeability obtained from the arithmetic average (100mD). This is more evident on the

late time period of test, when the pressure diffusion “reaches” the outer matrix reservoir.

4.3 The Well Test Curves Interpretation

4.3.1 The Partial Perforation Hump Effect

The geoskin appears on the derivative pressure curves as an early time

phenomenon, when the pressure response is related to the near wellbore patches

connected to the well. The geoskin is characterised by the 1/2 slope at the early time. A

tendency to a small geoskin effect, or skin due to geological properties, can be seen in

the derivative plots. The observed skin has a positive value, as shown in the zoomed

plot in Figure 4-2.

80


Mathematica code developed in this work. The lower set of curves refers to the pressure derivative plots; the upper set of curves refers to the pressure drawdown plots. In the

zoomed box: the early time well bore effect.

Three extra tests were made, with different values of the near well bore

permeability, maintaining the lens and outer reservoir permeabilities, to verify the

existence of the positive skin shown in the derivative plots of Scenarios 1, 2, 3 and 4.

The first test conducted was the Enhanced Pseudo-Fracture Channel Skin Test

model, as follows:

Table 4-2 – Enhanced Pseudo-Fracture Channel Skin Test: Porosity and Permeability values


Porosity 18.8% 10% 14%

Permeability 533 mD 533 mD; 10 mD nearby

the boundaries

100 mD (arithmetic)

81

The second test conducted was the Pseudo-Fracture Channel Skin Test, as

follows:

Table 4-3 – Pseudo-Fracture Channel Skin Test: Porosity and Permeability values


Porosity 18.8% 10% 14%


And, finally, the third test was the Partial Perforation Skin Test, as follows:

Table 4-4 – Partial Perforation Skin Test: Porosity and Permeability values


Porosity 18.8% 10% 14%


As expected, the Pseudo-Fracture Channel (PFC) Skin Test returned a flat curve

during the middle time. Enhanced Pseudo-Fracture Channel Skin (EPFC) Test returned

a negative skin value on the derivative plot (the EPFC curve was below the PFC curve).

The Partial Perforation (PP) Skin Test returned a positive skin value on the derivative

plot (the PP curve was above the PFC curve at the early time). This can be seen in

Figure 4-3.

82

Figure 4-3 – Enhanced Pseudo-Fracture Channel Skin Test, Pseudo-Fracture Channel Skin Test and Partial Perforation Skin Test: Pressure and Pressure Derivative versus

time.

This proves that the rapid positive skin shown in Figure 4-2 is provoked by the

lower permeability value of the matrix portion that corresponds to 10% of the radius,

and “contours” almost the whole length of the well bore. The more enhanced the

permeability, the lower the curve in the graph. The positive skin indicates that the

lower permeability portion of matrix near the well bore doesn`t positively contribute

much initially to flow, and is, from now on, named the Partial Perforation Hump Effect

in this work. The negative skin comes right after, and can be seen at the hump effect, or

geochoke.

As a matter of demonstration on how the pressure diffuses, the Partial Perforation

Skin Test (green curve in Figure 4-3) is shown below, so the Partial Perforation Hump

Effect can be seen in the reservoir:

83

Figure 4-4– Partial Perforation Skin Test: Pressure distribution at 0.1 hour of simulation. Pressure range: from 204 kgf/cm² (red) to 203 kgf/cm² (blue).

Figure 4-5– Partial Perforation Skin Test: Pressure distribution at 1 hour of simulation. Pressure range: from 204 kgf/cm² (red) to 201 kgf/cm² (blue).

Figure 4-6– Partial Perforation Skin Test: Pressure distribution at 10 hours of simulation. Pressure range: from 204 kgf/cm² (red) to 200 kgf/cm² (blue).

Figure 4-7– Partial Perforation Skin Test: Pressure distribution at 100 hours of simulation. Pressure range: from 204 kgf/cm² (red) to 199 kgf/cm² (blue).

Figure 4-8– Partial Perforation Skin Test: Pressure distribution at the end of the simulation. Pressure range: from 168 kgf/cm² (red) to 162 kgf/cm² (blue).

From these images of the Partial Perforation Skin Test, it’s possible to see that the

pressure drop happens preferentially around where the high permeable lens is, as if the

reservoir were partially perforated. This emphasizes the explaination why a positive

skin shows up at the early time on the derivative plot when a low permeable matrix

surrounds the lens on its bottom and top.

204 203

204 201

204 200

204 199

168 162

84

4.3.2 Geochoke

The geochoke happens due to an effective restriction (choke) in the near well bore

region, caused by ineffective communication between geological bodies [17]: channels

intersecting the well, in this case the high permeable lens, are quickly depleted, and

recharge takes time to be effective. The lens recharge from the low permeable

geological body is slow, resulting in a period of restricted flow.

A “hump” on the derivative curve characterises the geochoke [19]. “The hump is

a combination of a negative skin, a short radial flow and later expanding flow

(expantion of the kh product) in sequence.” (HAMDI, 2012) [17] .



zoomed box: the hump effect, or geochoke.

85

4.3.3 Radial Flow

The radial behaviour is characterised by a flat slope and happens during the

middle time. After Δt = 10 hours, the reservoir has a total equivalent radial behaviour

for all cases. This behaviour is a small effect, and can last longer or shorter depending

on the reservoir’s characteristics.



zoomed box: the radial flow behaviour.

4.3.4 Sealed Boundary

When the sealed reservoir has its boundaries achieved by the pressure diffusivity,

its pressure derivative curve at the late time shows a unity slope.

For Scenarios 1 and 2, the boundaries response show earlier (Δt = 400 ~ 500 h)

than for the other ones (Δt ≥ 1000 h).

86



zoomed box: the sealed boundary pressure drop behaviour.

4.4 The Well Test Analysis Comparison Here, the last analysis will be presented: the results obtained for Scenarios 1 to 4

and the homogeneous cases of 100mD and 26mD will have their Well Test curves

plotted, analysed and compared. Differently from what has been done previously, the

results to be confronted will be those from the Schlumberger Eclipse 100, that uses the

Finite Difference Method and doesn`t consider early time phenomena, and those from

the Wolfram Mathematica code developed for this work, which uses the Finite Volume

Method and considers early time phenomena.

The plots in Figure 4-12 and Figure 4-13 show the Eclipse 100 and the

Mathematica pressure and pressure derivative curves:

87

Figure 4-12 – Well Test Analysis for the Finite Volume Method code case, developed in Wolfram Mathematica for this work.

Figure 4-13 – Well Test Analysis for the Eclipse 100 case. The early time phenomena show a possible numerical storage.

88

Since the Schlumberger Eclipse 100 is a production software, meaning that it

doesn’t consider early time phenomena (e.g., storage), the results will be compared both

within the time limits from 0.1h to 200h. The pressure data with time were extracted

from both simulations, and plotted in Fekete F.A.S.T. Well Test, as shown in Figure

4-14 and Figure 4-15.

Figure 4-14 – Well Test curves for the Wolfram Mathematica code, developed with the Finite Volume Method, during the middle time.

89

Figure 4-15 – Well Test curves for the Schlumberger Eclipse 100 simulation, developed with the Finite Volume Method, during the middle time.

From the curves presented above, it`s possible to summarize the comparison

results with a table:

90

Table 4-5 – Scenarios 1 to 4 Well Test Behaviour, Expectations and Results from Eclipse 100 and Mathematica code. This table summarizes the derivative plots. k stands

for permeability.

Curve Characteristic Expectation Mathematica

Code Result

Eclipse 100

Result

Middle Time: 0.1 h to 2.0 h

Scenarios 1, 3

(green, red) Lens: 533mD Superposition Expected

Superposition

achieved

Superposition

achieved

Scenarios 2, 4

(blue, orange) Lens: 1066mD Superposition Expected

Superposition

achieved

Superposition not

achieved

Late Time: 2.0 h to 200.0 h

Scenarios 1, 2

(green, blue) Matrix: 100mD

Tendency to

Superposition Expected.

Curve lowering down

(due to K increase)

Superposition

achieved, Curve

lowering down

achieved

Superposition

achieved, Curve

lowering down

achieved

Scenarios 3, 4

(red, orange) Matrix: 26mD

Tendency to

Superposition Expected.

Curve heightening up

(due to K decrease)

Superposition

achieved, Curve

lowering down

achieved

Superposition

achieved, Curve

heightening up

achieved

Scenario 4 behaves as expected, most of the time, in both results from Eclipse 100

and the Mathematica code. Except that, maybe, Eclipse is not very sensitive to the high

permeable lens, when the outer reservoir matrix and/or the near well bore matrix have a

much lower permeability. This (in Eclipse 100) could be due to a delay in the fluid

provision from the low permeability matrix to the high permeable lens near the

wellbore.

All scenarios, in the late time, tend to its relative homogenous case. Scenarios 1

and 2 have outer matrix permeability equal to 100 mD, and, hence, have a derivative

curve tendency in the late time to overlap the derivative plot of the homogeneous case

of 100 mD (black curve). Scenarios 3 and 4 have outer matrix permeability of 26 mD

and their derivative curves tend to overlap the derivative plot of the homogeneous case

of 26 mD (grey curve).

This proves that both simulations have sensibilities that improve the well test

analysis in a certain way, and can both be used to understand the behaviour and

characteristics of the heterogeneous media. The code developed in Wolfram

Mathematica, though, has the advantages of using the Finite Volume Method, considers

91

well bore storage and can simulate any value of total flow rate at the well head,

characterising it a great tool for future well test studies with very high flow rate for non-

symmetrical boundaries reservoir, for every draw down period.

92

5. Conclusion

For the formations with high permeable lens intersecting the wellbore, with low

permeable matrix, an early hump can be observed in the WTA plots from the

Mathematica codes. The Partial Perforation Hump Effect, or the early hump, indicates a

positive skin. This positive skin is a small effect and might be due to the low

permeability matrix surrounding the wellbore, not the lens: a geochoke might be present

in the early time due to a geological fluid storage in the low permeable matrix. This

geological storage would be similar to a wellbore storage, but within the matrix. The

geochoke, or delay in the fluids supply from the low to the high permeable cluster,

could have as main cause the (apparent) geological storage during the early time in the

low permeable matrix. It is said to be apparent because it is not totally proven by real

tests.

The Partial Perforation Hump Effect has to be further interpreted. Another

possible situation to be analysed is that of having more than one high permeable lens

intersecting the wellbore. With this new approach, it’s possible to confront or not

SAGAWA’s affirmation, that more than one lens intersecting the wellbore behaves the

same way as one equivalent lens, and also would prove or not if the Partial Perforation

Hump Effect is a characteristic of a one lens only-case.

During the middle time, a Hump Effect is seen and characterised by a negative

skin. This is probably due to the high permeability of the lens, as explained by HAMDI.

The Hump Effect, again, characterises a geochoke and is expected when macro cross-

flow is present.

Regarding the code developed using the IMPES Method and the Finite Volume

Method, it can be seen from the WTA graphs that the early time effects are represented

with consistency to the expectations, even though it was tested in a simplified

heterogeneous carbonate model. In the elaborated code, wellbore storage is considered

and any rate of production can be used to simulate a well test. The code developed has

93

shown to fit the expectations for research, as it is reliable for one well bore only-

problems, heterogeneous and anisotropic media and the Finite Volume Method has an

advantage that would be interesting to be tested: a very heterogeneous medium with

irregular geometry intersected by a wellbore is fairly represented by this discretisation

method.

94

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97

7. Appendix

7.1 IMPES Method – Formulae Development for Oil and Water Diffusivity Equation The equations that govern the laminar flow of oil and water in a porous medium,

according to ROSA et al. (2011) [30] , are:

,

ˆo ro

o o o stdo o o

S kP g z q

t B B

φ ρµ

∂ ′′′= ∇ ⋅ ∇ − ∇ + ∂

kɺ

7.1

,

ˆw rw

w w w stdw w w

S kP g z q

t B B

φ ρµ

∂ ′′′= ∇ ⋅ ∇ − ∇ + ∂

kɺ

7.2

1o wS S+ = 7.3

( )cow w o wP S P P= − 7.4

Re-writing the above equations to a radial bi-dimensional coordinate system (r-z)

and despising the hydrostatic term, there equations are:

,

(1 ) 1w H ro o V ro oo std

o o o o o

S k k P k k Pr q

t B r r B r z B z

φµ µ

− ∂ ∂∂ ∂ ∂ ′′′= + + ∂ ∂ ∂ ∂ ∂ ɺ

7.5

,

1w H rw w V rw ww std

w w w w w

S k k P k k Pr q

t B r r B r z B z

φµ µ

∂ ∂∂ ∂ ∂ ′′′= + + ∂ ∂ ∂ ∂ ∂ ɺ

7.6

98

It is necessary to expand the transient term so the oil and water phases pressure

variation with time are explicit. Therefore, the transient terms of the above equations,

for oil and water, become, respectively:

1 2

(1 ) 1 1(1 )

1 1(1 )

(1 )

w ww

o o o o

o ww

o o o o o

o ww

S SS

t B t B B t B t

P SS

B P P B t B t

P SS C C

t t

φ φ φφ

φ φφ

− ∂∂ ∂ ∂= − + − = ∂ ∂ ∂ ∂

∂ ∂∂ ∂= − + − = ∂ ∂ ∂ ∂

∂ ∂= − −

∂ ∂

7.7

3 4

1

1 1

w w ww

w w w w

o w o ww w

w o o w w

S S SS

t B B t B t t B

P S P SS S C C

B P P B t B t t t

φ φ φ φ

φ φφ

∂∂ ∂ ∂= + + = ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂∂ ∂ = + + = + ∂ ∂ ∂ ∂ ∂ ∂

7.8

Where:

1 2

3

4

1 1

1 1

1

o o o o o

w o o w

w

C CB P P B B

CB P P B

CB

φ φφ

φ φ

φ

∂ ∂= + = ∂ ∂

∂ ∂= + ∂ ∂

=

7.9

Hence, the equations to be discretised are:

1 2 ,

1(1 ) o o H ro o V ro o

w o stdo o o o

P S k k P k k PS C C r q

t t r r B r z B zµ µ ∂ ∂ ∂ ∂∂ ∂ ′′′− − = + + ∂ ∂ ∂ ∂ ∂ ∂

ɺ 7.10

3 4 ,

1o w H rw w V rw ww w std

w w w w

P S k k P k k PS C C r q

t t r r B r z B zµ µ ∂ ∂ ∂ ∂∂ ∂ ′′′+ = + + ∂ ∂ ∂ ∂ ∂ ∂

ɺ 7.11

99

7.1.1 Oil phase equation discretisation

The totally implicit finite volume method (FVM) will be used to discretise the

equations:

0

1 2

,

(1 ) 2

12

o

t n eo w

wt s w

t n eH ro o V ro o

o stdt s wo o o o

P SS C C r dr dz dt

t t

k k P k k Pr q r dr dz dt

r r B r z B z

π

πµ µ

∂ ∂ − − = ∂ ∂

∂ ∂∂ ∂ ′′′= + + ∂ ∂ ∂ ∂

∫ ∫ ∫

∫ ∫ ∫ ɺ

7.12

Integrating:

0 01 2

,

(1 ) ( )2 ( )2

2 2

2 2

w C C C wC wC C

H ro o H ro o

e wo o o oe w

V ro o V ro oC C o std

n so o o on s

S C P P r r z C S S r r z

k k P k k Pr z t r z t

B r B r

k k P k k Pr r t r r t q

B z B z

π π

π πµ µ

π πµ µ

− − ∆ ∆ − − ∆ ∆ =

∂ ∂ = ∆ ∆ − ∆ ∆ ∂ ∂

∂ ∂ + ∆ ∆ − ∆ ∆ + ∂ ∂ ɺ t∆

7.13

Dividing by (∆V ∆t = 2π rC ∆r ∆z ∆t) :

0 0

1 2

,

( ) ( )(1 )

1 1

1 1

C C wC wCw

H ro o H ro o

e wC o o C o oe w

o stdV ro o V ro o

n so o o on s

P P S SS C C

t t

k k P k k Pr r

r r B r r r B r

qk k P k k P

z B z z B z V

µ µ

µ µ

− −− − =∆ ∆

∂ ∂ = − ∆ ∂ ∆ ∂

∂ ∂ + − + ∆ ∂ ∆ ∂ ∆

ɺ

7.14

The partial derivatives of the oil phase pressure can be obtained through the

traditional method of the Taylor series expansion:

100

0 0

1 2

,

( ) ( )(1 )

1 1

1 1

C C wC wCw

H ro E C H ro C W

C o o E C o o We w

o stdV ro N C V ro C S

o o N o o Sn s

P P S SS C C

t t

k k P P k k P Pr r

r r B r r r B r

qk k P P k k P P

z B z z B z V

µ µ

µ µ

− −− − =

∆ ∆ − −

= − ∆ ∆ ∆ ∆

− −+ − + ∆ ∆ ∆ ∆ ∆

ɺ

7.15

Where:

H ro H ro

oHe oHwo o o oe w

V ro V rooVn oVs

o o o on s

k k k k

B B

k k k k

B B

λ λµ µ

λ λµ µ

= =

= =

7.16

Therefore:

( ) ( ) ( ) ( )

0 0

1 2

,

( ) ( )(1 ) C C wC wC

w

o stde oHe w oHw oVn oVsE C C W N C C S

C E C W N S

P P S SS C C

t tqr r

P P P P P P P Pr r r r r r z z z z V

λ λ λ λ

− −− − =

∆ ∆

= − − − + − − − +∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

ɺ

7.17

Collecting the similar terms:

1 2

, 0 01 2

(1 )

(1 )

w e oHe w oHw oVn oVsC wC

C E C W N S

o stde oHe w oHw oVn oVs wE W N S C wC

C E C W N S

S C r r CP S

t r r r r r r z z z z t

qr r S C CP P P P P S

r r r r r r z z z z V t t

λ λ λ λ

λ λ λ λ

−+ + + + − = ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

−= + + + + + −

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ɺ

7.18

Or, simply:

2

oC C wC oe E ow W on N os S oC

CP S P P P P B

tτ τ τ τ τ− = + + + +

∆

7.19

Where:

101

1

, 0 01 2

; ; ; ;

(1 )

(1 )

e oHe w oHw oVn oVsoe ow on os

C E C E N S

woC oe ow on os

o std woC C wC

r r

r r r r r r z z z z

S C

tq S C C

B P SV t t

λ λ λ λτ τ τ τ

τ τ τ τ τ

= = = =∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

−= + + + +

∆−

= + −∆ ∆ ∆ɺ

7.20

7.1.2 Water phase equation discretisation

The totally implicit finite volume method (FVM) will be used to discretise the

equations:

0

0

0

3 4

,

2

12

2

t n eo w

wt s w

t n eH rw w V rw w

t s ww w w w

t n e

w stdt s w

P SS C C r dr dz dt

t t

k k P k k Pr r dr dz dt

r r B r z B z

q r dr dz dt

π

πµ µ

π

∂ ∂ + = ∂ ∂

∂ ∂∂ ∂= + + ∂ ∂ ∂ ∂

′′′ +

∫ ∫ ∫

∫ ∫ ∫

∫ ∫ ∫ ɺ

7.21

Integrating:

0 03 4

,

( ) ( )2

2 2

2 2

C C wC wC C

H rw w H rw w

w w w we we w

V rw w V rw wC C

w w w wn sn s

w std

C P P C S S r r z

k k P k k Pr z t r z t

B r B r

k k P k k Pr r t r r t

B z B z

q t

π

π πµ µ

π πµ µ

− + − ∆ ∆

∂ ∂ = ∆ ∆ − ∆ ∆ + ∂ ∂

∂ ∂ + ∆ ∆ − ∆ ∆ + ∂ ∂

′′′+ ∆ɺ

7.22

Dividing by (∆V ∆t = 2π rC ∆r ∆z ∆t):

0 0

3 4

,

( ) ( )

1 1

1 1

C C wC wCwC

H rw w H rw w

C w w C w we we w

V rw w V rw w

w w w wn sn s

w std

P P S SS C C

t t

k k P k k Pr r

r r B r r r B r

k k P k k P

z B z z B z

q

V

µ µ

µ µ

− −+

∆ ∆ ∂ ∂ = − + ∆ ∂ ∆ ∂

∂ ∂ + − + ∆ ∂ ∆ ∂

+∆ɺ

7.23

102

Where:

H rw H rw

wHe wHww w w we w

V rw V rwwVn wVs

w w w wn s

k k k k

B B

k k k k

B B

λ λµ µ

λ λµ µ

= =

= =

7.24

Through the Taylor traditional method of series expansion, the partial derivatives

of the water phase are obtained:

0 0

3 4

,

( ) ( )

( ) ( )

( ) ( )

C C wC wCwC

e wHe w wHwwE wC wC wW

C E C W

wVn wVswN wC wC wS

N S

w std

P P S SS C C

t tr r

P P P Pr r r r r r

P P P Pz z z z

q

V

λ λ

λ λ

− −+

∆ ∆

= − − − +∆ ∆ ∆ ∆

+ − − − +∆ ∆ ∆ ∆

+∆ɺ

7.25

But, reminding the capillary pressure concept, one has:

0( )w o cow wP P P S= − 7.26

Thereunto, expanding the terms:

103

00

3 4

0 0

0 0

( )( )

( ) ( )

( ) ( )

gC gCC CW

e wHe e wHe w wHw w wHwE cow wE C cow wC

C E C E C E C E

w wHw w wHw w wHw w wHwC cow wC W cow wW

C W C W C W C W

wVn

N

S SP PS C C

t tr r r r

P P S P P Sr r r r r r r r r r r r

r r r rP P S P P S

r r r r r r r r r r r r

Pz z

λ λ λ λ

λ λ λ λ

λ

−−+ =

∆ ∆

− − +∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

− + + −∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

+∆ ∆

0 0

0 0

,

( ) ( )

( ) ( )

wVn wVn wVnN cow wN C cow wC

N N N

wVs wVs wVs wVsC cow wC S cow wS

S S S S

w std

P S P P Sz z z z z z

P P S P P Sz z z z z z z z

q

V

λ λ λ

λ λ λ λ

− − +∆ ∆ ∆ ∆ ∆ ∆

− + + −∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

+∆ɺ

7.27

And collecting the similar terms:

3 4

0( )

gVn gVswC e wHe w wHwC wC

C E C W N S

gVn gVse wHe w wHwE W N S

C E C W N S

e wHe w wHwcow wE

C E C W

S C r r CP S

t r r r r r r z z z z t

r rP P P P

r r r r r r z z z z

r rP S

r r r r r r

λ λλ λ

λ λλ λ

λ λ

+ + + + + = ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

= + + + ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

− − ∆ ∆ ∆ ∆

0 0 0

0

, 0 03 4

( ) ( ) ( )

( )

gVn gVscow wW cow wN cow wS

N S

gVn gVse wHe w wHwcow wC

C E C W N S

wstd wCC wC

P S P S P Sz z z z

r rP S

r r r r r r z z z z

q S C CP S

V t t

λ λ

λ λλ λ

− − + ∆ ∆ ∆ ∆

+ + + + + ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

+ + +∆ ∆ ∆ɺ

7.28

Or, simply:

4wC C wC we E ww W wn N ws S wC

CP S P P P P B

tτ τ τ τ τ+ = + + + +

∆

7.29

Where:

104

3

0 0 0

;

;

( ) ( ) ( )

e wHe w wHwwe ww

C E C W

gVn gVswn ws

N S

wCwC we ww wn ws

gVne wHe w wHwwC cow wE cow wW cow wN

C E C W N

gVs

S

r r

r r r r r r

z z z z

S C

t

r rB P S P S P S

r r r r r r z z

z z

λ λτ τ

λ λτ τ

τ τ τ τ τ

λλ λ

λ

= =∆ ∆ ∆ ∆

= =∆ ∆ ∆ ∆

= + + + + ∆

= − − − − ∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

0 0

, 0 03 4

( ) ( ) gVn gVse wHe w wHwcow wS cow wC

C E C W N S

w std wCC wC

r rP S P S

r r r r r r z z z z

q S C CP S

V t t

λ λλ λ + + + + + ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

+ + +∆ ∆ ∆ɺ

7.30

7.1.3 The IMPES method applied to two-phase flow of oil and water

The discretised equation for the oil phase is:

2oC C gC oe E ow W on N os S oC

CP S P P P P B

tτ τ τ τ τ− = + + + +

∆

7.31

And for the water phase is:

4wC C wC we E ww W wn N ws S wC

CP S P P P P B

tτ τ τ τ τ+ = + + + +

∆

7.32

Where:

2 4; o

o o w

BC C

B B B

φ φ = =

7.33

Multiplying the discretised equation for the oil phase by

5

o

w

BC

B

=

7.34

And adding it to the water phase discretised equation, the result is:

105

5

5 5 5 5 5

( )

( ) ( ) ( ) ( )oC wC C

oe we E ow ww W on wn N os ws S oC wC

C P

C P C P C P C P C B B

τ ττ τ τ τ τ τ τ τ

+ =+ + + + + + + + +

7.35

Which, simplified, becomes:

C C e E w W n N s S CA P A P A P A P A P B= + + + + 7.36

Where:

5

5 5

5 5

5

( )

( ); ( )

( ); ( )

( )

C oC gC

e oe ge w ow gw

n on gn s os gs

C oC gC

A C

A C A C

A C A C

B C B B

τ ττ τ τ ττ τ τ τ

= +

= + = +

= + = +

= +

7.37

From the multiplication between the inverse of the coefficients matrix A and

matrix B, the pressure matrix P can be obtained. Matrix P is the new matrix of pressure

distribution along the reservoir.

Now, from either water or oil equation, it is possible to calculate the new water

and oil saturations. Therefore:

2

( )wC oC C oe E ow wf on N os S oC

tS P P P P P B

Cτ τ τ τ τ∆= − − − − −

7.38

106

7.2 Auxiliary Scenarios Graphs

7.2.1 Scenario 1

The first graph represents the oil flow rate variation, at standard conditions, with

time.

Figure 7-1 – Scenario 1: Oil flow rate variation, at standard conditions, with time.

The second graph shows the water flow rate variation, at standard conditions, with

time. Note that, as the oil flow rate decreases, the water flow rate increases, as the

reservoir is produced through the well.

0.01 1 100

149.86

149.87

149.88

149.89

149.90

149.91

Time Hdays L

Flo

wra

teHm

3 êdL

107

Figure 7-2 – Scenario 1: Water flow rate variation, at standard conditions, with time.

As a draw down simulation test, the total flow rate at the well head must be

maintained constant, as long as the sealed reservoir is under production. Therefore, to

keep the flow rate, the pressure at the bottom of the well must drop, as can be seen in

Figure 7-3 and Figure 7-4.

0.01 1 1000.09

0.10

0.11

0.12

0.13

0.14

Time Hdays L

Flo

wra

teHm

3 êdL

108

Figure 7-3 – Scenario 1: Pressure draw down at the bottom of the well, versus elapsed time: due to the constant flow rate at the surface, and lack of external inflow, the

pressure at the bottom of the well falls in order to keep the production at a constant rate. BHP stands for Bottom Hole Pressure.

Figure 7-4 – Scenario 1: ΔP versus elapsed time: the graph confirms the one fromFigure 7-3, showing that the pressure at the bottom of well bore decreases with time. ΔP =

(Initial pressure) – (Pressure at time).

0.01 1 100

1.0µ107

1.2µ107

1.4µ107

1.6µ107

1.8µ107

Time Hdays L

BH

PHP

aL

0.01 1 100

1.0µ107

5.0µ106

2.0µ106

3.0µ106

1.5µ106

7.0µ106

Elapsed Time Hdays L

DPHP

aL

109

7.2.2 Scenario 2

The water and oil flow rates at standard conditions are presented below.



The pressure drop at the bottom of the well was also measured, and presented in

Figure 7-7 and Figure 7-8.

0.01 1 100

149.88

149.90

149.92

149.94

Time Hdays L

Flo

wra

teHm

3 êdL

0.01 1 100

0.06

0.08

0.10

0.12

Time Hdays L

Flo

wra

teHm

3 êdL

110



Figure 7-8 – Scenario 2: ΔP versus elapsed time: the graph confirms the one from Figure 7-7, showing that the pressure at the bottom of well bore decreases with time. ΔP

= (Initial pressure) – (Pressure at time).

0.01 1 100

1.4µ107

1.5µ107

1.6µ107

1.7µ107

1.8µ107

1.9µ107

Time Hdays L

BH

PHP

aL

0.01 1 100

1.0µ106

5.0µ106

2.0µ106

3.0µ106

1.5µ106

7.0µ106


DPHP

aL

111

7.2.3 Scenario 3

The flow rates are presented in Figure 7-9 and Figure 7-10:



The pressure at the bottom of the wellbore with time for the third scenario is

plotted in Figure 7-11 and its pressure difference in Figure 7-12:

0.01 1 100

149.86

149.87

149.88

149.89

149.90

149.91

Time Hdays L

Flo

wra

teHm

3 êdL

0.01 1 1000.09

0.10

0.11

0.12

0.13

0.14

Time Hdays L

Flo

wra

teHm

3 êdL

112



Figure 7-12 – Scenario 3: ΔP versus elapsed time: the graph confirms the one from Figure 7-11, showing that the pressure at the bottom of well bore decreases with time.

ΔP = (Initial pressure) – (Pressure at time).

0.01 1 100

1.3µ107

1.4µ107

1.5µ107

1.6µ107

1.7µ107

1.8µ107

Time Hdays L

BH

PHP

aL

0.01 1 100

5.0µ106

2.0µ106

3.0µ106

1.5µ106

7.0µ106


DPHP

aL

113

7.2.4 Scenario 4

. The flow rates are presented in Figure 7-13 and Figure 7-14:



0.01 1 100

149.88

149.90

149.92

149.94

Time Hdays L

Flo

wra

teHm

3 êdL

0.01 1 100

0.06

0.08

0.10

0.12

Time Hdays L

Flo

wra

teHm

3 êdL

114

The pressure at the bottom of the wellbore with time for the fourth scenario is

plotted in Figure 7-15 and its pressure difference in Figure 7-16:



Figure 7-16 – Scenario 4: ΔP versus elapsed time: the graph confirms the one from Figure 7-15, showing that the pressure at the bottom of well bore decreases with time.

ΔP = (Initial pressure) – (Pressure at time).

0.01 1 100

1.60 µ107

1.65 µ107

1.70 µ107

1.75 µ107

1.80 µ107

1.85 µ107

1.90 µ107

Time Hdays L

BH

PHP

aL

0.01 1 100

1.0µ106

2.0µ106

3.0µ106

1.5µ106


DPHP

aL

115

7.3 Extra tests made from Scenario 1 Here, the extra tests with refined mesh (100x22) and no consideration to the

wellbore storage are shown for Scenario 1, in comparison to its original test, presented

in this work (50x11 mesh, and consideration of wellbore storage). This can be seen in

Figure 7-17.

116

Figure 7-17 – Scenario 1 variations: refined mesh case and no wellbore storage

case.

117

Graphically, Scenario 1 with and without consideration to the wellbore storage

effect has the same results. This confirms that the wellbore storage is indeed a small

effect. From the Scenario 1 with refined mesh, it is possible to see that the pressure drop

is even higher when the volume is smaller and approximates the size of the well radius

(Scenario 1 was re-meshed: originally it consists of 50 cells in the radial direction and

11 cells in the axial direction, while its refined mesh version consists of 100 cells in the

radial and 22 cell is the axial direction).

For the case with no wellbore storage effects, a comparison graph was generated,

as it can be seen in Figure 7-18.

Figure 7-18 – Pressure draw down evolution for the Wolfram Mathematica FVM code. The cases considered here are of no wellbore storage and consideration to wellbore

storage effect, both run for Scenario 1.

118

7.4 Schlumberger Eclipse 100 Code for Scenario 1

RUNSPEC

TITLE

Scenario 1 - MSc Tatiana Lipovetsky - PEC/COPPE/UFRJ

DIMENS

--NR NT NZ

51 2 13/

RADIAL

NONNC

-- command that doesn't allow flow between non-adjacent cells

WATER

OIL

METRIC

EQLDIMS

-- equilibium zones dimension - depth in P table - max depth in fluid table

1 100 120 1 1 /

TABDIMS

--specifies input pvt and saturation function table dimensions

1 1 20 20 1 30 /

WELLDIMS

--specifies the number of wells and groups in the model

-- max number of wells;

--max connections per well (layers);

-- The maximum number of groups in the model;

119

1 11 1* 1 /

START

1 'JAN' 2013 /

NSTACK

-- Linear Solver stack size

25 /

UNIFOUT

-- Indicates that outpout files are unified

UNIFIN

-- Indicates that input files are unified

GRID

==============================================================

INIT

GRIDFILE

2 0 /

BOUNDARY

-- area of grid to be printed

--I1 I2 T1 T2 K1 K2

1 50 1 2 2 12 /

120

-- COMPLETE THE CIRCLE IN THETA-DIRECTION

-------- K1 K2 completed

COORDSYS

1 13 COMP /

OLDTRAN

-- specifies block center transmissibilities

-- RADIAL GRID DEFINED USING INRAD AND OUTRAD

DRV

5.31937 7.43405 8.77609 9.77231 10.5693 11.2357 11.8097 12.3145 12.7656

13.1737 13.5466 13.8899 14.2084 14.5053 14.7836 15.0455 15.2929

15.5274

15.7503 15.9627 16.1657 16.36 16.5463 16.7255 16.8979 17.0641 17.2245

17.3796 17.5296 17.675 17.816 17.9529 18.0859 18.2152 18.3411 18.4637

18.5832 18.6998 18.8136 18.9247 19.0333 19.1395 19.2433 19.345 19.4445

19.542 19.6376 19.7313 19.8232 19.9135 20 /

INRAD

-- Inner radius

0.0889 /

DTHETAV

-- tamanho das células na direção theta

2*180.000

/

DZ

-- tamanho das células na direção Z

1326*4.54545

121

/

BOX

-- near wellbore upper part

-- I1 I2 T1 T2 K1 K2

1 9 1 2 2 6 /

PERMR

--mD

-- permeability values in the radial direction

90*10/

PERMTHT

-- permeabity values in the azimuthal direction

90*10/

PERMZ

-- specifies z-permeability values

90*0.1/

PORO

-- grid block porosity values

90*0.1/

ENDBOX

BOX

-- near wellbore lower part

-- I1 I2 T1 T2 K1 K2

1 9 1 2 8 12 /

PERMR


122

90*10/

PERMTHT


90*10/

PERMZ


90*0.1/

PORO


90*0.1/

ENDBOX

BOX

-- lens

-- I1 I2 T1 T2 K1 K2

1 9 1 2 7 7 /

PERMR


18*533/

PERMTHT


18*533/

PERMZ


18*53.3/

123

PORO


18*0.188/

ENDBOX

BOX

-- outer reservoir

-- I1 I2 T1 T2 K1 K2

10 50 1 2 2 12 /

PERMR


902*100/

PERMTHT


902*100/

PERMZ


902*10/

PORO


902*0.14/

ENDBOX

BOX

-- column 51

-- zero perm values == 0.01mD

124

-- I1 I2 T1 T2 K1 K2

51 51 1 2 1 13 /

PERMR


26*0.01/

PERMTHT


26*0.01/

PERMZ


26*0.001/

PORO


26*0.14/

ENDBOX

BOX

-- line 1


-- I1 I2 T1 T2 K1 K2

1 51 1 2 1 1 /

PERMR


102*0.01/

PERMTHT


125

102*0.01/

PERMZ


102*0.001/

PORO


102*0.14/

ENDBOX

BOX

-- line 13


-- I1 I2 T1 T2 K1 K2

1 51 1 2 13 13 /

PERMR


102*0.01/

PERMTHT


102*0.01/

PERMZ


102*0.001/

PORO


102*0.14/

126

ENDBOX

BOX

1 51 1 2 1 13 /

DZNET

-- net thickness value for each grid block

1326*1 /

TOPS

-- depths of top face of each grid block

102*1405/

ENDBOX

RPTGRID

DR DTHETA DZ /

EDIT

==============================================================

--ENDBOX

PROPS

==============================================================

--STONE2

127

-- só deve ser usado em rodadas trifásicas

SWOF

--Sw Krw Krow Pcow(bars)

0. 0 1 0

0.1 0 1 0

0.2 0.000203542 0.833706 3.97726

0.3 0.00549562 0.558518 0.953512

0.4 0.0254427 0.35172 0.49082

0.5 0.0698148 0.203542 0.316925

0.6 0.148382 0.104213 0.228596

0.7 0.270914 0.043965 0.176106

0.8 0.447181 0.0130267 0.141729

0.9 0.686953 0.00162833 0.11767

1. 1. 0. 0.10000

/

PVTW

--Pref Bw Cw uw Cwu

--barsa rm3/sm3 1/bars cP 1/bar

200 1.103 0.00004 1 0 /

ROCK

--Pref Cf

--barsa 1/bars

200 0.000044 /

DENSITY

-- oil water gas

--kg/m3

875 1000 732.7 /

128

PVDO

--used to specefy properties of oil above the Pb (undersaturated)

-- Pressure Bo uo

--bars rm3/sm3 cP

200 1.103 0003.48112

190.051 1.10465 0003.44962

180.101 1.10629 0003.41811

170.152 1.10794 0003.3866

160.203 1.10958 0003.3551

150.253 1.11123 0003.32359

140.304 1.11288 0003.29208

130.355 1.11452 0003.26058

120.405 1.11617 0003.22907

110.456 1.11782 0003.19757

100.507 1.11946 0003.16606

90.5573 1.12111 0003.13455

80.608 1.12275 0003.10305

70.6586 1.1244 0003.07154

60.7093 1.12605 0003.04003

50.7599 1.12769 0003.00853

40.8106 1.12934 0002.97702

30.8613 1.13098 0002.94551

20.9119 1.13263 0002.91401

10.9626 1.13428 0002.8825

1.01325 1.13592 0002.851

/

RPTPROPS

-- PROPS Reporting Options

--

/

129

SOLUTION

=============================================================

EQUIL

--datum; presdatum; contato; cap.pres no contato; contato g/o;

--type of initialization for Solubility; idem for wet gas;

--accuracy in volume calculations

--m barsa m barsa m m 0 0 0

1345.91 200 0 0 0 0 0 0 0 /

RPTSOL

'PRES' 'SOIL' 'SWAT' 'FIP=3' 'EQUIL' /

-- Initialisation Print Output

--

RESTART=2 SWAT FIP=3 FIPRESV EQUIL /

--RPTSOL

-- 'PRES' 'RV' 'RESTART=2' 'EQUIL' /

--is equivalent to

-- 1 4* 1 2 1* 1 /

SUMMARY

==============================================================

FOPR

FWPR

FWCT

FPR

130

FOPT

FWPT

TIMESTEP

PERFORMA

SCHEDULE

=============================================================

-------- THE SCHEDULE SECTION DEFINES THE OPERATIONS TO BE

SIMULATED

------------------------------------------------------------------------

-- CONTROLS ON OUTPUT AT EACH REPORT TIME

RPTSCHED

'PRES' 'FIP=3' 'WELLS=2' 'SWAT' 'SUMMARY=3' 'CPU=2'

'WELSPECS' 'NEWTON=2' 'RESTART=2' /

--DRSDT

--1.0 /

-- WELL SPECIFICATION DATA

--

-- WELL GROUP LOCATION BHP PI

-- NAME NAME I J DEPTH DEFN

WELSPECS

WELL1 CLUSTER1 1 1 1355 OIL /

/

131

-- COMPLETION SPECIFICATION DATA

--

-- WELL -LOCATION- OPEN/ SAT CONN WELL

-- NAME I J K1 K2 SHUT TAB FACT RAD

COMPDAT

WELL1 1 1 2 12 OPEN 0 1* 0.1788 /

/

-- PRODUCTION WELL CONTROLS - OIL RATE IS SET TO 1000 BPD

--

-- WELL OPEN/ CNTL OIL WATER GAS LIQU RES BHP

-- NAME SHUT MODE RATE RATE RATE RATE RATE

-- sm3/day barsa

WCONPROD

WELL1 OPEN ORAT 150 3* 8E6 /

/

-- SPECIFY UPPER LIMIT OF 1 DAY FOR NEXT TIME STEP

TUNING

-- days

0.000546 0.1 /

/

/

-- SPECIFY REPORT AT 1 DAY

TSTEP

1/

132

-- RESET OUTPUT CONTROLS TO GET FULL OUTPUT FOR LAST

REPORT

--RPTSCHED

-- PRES SWAT FIP=2 WELLS=5 VFPPROD=1 SUMMARY=2

CPU=2 WELSPECS NEWTON=2 /

--TIME

-- /

END

133

8. Annexes

8.1 Flow regimes By opening the well, pressure diffuses away from the wellbore deep into the

formation, bringing information about properties and characteristics of the reservoir [1].

Since well test analyses are made by assessing the pressure behaviour through time, it is

important to recognize and classify the different flow behaviours.

The differential equation of flow, as a function of the state equations (equations

that represent the rock and fluid compressibilities), is given by Equation 8.1, in three-

dimensional Cartesian coordinates.

ªª¯ RtÀμ 1> ªÁª¯U + ªªk ®tÂμ 1> ªÁªk³ + ªªc Rt¦μ 1> ªÁªcU = ªª: (�Á) 8.1

Where:

- c is the fluid compressibility;

- Á is the fluid density;

- � is the rock porosity;

- μ is the fluid viscosity;

- tÀ, tÂ, t¦ are the permeabilities in x, y, z directions; and

- : is time.

For simplification, �, μ, ><, t can be considered constant. The differential equation

of flow, in terms of pressure, becomes Equation 8.2, known as the hydraulic diffusivity

equation [30].

ªY5ª¯Y + ªY5ªkY + ªY5ªcY = �μ><t ª5ª: 8.2

Where

- P is the fluid pressure;

134

- >< = ∑ >/G/ + cf , which is the total compressibility (ci is fluid compressibility,

Si is fluid saturation, cf is formation compressibility) ; and

- �£ÃF = Ä, which is the hydraulic diffusivity constant.

In cylindrical coordinates, Equation 8.2 becomes:

1C ªªC RC ª5ªCU = �μ><t ª5ª: 8.3

8.1.1 Steady State

In the steady state flow regime, the pressure does not change with time [30]. This

happens when, for example, a gas cap or some types of water drive cause a constant

pressure effect, so there is pressure maintenance in the producing formation. Therefore,

ª5ª: = 0 8.4

8.1.1.1 Pseudo Steady State

The pseudo steady regime is characterized by a closed reservoir [30]. The

production rate is constant, meaning that the drop of pressure becomes constant for each

unit of time:

ª5ª: = >@bi:gb: 8.5

8.1.2 Transient State

The transient responses are observed where there is a transition from one state to

another, meaning that they are observed before constant pressure or boundary effects

are reached [30]. The pressure variation with time is a function of the well geometry and

the reservoir properties:

ª5ª: = d(¯, k, c, :) 8.6

135

8.1.3 Closed reservoir: pseudo steady state regime

After a period of time producing the well, when all reservoir boundaries have

been reached by its fluids transient pressures, the flow regime changes to pseudo steady

state, which means that the rate of pressure decline at any point of the reservoir is

constant and proportional to time [30]:

ª5ª: = >@bi:gb: 8.7

While the reservoir´s behaviour is infinite acting, the pressure profile expands

around the well during the production. In the radial flow case, the pressure at well

bottom drops with the logarithm of time. When all boundaries have been reached, the

shape of the pressure profile becomes constant with time, and it simply drops as the

reservoir is being depleted, meaning that during the pseudo state period, the well bottom

flowing pressure is a linear function of the elapsed time. During the shut-in, the

reservoir pressure tends to stabilize again, reaching the average reservoir pressure 5�,

which is usually lower than the static reservoir pressure (Pi) [13]. The pseudo steady

state behaviour and the drawdown and builup pressure responses in a closed reservoir

are shown in Figure 8-1 and Figure 8-2, respectively.

Figure 8-1 – Pressure profiles of a circular closed reservoir. t1: the boundaries are not reached, infinite reservoir behaviour; t2: boundaries reached, end of infinite acting; t3

and t4: pseudo steady state regime, the pressure profile drops. (BOURDET, 2002)

136

Figure 8-2 – Closed system drawdown and buildup responses. Linear scale. (BOURDET, 2002)

The pseudo steady state regime is analysed with a plot of pressure versus elapsed

time Δt on a linear scale, during the drawdown. At late time, the straight line of slope

m* is used to estimate the reservoir pore volume �hA (porosity-reservoir height-

reservoir area):

∅hA = 0.234 ÆÇ><n∗ 8.8

Where:

- ∅hA is the reservoir pore volume;

- q is the flow rate;

- B is the formation volume factor;

- ct is the total compressibility; and

- m* is the slope of the pseudo steady state regime plot during drawdown.

8.2 Well Test Interpretation Routine Here, a summary of the interpretation methodology is briefly presented. For more

information on non-presented scenarios, refer to BOURDET (2002) [13], chapter 10.

As explained in earlier sections, well test analysis is a process which involves,

basically, three steps:

(Drawdown) (Buildup)

137

1) Identification of the interpretation model: the derivative plot is the primary

identification tool.

2) Calculation of the interpretation model: the log-log pressure and derivative

plot is used to make the first estimates.

3) Verification of the interpretation model: the simulation is adjusted on the three

usual plots: log-log, semi-log superposition and test history on linear scale.

As opposed to the test history plot, log-log and superposition scale plots focus on

a single test period.

The main purpose of the semi-log superposition match is to refine the initial log-

log results. On log-log scales, the pressure ∆5 curve is not very sensitive to small

variations in the response and, on the derivative curve, the constant skin factor is only

present on early time data. The test history plot can indicate discrepancies in the data

such as in the rate history, or in the start of the analysed period.

In Table 8-1, a summary of usual log-log responses is presented.

Table 8-1 – Summary of important usual log-log responses, extracted from BOURDET (2002), Geochoke image extracted from CORBETT et al. (2005)

Well Models

Name and characteristic regimes Log-log pressure and derivative curves

Wellbore storage and skin

1 Wellbore storage, C

2 Radial, kh and S

Reservoir Models

Double porosity, restricted

interporosity flow

1 Radial fissures, kh

2 Transition (storativity), λ and L

3 Radial fissures + matrix, kh and S

138

Double porosity, unrestricted

interporosity flow

1 Transition λ

3 Radial fissures + matrix, kh and S

Double permeability, same skin (S1

= S2)

1 No crossflow

2 Transition (storativity), λ(kv), κ

and L

3 Radial, kh1+kh2 and ST

Geochoke (Double permeability

reservoir) or Hump Effect

1 Crossflow

2 High k layer intersecting the

wellbore

Boundary Models

Closed system centred

- Drawdown

1 Radial, kh and S

2 Pseudo Steady state, A

- Build-up

1 Radial, kh and S

2 Average pressure, 5� and A

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