AES/PE/12-14 Novel Simulator for Wireline Mini-Fracture

AES/PE/12-14 Novel Simulator for Wireline Mini-Fracture Testing

10/08/2012 B.M. Sintra Magalhaes

Title : Novel Simulator for Wireline Mini-Fracture Testing

Author(s) : B.M. Sintra Magalhaes

Date : August 2012

Professor(s) : Prof. Dr. P.L.J. Pacelli Zitha

Supervisor(s) : Dr. Auke Barnhoorn

MSc. Surej Kumar Subbiah

TA Report number : AES/PE/12-14

Postal Address : Section for Engineering and Geosciences

Department of Applied Earth Sciences

Delft University of Technology

P.O. Box 5028

The Netherlands

Telephone : (31) 15 2781328 (secretary)

Telefax : (31) 15 2781189

THIS WORK WAS PERFORMED IN:

Schlumberger Petroleum Services C.V

Parkstraat 83, 2514 JG

Den Haag, The Netherlands

Copyright © 2012 B.M. Sintra Magalhaes - Schlumberger- TU Delft Section for Engineering and Geosciences

Alle rechten voorbehouden. Niets uit deze uitgave mag worden verveelvoudigd, opgeslagen in een geautomatiseerd gegevensbestand,

of openbaar gemaakt, in enige vorm of op enige wijze, hetzij elektronisch, mechanisch, door fotokopieën, opnamen, of op enige

andere manier, zonder voorafgaand schriftelijke toestemming van de uitgever.

All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph

of this publication may be reproduced, copied, transmitted or save with the written permission of the author and parties involved or in

accordance with the publisher.

iii

Acknowledgements

In November 2011 I entered in Surej Kumar office to discuss the opportunities for a graduation project in

Schlumberger. While discussing with him the idea of developing a Mini-Frac simulator got my attention.

This assignment gave me the opportunity to apply many concepts learnt during my master studies and concise

them into one solid project.

I would like to thanks everyone who added to the value of the work described in this thesis and I would like

to express my deepest thanks and gratitude to my company advisor MCs. Surej Kumar for being an

outstanding advisor and for all his assistance during this project. His constant encouragement, support and

invaluable suggestions in rock mechanics contributed to the success of this work.

I also would like to show my gratitude to my faculty supervisors Prof. Dr. P.L.J. Pacelli Zitha and Dr. Auke

Barnhoorn for their constant feedback, ideas, guidance and support from the initial to the final stage of this

project, which enabled me to develop an understanding of the subject.

I acknowledge and thank Schlumberger Petroleum Services for allowing me to use their installations to

develop this idea.

Special thanks go to Paul Acquatella and Jonatan Flores for their cooperation in modelling and friendship

during my research. I am grateful to my colleagues in Petroleum Technology who indirectly contribute to this

project giving me distraction and good moments whenever I needed.

Last but not least I would like to thank God, my parents, Manuel Magalhaes and Fernanda Sintra, my sister

Paula Magalhaes and my girl friend Sarai Barrios for their understanding and love during the past years.

Their support and encouragement during these two years made this dissertation possible.

Boris Sintra Magalhaes

Den Haag, July 2012

Master of Science Thesis August 2012

i

Abstract

Wireline Mini-Fracture testing jobs consist of a short duration, small volume fracturing operation inside an

open-hole borehole, where a certain amount of fluid is injected into the formation at constant rate using a

Wireline Modular conveyed tool as a source of hydraulic power to pressurize the wellbore.

The tool is configured with an inflatable straddle packer and an internal pump, which inflate/deflates the

packers and supplies pressure to the formation until a hydraulic fracture is induced.

This procedure is used to determine in–situ formation breakdown and closure pressure also known as

minimum horizontal closure pressure. This provides vital information regarding hydraulic fracture design,

water and gas injection management, fault re-activation, wellbore stability, sand production, rock mechanical

properties, casing string design, cap and base rock integrity and gas storage design.

Geomechanical and operational parameters such as, elastic properties, poro-elasticity, rock strength,

formation pore pressure, far field horizontal stress, permeability/porosity distributions, borehole fluid

properties among others, influences the performance of the Mini-Frac Jobs.

In many cases poor understanding of the reservoir response to the fracture process, caused that the hydraulic

fracture did not propagate deep into the formation. In other cases the pressure applied to the formation might

be insufficient to break down the formation, leading to unsatisfactory application of the Mini-Fracture

technique in the process.

The objective of this thesis is to develop a Mini-Facture application simulator that uses the geomechanical

and operational parameters that control the performance of a Mini-Fracture job and estimate the possibility of

the occurrence of a tensile failure in the formation. The simulator is then validated by comparing its output

with the results of stress test done in the field.

With this simulator petrotechnical professionals and field engineers will have a platform that simulates the

pressure responses and fracture initialization during Mini-Frac treatments, incorporating all the variables

affecting a Wireline Mini-Fracture job, helping the design engineer to make key decisions about the ultimate

or required fracture plan.

Furthermore the simulator will reduce the uncertainties that limit the reliability of the Wireline Mini-Fracture

treatment by allowing the selection of appropriate tool configuration based on the job objectives and the

geological environmental conditions.

Finally this project demonstrates that combining the appropriate constitutive relations that reflect the coupling

among the tool operational performance with wellbore flow, reservoir and geomechanics modelling a Mini-

Fracture simulator can be developed.

Novel Simulator for Wireline Mini-Fracture Testing

ii

Table of Content

1 Introduction ................................................................................................................................................ 1

2 Wireline Mini-Fracturing Tool .................................................................................................................. 3

2.1 Wireline MDT Configuration and Operational Parameters ................................................................ 3

2.2 Stress Test Operational Procedure ...................................................................................................... 6

2.3 Stress Test Interpretation .................................................................................................................... 7

2.4 Thesis objectives .............................................................................................................................. 10

3 Wellbore Flow Modelling ........................................................................................................................ 12

3.1 Physical Model ................................................................................................................................. 12

3.2 Governing Equations ........................................................................................................................ 13

3.3 Boundary Conditions ........................................................................................................................ 13

3.4 Analytical solution ............................................................................................................................ 13

3.5 Pressure Drop in the Wellbore During a Mini–Frac Testing ............................................................ 14

3.5.1 Effect of Flow Rates ................................................................................................................. 14

3.5.2 Effect of Fluid Viscosity........................................................................................................... 15

3.5.3 Effect of Interval Height ........................................................................................................... 16

3.6 Fluid Velocity profile over the Interval ............................................................................................ 16

4 Near-Wellbore Reservoir Flow Modelling .............................................................................................. 18

4.1 Fluid Properties ................................................................................................................................ 18

4.1.1 Fluid Compressibility ............................................................................................................... 18

4.1.2 Rheological Properties ............................................................................................................. 19

4.1.3 Porous Media Rheological Properties ...................................................................................... 20

4.2 Physical Model ................................................................................................................................. 21

4.3 Governing Equations ........................................................................................................................ 22

4.4 Boundary Conditions ........................................................................................................................ 23

4.5 Self-Similarity Solution .................................................................................................................... 23

4.6 The Ei-Function Solution ................................................................................................................. 24

4.7 Numerical Solution ........................................................................................................................... 25

4.8 Models Predictions ........................................................................................................................... 26

4.8.1 Self-Similarity Prediction ......................................................................................................... 26

4.8.2 Ei-Function Prediction .............................................................................................................. 27

4.8.3 Numerical Prediction ................................................................................................................ 29


iii

5 Rocks Mechanics Modelling ................................................................................................................... 31

5.1 Terminology and Rock Mechanics Concepts ................................................................................... 31

5.1.1 Basic Concepts ......................................................................................................................... 31

5.1.2 Stresses in Undisturbed Ground ............................................................................................... 32

5.1.3 Pore Pressure ............................................................................................................................ 33

5.1.4 Effective Stress in a Rock ......................................................................................................... 33

5.2 Estimation of Breakdown Pressure................................................................................................... 33

5.2.1 Input Data ................................................................................................................................. 35

5.2.2 Dynamic Young Modulus and Poisson Ratio ........................................................................... 36

5.2.3 Calibration of Dynamic Parameters ......................................................................................... 37

5.2.4 Vertical or Overburden Stress Estimation ................................................................................ 37

5.2.5 Minimum Horizontal Stress...................................................................................................... 37

5.2.6 Maximum Horizontal Stress ..................................................................................................... 38

5.2.7 Stresses around Boreholes and Failure Criteria. ....................................................................... 39

5.2.8 Non-vertical borehole stress analysis and failure criteria. ........................................................ 41

5.3 Simulation of rock mechanics model ............................................................................................... 43

6 Mini-Fracture Simulator User Interface................................................................................................... 45

6.1 Input Data ......................................................................................................................................... 45

6.1.1 Uncontrollable Variables .......................................................................................................... 46

6.1.2 Controllable Variables .............................................................................................................. 46

6.2 Software User Interface .................................................................................................................... 47

6.2.1 Step # 1, Formation Parameters ................................................................................................ 48

6.2.2 Step # 2, Geomechanical Model ............................................................................................... 48

6.2.3 Step # 3, Tool Parameters ......................................................................................................... 49

6.2.4 Step # 4, Rheology Model ........................................................................................................ 49

6.3 MDT Tool Performance Module ...................................................................................................... 50

6.4 Wellbore Flow Module ..................................................................................................................... 50

6.5 Near-Wellbore Reservoir Module .................................................................................................... 51

6.6 Geomechanical Module .................................................................................................................... 52

6.7 Main Module & Simulator Answer .................................................................................................. 52

7 Results...................................................................................................................................................... 55

7.1 Breakdown Pressure Estimation ....................................................................................................... 55

7.2 Maximum MDT Build-up Pressure Estimation ................................................................................ 58


iv

7.3 Mini-Fracture Test Results ............................................................................................................... 58

8 Discussion ................................................................................................................................................ 75

8.1 Correct Predictions ........................................................................................................................... 76

8.2 Incorrect Predictions ......................................................................................................................... 78

8.2.1 Permeability Sensitivity Analysis ............................................................................................. 79

8.2.2 Porosity Sensitivity Analysis .................................................................................................... 79

8.2.3 Compressibility Factor Sensitivity Analysis ............................................................................ 80

8.2.4 Fluid Viscosity Sensitivity Analysis ......................................................................................... 81

8.2.5 Poisson’s Ratio Young’s Modulus Sensitivity Analysis .......................................................... 82

8.2.6 Tensile Strength Sensitivity Analysis ....................................................................................... 83

8.3 Simulator Limitations and Assumptions .......................................................................................... 84

9 Conclusion and Recommendation ........................................................................................................... 85

9.1 Conclusion ........................................................................................................................................ 85

9.2 Recommendations ............................................................................................................................ 86

10 References ................................................................................................................................................ 87

11 Appendix.................................................................................................................................................. 90

11.1 Derivation of the Wellbore Flow Model .......................................................................................... 90

11.2 Derivation of the Analytical Self-Similarity Solution ...................................................................... 93

11.3 Derivation of the Numerical Solution .............................................................................................. 96

11.4 Instructions to Install and use the Mini-Frac Simulator ................................................................... 98


v

List of Figures

Figure 2.1 MDT string for Mini-Fracture job (Carnegie et al., 2000). ............................................................... 3

Figure 2.2 Schematic draw of MRPA (Zacharia, 2007). .................................................................................... 4

Figure 2.3 Pump performance curve. ................................................................................................................. 6

Figure 2.4 Idealized Mini-Frac pressure and flow rate versus time profile (Zacharia, 2007). ........................... 8

Figure 2.5 Pressure and flow rate versus time profile obtained during a Mini-Frac test in shale formation.

Legend: 1 inflate packers, 2 leak off test, 3 breakdown, 4 fracture propagation and fall off, 5,6,7 and 9

fracture reopening / propagation / falloff cycles, 8 rebound to closure fracture, 10 deflate packers

(Dominique, 2004). ............................................................................................................................................ 9

Figure 2.6 Example of reconciliation plot of a Mini-Fracture test (Carnegie, 2000). ........................................ 9

Figure 1.1 Iterative coupling among geomechanics modelling, reservoir and wellbore flow, and tool

performance simulation. ................................................................................................................................... 11

Figure 3.1 Transversal section of the simplify model. ..................................................................................... 12

Figure 3.2 Pressure drop over the wellbore with different flow rates. ............................................................. 15

Figure 3.3 Pressure drop over the wellbore, varying fluid viscosity. ............................................................... 15

Figure 3.4 Pressure drop over the wellbore varying interval height. ................................................................ 16

Figure 3.5 Fluid velocity profile over the interval. ........................................................................................... 17

Figure 4.1 Comparison between rheology models and mud viscometer readings. .......................................... 20

Figure 4.2 Ideal radial flow reservoir model. ................................................................................................... 22

Figure 4.3 Discretization of radial symmetry model. ....................................................................................... 25

Figure 4.4 Self-Similarity result, first scenario. ............................................................................................... 26

Figure 4.5 Self-Similarity result, second scenario. ........................................................................................... 27

Figure 4.6 3D plot of unstable Ei-Function solution. ....................................................................................... 28

Figure 4.7 3D plot of unstable Ei-Function solution, increasing number of time steps. ................................... 28

Figure 4.8 Pressure in the reservoir using the numerical model. ...................................................................... 29

Figure 4.9 3D plot of numerical model simulation. ......................................................................................... 30

Figure 4.10 Comparison between analytical and numerical solution. .............................................................. 30

Figure 5.2 Stressed in undisturbed ground (Zacharia, 2007)............................................................................ 32

Figure 5.3 Breakdown pressure calculation workflow. .................................................................................... 34

Figure 5.4 Stress at borehole wall in the presence of far field stresses and pore pressure (Zacharia, 2007). ... 39

Figure 5.5 Minimum and maximum stress distribution at wellbore wall. ........................................................ 40

Figure 5.6 Tensile failure and fracture propagation in vertical wells. .............................................................. 41

Figure 5.7 Stress coordinate system for deviated borehole (Fjaer et al., 2008). .............................................. 42

Figure 5.9 Example of Geo-Mechanical simulator output. .............................................................................. 44

Figure 6.1 Mini-Fracture Simulator structure. .................................................................................................. 45

Figure 6.2 Example input data worksheet. ....................................................................................................... 47

Figure 6.3 Mini-Fracture simulator, user-interface window. ........................................................................... 47

Figure 6.4 Step # 1, Simulator formation parameters input windows. ............................................................. 48

Figure 6.5 Step # 2, Simulator geomechanical input windows. ...................................................................... 48

Figure 6.6 Step # 3 Simulator tool operational input windows. ....................................................................... 49

Figure 6.7 Step # 4 Simulator rheology parameters. ........................................................................................ 49

Figure 6.8 MDT Tool Performance Module structure...................................................................................... 50

Figure 6.9 Wellbore Flow Module structure. ................................................................................................... 51


vi

Figure 6.10 Near-Wellbore Reservoir Module structure. ................................................................................. 51

Figure 6.11 Geomechanical Module structure. ................................................................................................ 52

Figure 6.12 Main Module structure. ................................................................................................................. 52

Figure 6.13 Main User Interface Window Results Display. ............................................................................. 53

Figure 7.1 Breakdown pressure computed using the Mini-Fracture simulator and SLB-MEM. ..................... 56

Figure 7.2 Difference in results for Well # 1, (frequency distribution). ........................................................... 56

Figure 7.3 Difference in results for Well # 2, (frequency distribution). ........................................................... 57

Figure 8.1 Validation summary of Mini-Fracture simulation predictions against real field data. ................... 75

Figure 8.2 Pressure difference between the simulated maximum build-up pressure and the breakdown

pressure for each test. ....................................................................................................................................... 77

Figure 8.3 Simulation of test 1.2 using different types of pump. ..................................................................... 77

Figure 8.4 Difference in pressure between interval and borehole for each test. ............................................... 78

Figure 8.5 Permeability sensitivity analysis. .................................................................................................... 79

Figure 8.6 Porosity sensitivity analysis. ........................................................................................................... 80

Figure 8.7 Compressibility factor sensitivity analysis. ..................................................................................... 80

Figure 8.8 Fluid viscosity sensitivity analysis. ................................................................................................. 81

Figure 8.9 Young’s modulus sensitivity analysis. ............................................................................................ 82

Figure 8.10 Poisson’s ratio sensitivity analysis. ............................................................................................... 82

Figure 8.11 Tensile strength sensitivity analysis. ............................................................................................. 83

Figure 8.12 Test 2.6 and 2.8 results, using tensile strength equals to zero. ...................................................... 83


vii

List of Tables

Table 2.1 High performance packers portfolio (Yeldell, 2011). ........................................................................ 5

Table 2.2 Pump types specification (Stevens, 2009). ......................................................................................... 5

Table 2.3 Stress test operational procedure (Desroches et al., 2005). ................................................................ 7

Table 2.4 Parameter identification for stress test interpretation (Desroches et al., 2005). ................................. 8

Table 3.1 Input parameters for wellbore flow modelling. ................................................................................ 14

Table 3.2 Input parameters to compute the velocity profile. ............................................................................ 17

Table 4.1 Mud rheological properties used for modelling. .............................................................................. 19

Table 4.2 Coefficients for Herschel-Bulkley model calculated using viscometer data. ................................... 20

Table 4.3 Parameters used to test models predictions. ..................................................................................... 26

Table 5.1 Relevant Wireline logs description. ................................................................................................. 35

Table 5.2 Output of geomechanical simulator. ................................................................................................. 43

Table 6.1 Uncontrollable variables use in the Mini-Fracture simulator. .......................................................... 46

Table 6.2 Controllable variables use in the Mini-Fracture simulator. .............................................................. 46

Table 7.1 Stress test results Well # 1...........................................................................................................55

Table 7.2 Stress test results Well # 2. ............................................................................................................... 55

Table 7.3 Breakdown pressure estimation using the Mini-Fracture simulator and SLB-MEM. ...................... 57

Table 7.4 Maximum build-up pressure comparison. ........................................................................................ 58

Table 7.5 Results comparison between simulated and real stress test for Well # 1. ........................................ 59

Table 8.1 Detailed simulator versus real field stress test predictions. .............................................................. 76

Table 8.2 Values used for the sensitivity analysis. ........................................................................................... 78

Table 11.1Dimensionless Parameters. .............................................................................................................. 91


viii

List of Nomenclature

English Notation

A = Area [m

2]

Ceff = Total Isothermal Compressibility factor [1/Pa]

Cf = Fluid Isothermal Compressibility factor [1/Pa]

Cp = Pores Isothermal Compressibility factor [1/Pa]

E = Young’s Modulus [Pa]

Edyn = Dynamic Young’s Modulus [Pa]

Efr = Static Young’s Modulus [Pa]

Ei = Line-Source Solution [-]

Gr = Gamma Ray [API]

H = Interval [m]

Hpl = Blake–Kozeny Coefficient [-]

k = Permeability [mD]

KHB = Herschel-Bulkley Consistency Factor [cP]

m = Herschel-Bulkley Flow Behaviour Index [-]

MDT = Wireline Modular Dynamic Tested [-]

MEM = Mechanical Earth Model [-] MRCF = MDT Flow Control [-]

MRPA = MDT Dual-Packer [-]

MRPO = MDT Pump-Out [-]

P = Actual Pressure [Pa]

Pp = Pore Pressure [Pa]

Pref = Initial Pressure [Pa]

Pwfract

= Borehole fluid Pressure for fracturing [Pa]

Q = Flow Rate [cc/sec]

R = Radius [m]

Rt = Tool Radius [m]

Rw = Well Radius [m]

t = Time [sec]

T0 = Tensile Strength [Pa]

UCS = Unconfined Compressive Strength [Pa]

V = Volume [m3]

Vref = Reference Volume at Initial Pressure [m3]

Vtotal = Matrix plus Pore Volume [m3]

Greek Notation

α = Biot’s Constant [-]

γ = Euler Constant [-]

eq

= Equivalent Shear Rate [1/s]

ε = Strain [milli strain]

µ = Viscosity [Pa.s]

νp = Compressional Velocity [ft/us]

νs = Shear Velocity [ft/us]

ξ = Boltzmann Transformation [-]

ρ = Fluid Density [g/cm3]

σ1 = Great Principal Stress [Pa]

σ2 = Intermediate Principal Stress [Pa]

σ3 = Least Principal Stress [Pa]

σv = Vertical or Overburden Stress [Pa]

σH = Maximum Horizontal Stress [Pa]

σh = Minimum Horizontal Stress [Pa]

σx = Normal Stress in x Direction [Pa]

σy = Normal Stress in z Direction [Pa]

σz = Normal Stress in z Direction [Pa]

σθ = Circumferential (hoop) Stress[Pa]

σr = Longitudinal Stress [Pa]

σz = Radial Stress [Pa]

τ = Shear Stress Exerted by the Fluid [Pa]

τHB = Herschel-Bulkley Yield Point [Pa]

υ = Poisson’s Ratio [-]

υrf = Static Poisson’s Ratio [-]

υDyn = Dynamic Poisson’s Ratio [-]

φ = Porosity [m3/m3]

ω = Angular Position [deg]

ψ = Well inclination [deg]

ϑ = Well azimuth [deg]


1

1 Introduction

The latest statistical reviews of world energy shows that global oil production has struggled to keep up with

increasing demand. The worldwide oil consumption grew by 3.1 % in 2011, and gas consumption increased

by 7.4 %, the most rapid increased since 1984. In the same period the oil and gas production grew only 1.8%

and 7.3% respectively (BP Statistical Review, 2011).

The above facts raised the concern that in the coming decades the world supply of oil and gas might fall

below the level required to meet the demand and a severe energy crunch might be inevitable without an

expansion of oil and gas production.

For this reason, oil companies are constantly developing techniques, or improving existing techniques, for

increasing the amount of hydrocarbon that can be extracted from a reservoir. One of the techniques that

experienced large development in the recent years is hydraulic fracturing. This technique has been used for

decades to boost the production rate of oil and gas from subsurface reservoirs (Charlez, 1997). Hydraulic

fracturing is applied in reservoirs formed by sandstone, limestone or dolomite rocks but also they can include

shale or coal beds, classified as “unconventional reservoirs”.

Hydraulic fracturing proved to be particularly suitable for improving production from low permeability or

low pressure oil and gas reservoir (Carnegie et al., 2000). Inducing fractures provides a conductive path

connecting a larger volume of the reservoir to the well, increasing the volume of oil and gas that can be

recovered from the targeted formation.

For a hydraulic fracture job to be successful a clear understanding of the far field horizontal stress profile is

needed. This information is of vital importance for planning hydraulic fracturing jobs. Several techniques

have been used to measure horizontal stress magnitudes and directions through data obtained from core

samples and directly in the wellbore (Desroches et al., 1999).

These techniques include stress measurement in core samples based on strain test methods, such as

Differential Strain Curve Analysis and Anelastic Strain Recovery. Another technique is based on sonic

measurements that rely on semi-empirical relationships between rock properties and stresses near the

wellbore face, to inferred important rock modulus and hence calculate a rough estimate of horizontal stress.

But the most reliable and accurate approaches of in-situ stress are accomplished by down-hole techniques

which actually fracture the formation (Desroches et al., 2005). The most common examples are the Leak-Off

Test (LOT), and the Mini-Fracture technique.

LOT is a procedure performed mainly to determine the safe mud weight to drill a well section. The test is

done by pumping fluid into an open-hole section until a certain pressure is reached. With this procedure, an

indication of far field stress can be obtained if the pressure during the LOT is increased until a fracture

develops (Carnegie et al., 2000).

Nevertheless, LOTs suffer from several drawbacks which might lead of inaccurate estimates of fracturing

pressure and the far-field stresses. These drawbacks include, mud compressibility and the lather length of the

open-hole section (Carnegie et al., 2000). Furthermore, there is a considerable possibility that the test will

open and reopens the formation at the weakest locations, which might not be the desired depth for analysis.

http://en.wikipedia.org/wiki/Crude_oil

http://en.wikipedia.org/wiki/Oil_field


2

The Mini-Fracture technique consists of the creation of a mini-hydraulic fracture produced by injecting fluid

into a limited section of open-hole wellbore intervals that are isolated by sealing packer elements (Zacharia,

2007). This stress test technique enables accurate estimates of the stresses around the wellbore by measuring

the response obtained during the initiation, propagation and closure pressure of hydraulic fracture.

It also overcomes the disadvantages related to the LOT. The fracturing fluid is pumped very close to the

section to be tested by a down-hole pump which contains the pressure gauges. This reduces the

compressibility effect observe in the LOT. Furthermore unlike LOT, the volume of fluid pumped is very low

restricting the fracture from growing vertically beyond the tested section.

Once the fracture has been created, it is closed and reopened through several injection-shut in cycles. The in-

situ minimum stress value is estimated from the analysis of the pressure response obtained during reopening,

propagation and closure of the induced fracture (Carnegie et al., 2000).

The Mini-Frac technique was used many times in the field to estimate in-situ stresses. However, often the

Mini-Frac tests suffered either to insufficient pressure build-up for fracture initiation or resulted in fractures

that did not propagate very deep into the formation. Poor control of Mini-Frac tests was too a large extent due

to poor understanding of the fracturing process and to lack of a tool for properly designing the Mini-Frac

treatments.

Lack of understanding on the geomechanical and operational parameters involved in Mini-Fracture tests (i.e.

elastic properties, poro-elasticity, rock strength, formation pore pressure, far field horizontal stress, maximum

pump pressure, etc), affected the quality of the data of the stress test jobs, putting the reputation and

application of this technique at risk.

As a result of this concern, a proposal on the creation of a platform that emulates the pressure responds in the

reservoir and fracture initialization during a Mini-Frac treatment was made.

The objective of this thesis is to model Mini-Frac processes taking into account both wellbore and near

wellbore reservoir fluid flow and geomechanics aspects, to develop a Mini-Frac simulator which will be used

by field engineers to design the Mini-Frac tests. The model and simulator tool coupling will enable a better

definition of the operational parameters needed to optimize the performance of a Mini-Frac job and more

accurate estimate of the tensile failure in the formation.

This report consists in 10 chapters and one appendix. Chapter 2 explains the concepts of a Mini-Frac job, how

the tool is configured, operational procedures required to perform the test, the basic interpretation of stress

test and a description of the modelling of the tool performance. Chapter 3 explains the modelling of the

pressure and fluid velocity drop around the wellbore. Chapter 4 illustrates the modelling of radial flow in the

reservoir when fluid moves in porous media. Chapter 5 explains the steps necessary to model the rocks

mechanics required to calculate the breakdown pressure. Chapter 6 gives details of the structure of the Mini-

Fracture simulator. Chapter 7 compares the simulator results with real field data for validation. Chapter 8

discusses the results and interprets the simulation predictions. Chapter 9 concludes the whole report and

suggests recommendations for further studies and chapter 10 is the reference that we have used. The

appendices contain the derivations of the analytical and numerical solution used in the simulator.


3

2 Wireline Mini-Fracturing Tool

The Wireline Mini-Fracturing tool is derived from the Wireline-Conveyed Modular Dynamic Testing (MDT)

tool developed by Schlumberger. It is used to perform a complete in-situ stress analysis in the near-wellbore

area. The MDT enables injection of fluids at low flow rates into the formation and isolates the formation for a

better control of the fracture propagation. The tool uses motorized valves, is of modular design and entirely

software controlled (Zacharias, 2007).

It was primary designed to either recover formation fluid samples or to measure pore pressure and formation

permeability. However, applying certain modifications on the tool configuration, allow it use for stress testing

(Plumb, 1996).

2.1 Wireline MDT Configuration and Operational Parameters

Figure 2.1 shows a typical configuration of a MDT string for a stress test where the valves and the fluid path

between the borehole and the interval are illustrated. The configuration used for stress testing includes the

packer module, pump-out module and the flow control module.

Figure 2.1 MDT string for Mini-Fracture job (Carnegie et al., 2000).


4

To model the MDT performance during a stress test job we need to understand the most critical tool

parameters that influence the outcome of the Mini-Fracture simulator. These parameters are the maximum

allowable differential pressure, the spacing between seals, the maximum build-up pressure, the maximum

injection flow rate and the pump performance curve.

These five parameters are controlled by two modules in the MDT string namely: dual packer module (MRPA,

MDT Dual-Packer) and the pump-out module (MRPO, MDT Pump-out).

The MRPA provides two inflatable packer elements to isolate the borehole interval for testing. Fluid is

pumped to about 1000 psi above hydrostatic pressure to inflate the packer elements, once the interval is seal

fluid is pumped into the interval to induce a fracture between the two packers. This will create a differential

pressure between the interval and the wellbore (Yeldell, 2011).

The Dual Packer Module is equipped with a dual system of sensors (Strain Gauge and CQG Crystal Quartz

Gauge). Pressure in the packers and in the interval are recorded simultaneously and can be displayed for

analysis in real time (Fourmaintraux et al., 2005). Figure 2.2 shows schematically the main MRPA

components and the location of the injection interval between the packers.

Figure 2.2 Schematic draw of MRPA (Zacharia, 2007).

The maximum allowed differential pressure and the spacing between seals are specified depending on the

packer (Seal) that is been used on the string. The packer portfolio used to seal the formation is shown in

Table 2.1 and the selection will depend on the hole size and the maximum wellbore temperature.

The length of the test interval (i.e., the spacing between the packers) for standard jobs is 3.2 ft [0.98 m] and

can be extended by 2, 5, or 8 ft [0.61, 1.52, or 2.44 m]. The tool radius will also vary from 5.7” to 6.7” [ 0.15

to 0.17 m] depending on the type of packer and filter screens selected (Yeldell, 2011).


5

Table 2.1 High performance packers portfolio (Yeldell, 2011).

Packer Type Hole size

[inches]

Max Tem

[DegF]

Pressure rating

[kpsi]

Max Diff Pressure

[kpsi]

SIP-A3A-5in 6.0 350 20 5

IPCF-BA-500 6.0 410 14 3

IPCF-PA-700 8.5 350 20 3

IPCF-BA-700 8.5 410 14 3

IPCF-PC-700 8.5 350 20 5

IPCF-H2S-700 8.5 350 20 3

SIP-A3A-6.75 8.5 350 20 3

SIP-A3A-8.5 12-1/4 350 20 3

The MRPO is the wireline pump used to inflate the rubber elements of the dual packer module and to

pressurize the test interval in order to create the hydraulic fracture. There are different types of pumps with

different features that can be use to increase the hydraulic pressure between the two packers. The maximum

pressures build-up and maximum flow rate will depend on the type of pump that is going to be use (Stevens,

2009).

Variable displacement pump is the most commonly used pump on MDT strings. The main characteristic of

this pump is that the displacement/flow rate decreases as the pressure increases. This can cause some

disadvantages at the moment of interpret the data, as ideally the flow rate should be constant when fluid is

injected into the interval. Table 2.2 summarizes the different pump options and their specifications.

Table 2.2 Pump types specification (Stevens, 2009).

Standard Pump High Pressure X High Pressure XX High Pressure

Max. Differential

Pressure [psi]

4956

6092

8360

11760

Flow Rate

[cc/sec]

8.2-32.8

6.3-24.6

4.4-18.3

0.8-16

In the final Mini-Fracture simulator, the user will have the option to select the type of pump, packer and

interval length to match the configuration used in the MDT string and the desirable depth for stress testing.

These controllable operational parameters can be adjusted to achieve the optimum combination that will

allow the tool to create a fracture.

Irrespective of the type of pump used, in order to perform a stress test other considerations need to be taken

prior to performing the job (Zacharias, 2007). The flow rate provided by the MRPO should be higher than the

leak off rate of the formation otherwise, the pressure in the interval won’t build-up. The breakdown pressure

should be achieved before the minimum flow rate of the pump is attained, otherwise the pump will stall and

fracture won’t be initiated. If the fracture is not initiated with a standard displacement unit, an extra high

pressure displacement unit will need to be installed in the pump. The fracture fluid (wellbore mud in most of

the cases) should have an API fluid loss in the range of 5 to 20 cc. Lower values indicate mud that can plug

the formation while higher values will not propagate the fracture.

To introduce the typical performance curve from these pumps into the simulator, real field data was used to

create a curve that shows how the flow rate decreases when the interval pressure increases. This was achieved

by plotting a graph of the interval differential pressure against flow rate for several stress tests.


6

Subsequently an exponential tender line was set to fit the curves and generate a mathematical expression that

can be included in the simulator. Figure 2.3 shows the plot of 10 different test used to create the fitting curve.

Figure 2.3 Pump performance curve.

The flow control module (MRFC, MDT Flow Control) is an optional module that can be used to perform a

pressure rebound test in impermeable layers. This test is carried out if a fracture closure does not occur after

the pump is stopped. The MRFC withdraw fluid from the interval at constant rate to facilitate fracture

closure. A gamma ray tool is also present in the tool string configuration for depth correlation control and

also placing the tool in the desire target depth.

2.2 Stress Test Operational Procedure

Before performing a stress test, a series of open-hole logs are run in the well to carry out a petrophysical

analysis which is required for selecting the intervals for stress testing.

A sample suite of logs that are run before the stress testing can include density–neutron logs, plus a

geochemical log that helps to accurately estimate the mineralogy of the zone of interest for the test. An image

log can also be lowered into the borehole to identify the laminations, natural and drilling induced fractures

and their extent, along with an advanced sonic log to estimate the degree of horizontal and vertical stress

anisotropy. Sometimes a third run is made to acquire side wall core plugs. The MDT tool is then ran to

perform the in-situ stress testing (Ramakrishnan et al., 2008).

Once the tool has been properly positioned, the objective is to create a control fracture in a desired zone and

measure the related pressure response. The fracture is then re-opened and closed to assure repeatability of the

measurements. These repeated cycles also help the fracture to grow deep in the formation to sense the far

field stresses accurately (Koksal, et al., 2009). Table 2.3 shows the details of the stress test operational

procedure, which consist of six steps.


7

Table 2.3 Stress test operational procedure (Desroches et al., 2005).

Step Description Procedure

1

Packer Inflation

The interval to be tested is isolated by inflating the straddle packers.

Pressure in the interval may rise as the packers inflate and compress

the fluid in the interval.

2

Filtration or Leak off Test.

Pressure is increased in steps to a level below what is necessary to

break the formation with the aim of testing if the packer seal is holding

and also to test the leak off of the formation within the interval.

3

Formation Breakdown

The interval is then pressurized at constant flow rate by pumping fluid

against the borehole wall. The interval pressure and the inflated

pressure will keep on increasing until a tensile fracture begins to

develop. Once a fracture is initiated (Break Down Pressure), it can be

identified by a sharp decrease in interval pressure followed by a

pressure plateau or by a pressure plateau only (in ductile formations).

4

Pressure Fall-Off

The injection is then stopped and the pressure is allowed to fall-off to

a pressure level that ensures that the fracture is closed.

5

Fracture Reopening

A series of injection/falloff cycles, further propagate and close the

fracture.

6

Packer Deflates

Once the operator is sure that good quality data has been acquired, the

packers are deflated and the stress testing interpretation can be done.

2.3 Stress Test Interpretation

Reservoir geomechanics specialists are usually interested in the so-called formation principal stress. There are

three and frequently referred to as great principal stress σ1, intermediate principal stress σ2 and least principal

stress σ3.

There are several techniques that have been developed to estimate the magnitude of the principal stresses

from pressure records. The method presented in this thesis consist of first identifying for each injection cycle

the fracture breakdown pressure, fracture propagation pressure, instantaneous shut-in pressure (ISIP), fracture

closure pressure and fracture re-opening pressure (Desroches et al., 2005). Table 2.4 shows how to recognize

the pressure parameters used for the stress interpretation.


8

Table 2.4 Parameter identification for stress test interpretation (Desroches et al., 2005).

Parameter Description

Break Down Pressure

Is the pressure at which the formation breaks by tensile stress and a fracture is

created. It can be recognize by a quick drop of pressure while fluid is still being

pumped into the formation.

Propagation Pressure

If a pressure plateau is reached while pumping at constant flow rate (after the

fracture is initiated), this is an indication that a stable fracture is propagating and

so the propagation pressure can be determined.

Instantaneous Shut in

Pressure (ISIP)

This is the pressure reached just after the injection is stopped.

Closure Pressure

This is the pressure at which the fracture closes after the injection is stopped.

The Reopening Pressure

This is the pressure at which a pre-existing fracture opens up. This corresponds to

a change in the stiffness of the tested interval.

Figures 2.4 and 2.5 represents the idealized and the real field Mini-Fracture pressure and flow rate versus

time profile respectively. Both plots illustrate the different pressures signatures associated with each injection

cycle.

Figure 2.4 Idealized Mini-Frac pressure and flow rate versus time profile (Zacharia, 2007).


9

Figure 2.5 Pressure and flow rate versus time profile obtained during a Mini-Frac test in shale formation.

Legend: 1 inflate packers, 2 leak off test, 3 breakdown, 4 fracture propagation and fall off, 5,6,7 and 9

fracture reopening / propagation / falloff cycles, 8 rebound to closure fracture, 10 deflate packers

(Dominique, 2004).

After all parameters have been recognized, the interpretation procedure is followed by a “reconciliation

phase”. This part of the interpretation is based on the idea that no single parameter (closure pressure, ISIP,

etc) determined on a single hydraulic fracture cycle is by itself representative to calculate the stresses of the

subsurface. In the reconciliation procedure the parameters determined for each cycle are plotted together on a

single plot to define their pertinence and validity. The assumption is that the induced fracture grows from one

cycle to the next. This growth process has to ensure consistency from one cycle to another cycle showing a

marked trend between them (Carnegie et al., 2002). Figure 2.6 Example of reconciliation plot.

Figure 2.6 Example of reconciliation plot of a Mini-Fracture test (Carnegie, 2000).


10

Once the geomechanics specialist is satisfied with the interpretation of each event, a band is selected where

the closure stress is likely to be. If the closure stress value stabilizes on the last cycles, a value can then be

selected within the closure stress confidence band and this value can be estimated as a reliable measure of the

minimum principal stress σ3.

2.4 Thesis objectives

This master thesis will focus on the formation breakdown step and it will also investigate under which

conditions the tool is not able to reach the breakdown pressure in certain occasions. A Mini-Fracture

simulator will be built to reproduce this step and compare the maximum build-up pressure that the MDT tool

can deliver with the breakdown pressure that is needed in order to create a tensile failure. The other steps

involved in the analysis of stress test interpretation are not going to be developed in this the project as they

are beyond the scope of this thesis.

To properly combine all interacting aspects in the process of creating a hydraulic fracture around the

wellbore, the simulator fully couples the tool operational performance with wellbore flow, reservoir and

geomechanics modelling. Coupling simulation was first introduced in 1980s and several other fracture

propagation models have since evolved (Settari et al., 1998).

The idea behind the Mini-Fracture simulation is to use data from a wireline logs and core samples to create a

geomechanic model. Subsequently, the simulator uses this model to compute the pressure required to create a

tensile failure. At the same time, the simulator numerically solves the partial differential equation for fluid

flow in the reservoir (with finite difference method) as fluid is being injected into the reservoir by the tool.

The flow and stress coupling is implemented by interaction functions between geomechanical aspects of the

porous media and reservoir fluid flow properties (i.e., permeability, porosity).

Moreover, the simulator uses the tool parameters and limitations (i.e., maximum allowed interval pressure,

injected flow rate, pump efficiency curve), to compute how the pressure changes in the reservoir affect the

performance of the tool used to fracture the formation. Tool modelling, geomechanics simulation, reservoir

and wellbore flow are computed together in a modular manner by a coupling algorithm that is used as a main

process to iteratively estimate the possibility of a tensile failure occurrence in the formation as show in Figure

2.7.

Additionally, this simulator helps the design engineer to make key decisions about the ultimate and required

fracture plan, by selecting the appropriate tool configuration based on the job objectives and geological

environmental variables that affect the Mini-Fracture job. Therefore the uncertainty elements that limit the

reliability of the tool are reduced.

To develop this simulator the project was divided in four parts.

The first part of the project focuses on determining (from literature) the appropriate constitutive equations

and solutions that represent the best physical characteristics that are involved during a Mini-Fracture process.

This process starts by evaluating how the fluid changes in pressure and velocity as it moves from the tool

output to the wellbore wall. For this, an analytical model of the radial flow between two parallel discs

separated by a finite distance was considered.

Subsequently a complete analysis of flow in porous media was prepared. This analysis will help to estimate

the pressure profile in the formation and also to understand the performance of the fluid when it flows from

the wellbore wall through the reservoir.


11

To conclude the first part of the project, the stress field and rock mechanic aspects near the wellbore region is

studied. This provides understanding on how to model the rock failure mechanics and compute the

breakdown pressure.

The second part of the project involves the development of a mathematical formulation that is used in the

model to describe a Mini-Fracture simulator, which will match the treatment behaviour.

The third part of the project implements the numerical formulation and builds the simulator using Matlab to

model the problem. The final product will consist of standalone software that can be installed in any target

computer to predict the performance of the stress test tool during a Mini-Fracture job.

Finally, the last step of the process is to validate the predictive capabilities of the model using field data as a

reference and conducting parametric analysis on different properties that influences the formation breakdown.

Figure 2.7 Iterative coupling among geomechanics modelling, reservoir and wellbore flow, and tool

performance simulation.


12

3 Wellbore Flow Modelling

The objective of this chapter is to derivate the expression for the pressure drop and the velocity profile along

the wellbore as a function of geometrical and physical parameters. The analytical model used to describe this

process was analysed by adapting the work of Medina (2009), based on the examination of the radial flow

between two parallel discs separated by a finite distance. This solution will be derived from the continuity

equation and Navier-Stokes equation in radial coordinates (Bird, 1987).This mathematical expression will be

use to create the wellbore flow module employed in the simulator.

Furthermore, a sensitivity analysis will be carried out using the parameters commonly employed in a Mini-

Fracture job. This helps to understand the effect of changing flow rates, fluid viscosity and interval height on

the Wellbore Flow simulator output.

3.1 Physical Model

Figure 3.1 shows schematically the set-up of the tool that will be used to derivate the solution. The fluid

enters vertically into the center of the disk and change direction moving horizontally throughout the radius of

the disc. The coordinate system is placed in the centre between the two packers, while the geometrical

parameters to be considered are the tool radius Rt, the wellbore radius Rw and the distance 2H between

packers.

We considered steady-state, radially symmetric, laminar and incompressible flow of a Newtonian fluid,

having viscosity µ from the tool radius Rt to the output disc radius Rw. As the cross-sectional area varies

proportionally with the radius, the fluid velocity also varies in the direction of the r coordinate. Moreover,

from the condition of non-slip, it can be inferred that the velocity profile also depends on the z coordinate i.e.

r ( , )V V r z

Figure 3.1 Transversal section of the simplify model.


13

3.2 Governing Equations

As already mentioned the above model corresponds to the flow of viscous fluid between two parallel

concentric discs separated by a finite distance (Medina, 2009). To find the pressure and velocity profile we

solve the corresponding continuity equation and the Navier-Stokes equation in cylindrical polar coordinates:

( ) 0rrVr

(3.1) 2

2

r rr

V VpV

r r z

(3.2)

3.3 Boundary Conditions

The boundary conditions applied are based on the assumption of non-slip. This means that the radial

velocities at positions z=-H and z=H are zero. Additionally, the pressure of the fluid at the tool radius Rt is Pt

while the pressure at the wellbore radius Rw is Pw.

Summarizing these boundary conditions we get:

( ) 0

( ) 0

( )

( )

r

r

t

w

V z H

V z H

P r R Pt

P r R Pw

3.4 Analytical solution

The detail derivation of the analytical solution is presented in Appendix A. The equations are solved defining

the function as:

rrV (3.3)

Substituting the first integral of the continuity equation into the Navier-Stokes equation we obtain the

following differential equation.

2 2

3 2

1dp d

r dr r dz

(3.4)

In this equation the variables can be separated since p is only function of r and ϕ is only function of z. By

integration of equation 3.4 with respect to r over the interval Rt to Rw we obtain the following second order

differential equation for ϕ(z).

2

2

2 2 2

1 1( ) ( ) 0

2

wt w

t w t

R dz P P Ln

R R R dz

(3.5)


14

Considering that the term 2 is vanishingly small, the integration of the equation is the trivial. Taking the

boundary conditions into consideration, the velocity profile of the fluid in the wellbore can be written as:

22( )

( , ) 11

2

t wr

w

t

p p H zV r z

HRr Ln

R

(3.6)

which gives a parabolic profile with respect to z and a hyperbolic profile with respect to r. Introducing the

change in volume over the coordinate r, we can obtain the average velocity by integrating:

1

4 4r r

A

Q QV V dA

A rH rH

(3.7)

where Q is the flow rate. Re-arranging equation 3.7 and substituting rV from equation 3.6 we obtain the

pressure profile over the interval.

3

3

4t w

t

Q rp p Ln

H R

(3.8)

3.5 Pressure Drop in the Wellbore During a Mini–Frac Testing

This section aims to illustrate how to use the analytical solution to estimate the difference in pressure between

the tool output and the wellbore wall during Mini-Fracture job using a Matlab code. The input data used in

this section corresponds to the values commonly employed during Mini-Fracture jobs. These values are based

on tool ratings and fluids properties. Several runs of the code were done by varying the main physical

parameters. Table 3.1 summarize the range of values used in the model.

Table 3.1 Input parameters for wellbore flow modelling.

Parameter Symbol Units Minimum Value Maximum Value

Viscosity µ [cP] 30 100

Interval H [m] 1 3.5

Tool radius Rt [in] 5.75 5.75

Wellbore radius Rw [in] 8.5 12.5

Flow Rate Q [cc/sec] 0 30

3.5.1 Effect of Flow Rates

Figure 3.2 shows the pressure for flow rates varying from 0 cc/sec to 30 cc/sec. The data in the figure

confirms that the pressure drop is proportional to the flow rate. The faster the fluid moves the larger the

differential pressure. The pressure gradient is not linear. It has a logarithmic distribution with higher gradients

on the values close to the tool radius.


15

Figure 3.2 Pressure drop over the wellbore with different flow rates.

3.5.2 Effect of Fluid Viscosity

Figure 3.3 shows the pressure drop over the interval for fluids viscosities varying from 1 cP to 1000 cP. It can

be seen that an increase in viscosity leads to a higher pressure drops over the interval while very low

viscosities creates very little pressure drop. This is logical because the viscosity term is on the numerator of

the equation 3.8.

Figure 3.3 Pressure drop over the wellbore, varying fluid viscosity.


16

3.5.3 Effect of Interval Height

Figure 3.4 shows the pressure drop over the interval with different distance between packers. As expected

from equation 3.8 the pressure drop diminishes when the height of the gap between packers increased. This is

logical as the term H is dividing the solution and elevated to the exponent three. This means that the

parameter H influence more the result than the rest of the parameters.

From this analysis it can be concluded that the pressure drop over the wellbore interval is extremely small (in

the order of X-10 psi) hence, can be neglected. This is mainly because the flow rates used on a mini

fracture job are very small, and the distance between the tool and the well is also very small. It can also be

concluded that the pressure drop gives a logarithmic distribution with a higher gradient at the proximity of the

tool. It has to be notice that the assumption of having viscosity µ from the tool radius Rt to the output disc

radius Rw it may not represent the rheology properties of the fluid used to create the fracture, but as the results

showed that the pressure difference is so small, further investigation using the non-linearity approach was not

carried out.

Another observation is that the pressure drop gradient is proportional to the flow rate and inversely

proportional to the interval height. Furthermore, it can be concluded that the interval height is the parameter

that affects the pressure drop the most. This means that the lowest pressure drop will be achieved when the

distance between packers is the maximum possible.

Figure 3.4 Pressure drop over the wellbore varying interval height.

3.6 Fluid Velocity profile over the Interval

From equation 3.6 is observed that the velocity profile is bi-dimensional as it depends on two variables z and

r. It is possible to determine the velocity profile if the geometrical parameters are known. Table 3.2 shows the

parameters used to model the velocity profile over the interval.


17

Table 3.2 Input parameters to compute the velocity profile.

Parameter Symbol Units Value

Viscosity µ [cP] 30

Interval H [m] 1

Tool radius Rt [in] 5.75

Wellbore radius Rw [in] 12.2

Flow Rate Q [cc/sec] 20

Figure 3.5 shows the velocity profiles as a function of the interval height for different radius. It can be

observed that the velocity profile is symmetrical for values of z=0, while the velocity is higher for distances

closer to the tool and decreases as the fluid moves away from the tool towards the wellbore wall.

The maximum velocity is at the outlet of the tool and the minimum velocity is at the wellbore. This is due to

the fact that the transversal section increase as the radius increases. Therefore to maintain the conservation of

mass, the velocity must also decrease.

Figure 3.5 Fluid velocity profile over the interval.


18

4 Near-Wellbore Reservoir Flow Modelling

This chapter deals with the analysis of fluid flow in the reservoir around the near-wellbore area. The main

objective is to develop a mathematical formulation to describe fluid flow in a reservoir during a Mini-

Fracture job.

The first step of this analysis is to classify the fluid properties used for fracturing. The second step consisted

of defining the physical model and reservoir geometry. Subsequently, the boundary conditions and

governmental equations are defined to achieve a general applicable equation (Radial Diffusivity Equation) for

a specific flow regime. These equations are commonly referred to as field equations.

To find the solution to the Radial Diffusivity equation, three different approaches are used. These approaches

are the Self-Similarity Solution, the Ei-Function Solution and the Numerical Solution.

Finally, these three solutions are validated using real operational field data parameters. The aim is to obtain a

method that provides sensible solutions with a high degree of accuracy. The results are compared and the

most stable and reliable solution is selected for the Mini-Fracture Simulator.

4.1 Fluid Properties

This section defines the properties of the fluid used to create a fracture. It will be also analyse their fluid flow

principles, discuss the measurement of their rheological properties and define the rheological models that will

be implemented in the simulator. To build an integrated simulator model, fluid commonly used in Mini-

Fracture jobs will be considered.

The fluid commonly used in Mini-Fracture jobs is drilling fluid. To derive the mathematical expression that

describes the rheology of this fluid, the isothermal compressibility and the strain-stress relation of the fluid

were taken into consideration.

4.1.1 Fluid Compressibility

Drilling mud and crude oil used for Mini-Fracture tests are slightly compressible fluids. The isothermal

compressibility of fluids is given by the following expression (Dake, 1978):

1 ii

i T

Vc

V p

(4.1)

This equation can be integrated to express the isothermal compressibility as:

(Pr )fc ef P

refV V e

(4.2)

where Vref and Pref are the reference volume at initial pressure, respectively and Cf is the isothermal

compressibility factor of the fluid.


19

4.1.2 Rheological Properties

Based on their rheological properties, fluids can be divided into two main classes, namely Newtonian fluids

and non-Newtonian fluids (Bird, 1987). The viscosity of Newtonian fluids depends on temperature but is

independent of flow rate and strain rate. Therefore, Newtonian fluids are characterized by a linear stress-

strain relation (See Figure 4.1), which can be expressed as:

xyx

du

dy

(4.3)

where τyx is the shear stress exerted by the fluid, μ is the fluid viscosity, and xdu

dy is the velocity gradient

perpendicular to the direction of the strain rate.

Non- Newtonian fluids on the contrary are characterized by a non-linear strain-stress relation (Tarek, 2001).

Therefore the viscosity is a function of the shear rate. Several models are used to describe the behaviour of

non-Newtonian fluids. The most common models are Power Law Model, Bringham Plastics Model and

Herschel-Bulkley Model.

Drilling fluids present a complex molecular structure and are generally pseudoplastic in nature. They are

termed shear-thinning because of their tendency to decrease in viscosity as the shear rate increases. To

determine the model that best fits the drilling fluid characteristics, six values of shear rate and shear stress

from mud viscometer analysis (3, 6, 100, 200, 300, 600 rpm) were used. The mud data was gathered from a

well where Mini-Fracture job had been performed. See Table 4.1

Table 4.1 Mud rheological properties used for modelling.

Viscometer speed [RPM] Shear Stress [lbf/100 ft2]

3 14

6 16

100 30

200 38

300 44

600 60

The three rheological models were simulated while a non-linear square routine was built to solve the non-

linear square curve required to fit the Herschel-Bulkley model to the real viscometer data, so that the

Herschel-Bulkley coefficients can be calculated. Figure 4.2 compares the fit of the viscometer readings with

the different rheology models.

From this analysis, it was concluded that the most accurate model to simulate the drilling fluid is the

Herschel-Bulkley model. It encompasses the yield behaviour of the Bringham model and also allows the

power law models for shear-thinning, giving a better fit to typical mud performance. The equation for a

Herschel-Bulkley model is:

m

yx HB HBk (4.4)

where τHB is the Herschel-Bulkley yield point, kHB is the Herschel-Bulkley consistency factor and m is the

Herschel-Bulkley flow behaviour index.


20

The consistency factor kHB describes the thickness of the fluid and is analogous to the apparent viscosity

(when the consistency factor increases, the drilling fluid becomes thicker). The flow behaviour index m

indicates the degree of non-Newtonian behaviour. Pseudo-plastic type of fluids are characterize by values of

m<1. Table 4.2 shows the Herschel-Bulkley coefficients values obtained from the previous analysis.

Table 4.2 Coefficients for Herschel-Bulkley model calculated using viscometer data.

Parameter Symbol Units Value

Yield point HB [lbf/100ft

2] 11.85

Consistency factor HBk [cP] 499.78

Flow behaviour index m [-] 0.55

Figure 4.1 Comparison between rheology models and mud viscometer readings.

The Herschel-Bulkley model clearly gives the best fit required to describe the viscosity for the drilling fluids.

We are now interested in calculating how the viscosity of the fluid changes when it is injected into the

formation at different velocities. This is described in the next section.

4.1.3 Porous Media Rheological Properties

The rheological behaviour of mud through porous media can be expressed in terms of the “apparent

viscosity” of the fluid. Blake-Kozeny modelled a fluid behaviour in porous media using the power law model

(Bird, 1987). The apparent viscosity can be expressed as:

1m

app plH u (4.5)


21

where u is the superficial fluid velocity Hpl is the Blake–Kozeny coefficient and m is the flow behaviour

index.

Now to calculate the apparent viscosity using Herschel-Bulkley equation, we need to modify equation 4.5 to

integrate the Herschel-Bulkley yield point. Equation 4.5 can be re-written as:

1mHBapp pl

eq

H u

(4.6)

where eq

is the shear rate given in [1/s]. And it can be estimated as (Hirasaki and Pope 1974):

1 3

8eq

m u

m k

(4.7)

If the coefficients τHB and m are calculated from the viscometer data using Herschel-Bulkley, then Hpl can be

estimated as:

1

29 3

15012

m nHB

pl

K mH k

m

(4.8)

where k and φ are the permeability and porosity of the formation.

In the Mini-Fracture simulator, Newtonian and non-Newtonian models are incorporated even though it is

recommended to use the non-Newtonian model, since it provides the most accurate representation of the

reality. This is done with the finality of introduce a single value of viscosity in the simulator, in case that no

viscometer data is available.

4.2 Physical Model

Figure 4.2 shows a sector of the near-wellbore model. We considered radial flow in a homogeneous and

isotropic reservoir which is initially fully saturated with a single fluid. Gravitational effects are negligible and

the fluid is injected with a uniform pressure.

Since the injection volumes and times are small this study will consider transient flow regime, for which

pressure is also function of time.

Injection of the fluid creates a pressure disturbance which propagates away from the wellbore at a rate

determined by the permeability, porosity, fluid viscosity and total (rock and fluid) compressibility. The

transient flow regime describes the period, when the outer boundary has no effect on the pressure behaviour

in the reservoir and the reservoir will behave like it is infinite in size (Tarek, 2001).

This regime fits perfectly in the model, as the amount of fluid injected during a Mini-Fracture job is small (≈

0.2 m3), making the pressure disturbance to moves only a short distance away from the wellbore such that it

does not reach the reservoir boundary.


22

Figure 4.2 Ideal radial flow reservoir model.

4.3 Governing Equations

The development of the fundamental equations for flow modelling in near-wellbore reservoirs starts with the

continuity equation and the Darcy’s law. Equation 4.9 provides the principle of conservation of mass in 1D

radial flow.

1 k pr

r r r t

(4.9)

where k is the permeability, µ the fluid viscosity, ρ the fluid density and φ the porosity.

By introducing the isothermal compressibility term and assuming the compressibility to be small we derive

the radial diffusivity equation for radial 1D flow (Tarek, 2001).

1 effcp pr

r r r k t

(4.10)

The solutions to the diffusivity equation are designed to provide pressure changes throughout the radial

system, knowing the flow rate that has been injected at each moment (i.e., at the injector well). In other words

pressure and its time gradient are functions of position and time.

Several solutions have been proposed to solve this equation. In the next section will be explained the different

solutions tried in the simulator, their validities to represent a stress test job and the best fit solution selected

for the Mini-Fracture simulator. The detailed derivation of these solutions is presented in the Appendix 11.2

and 11.3 of this document.


23

4.4 Boundary Conditions

The above equation is solved subject to certain initial and boundary conditions (Tarek, 2001). In this study it

is assumed that initially the reservoir is at an uniform pressure Pi, or P(r,0)= Pi. The boundary condition

states that there is no flow across the outer boundary and that the reservoir behaves as if it is infinite in size

re=∞, for all t at r=∞. The line source inner boundary condition is also given as:

For t>0 0

lim2r

p qr

r kh

(4.11)

4.5 Self-Similarity Solution

Due to the implicit pressure dependence on the density, viscosity and compressibility, the diffusivity equation

is a partial differential equation that can be solved using the Boltzman transformation (Carslaw et al., 1946).

We seek a self-similar equation by defining a new variable ,f r t such that the pressure is expressed in

terms of independently and not anymore expressed in terms of radius r and time t. The partial differential

equation will thereby be converted into an ordinary differential equation which can be solved analytically. To

this end we used the he Boltzmann transformation defined as:

2

4

effc r

k t

(4.12)

In terms of the Boltzmann variable, the equation of motion can be rewritten,

0p p

(4.13)

where p is the pressure in Boltzmann coordinates. The initial and boundary conditions must also be rewritten in

terms of . In terms of this variable the initial condition and the infinite reservoir boundary condition converge to

a single boundary condition for the transformed equation (Fair, 1992). Also, the line source approximation is

needed so that the other boundary condition is not depending on radius.

0lim 2

2

p q

kh

(4.14)

Applying the above boundaries conditions, carrying out with the calculations and rearranging the terms yields

an ordinary differential equation that can be introduced into the diffusivity equation for further integration.

exp

4

p q

kh

(4.15)

By replacing back the definition of the equation 4.16 is obtained.


24

2

'ln4 4

eff w

w i

c rqp p

kh kt

(4.16)

Note that the equation 4.16 is only valid for terms:

2

0.014

effc r

kt

and ' =exp( ). (4.17)

where is defined as Euler constant =0.5772

4.6 The Ei-Function Solution

Matthews and Russell (1967) proposed a solution to the diffusivity equation that is based on the same

assumptions explained in section 4.2.

294870.6

( , )eff

i i

c rqp r t p E

kh kt

(4.18)

In which is introduced the mathematical function Ei that is call exponential integral and is defined by:

2 3

( ) ln1! 2(2!) 3(3!)

u

i

x

e du x x xE x x

u

(4.19)

The Ei solution is commonly referred to as the line-source solution. When its argument x is less than 0.01, the

exponential integral Ei can be approximated by the following equation:

( ) ln(1.781 )iE x x (4.20)

where the argument x is given by:

2948 effc r

xkt

(4.21)

Equation 4.18 approximates the Ei function within less than 0.25% error. To approximate the Ei function for

values of range 0.01<x<10 the equation 4.22 given below is used.

2 3 2 3

1 2 3 4 5 6 7 8( ) ln( ) ln( ) ln( ) /iE x a a x a x a x a x a x a x a x (4.22)

with the coefficients a1 through a8 having the following values:

a1=-0.33153973 a2=-0.81512322 a3=5.22123384(10-2

) a4=5.9849819(10-3

)

a5=0.662318450 a6=-0.12333524 a7=1.0832566(10-2

) a8=8.6709776(10-4

)


25

The above relationship approximates the Ei values with an average error of 0.5%. For values of x> 10, Ei(x)

can be set to 0 for all practical reservoir engineering calculations.

4.7 Numerical Solution

To develop the numerical model we considered the 1D radial geometry presented in section 4.2 where r ϵ

[rw,re]. In this geometry, the pressure in each cell can be identified in the radial coordinate. The flow domain

is discretized in nr equidistant grid-blocks, and assigning average pressure Pi to the centre of each grid block i

(Vuik, 2007).

Figure 4.3 Discretization of radial symmetry model.

For simplification and convenience the radial grid was uniformly spaced although a grid refinement in the

immediate vicinity of the well will result in a finer estimation of the pressure profile. The numerical scheme

is now obtained by approximating the derivatives in 4.10 by central difference quotients.

12

1,..., e wi r

r

r rr i r i n r

n

(4.23)

The time is discretised in identical time step of t seconds.

, 0,1,2,...t i t i (4.24)

To describe the pressure propagation throughout the grid, pressures at the different grid-blocks are evaluated

at times t=[n+1]t which numerically simulate the upwind direction in a flow field. Finally, the boundary

conditions are introduced to evaluate the first and the last grid block and express a solution in terms of

vectors and matrix. For n=0,

0

initpp (4.25)

The pressure is equal to the initial reservoir pressure vector set by the initial condition. For n>0,

11n nt p D A Dp e (4.26)

Here P is the pressure vector, D and A are column vector and e is the column vector accounting for the

boundary conditions, with the matrix entries given by:

(4.27)

(4.28)

(4.29)

, ,i i i i

ii i

i i

a

d

e

A

D

e


26

4.8 Models Predictions

To check the validity of the analytical and numerical solutions to the radial diffusivity equation, the three

solutions were modelled using Matlab. Table 4.3 summarized the parameters and the range of values

commonly measured during Mini-Fracture jobs.

Table 4.3 Parameters used to test models predictions.

Parameter Symbol Units Low Value High Value

Viscosity μ [cP] 1 100

Interval H [m] 1 3

Well Radius rw [m] 0.10 0.15

Reservoir Radius re [m] 1 10

Flow Rate Q [cc/sec] 1 30

Time t [sec] 100 1000

Permeability k [mD] 0.1 500

Porosity [%] 5 30

Compressibility Ceff [1/Pa]x10^10 3 5

4.8.1 Self-Similarity Prediction

Using the Boltzmann transformation, the analytical solution of the diffusivity equation was found. Due to the

non-linear character of the problem, two scenarios were modelled. In the first scenario the parameters were

chosen such that condition 4.17 is met. In the second scenario the parameters of time and permeability were

changed to create a condition such as criterion 4.17 is violated.

Figure 4.4 shows the pressure profile modelled over 10 m of reservoir length and 500 seconds of time

iteration when condition 4.17 is met. It is observed that for long-term predictions the solution gives stable

results throughout the reservoir.

Figure 4.4 Self-Similarity result, first scenario.


27

Figure 4.5 shows the result of the pressure profile for the case where condition 4.17 is not met. It is noticed

that at initial times and long distance from the well the pressure in the reservoir is lower than the initial

pressure Pi= 3000 psi. This clearly demonstrates the important role that the time component plays in the

accuracy of the approximate self-similarity analytical solution.

Figure 4.5 Self-Similarity result, second scenario.

The above examples had shown that the condition for the linearization on equation 4.10 limits the

applicability of the solution if the non-linearity is too large. If this analytical solution is build in the Mini-Frac

simulator, the results will be constrained by the non-linearity range. The pumping time during a Mini-

Fracture job is usually low in the range of hundreds of seconds, and the permeability values can be extremely

small when tests are performed in shale or low permeable areas. For this reason, a more general method

should be implemented in the simulator.

4.8.2 Ei-Function Prediction

Figure 4.6 shows a 3D plot of the pressure versus time and radius obtained using equation 4.18. The data

clearly show that the Ei-Function solution is unstable (i.e., the pressure fluctuates), at the initial injection

times but with oscillation decreasing over time. This scheme notoriously suffers from oscillations which can

produce non-physical results near steep gradients. This demonstrates that the Ei-function prediction is

sensitive close to the wellbore in transient conditions.

For large time steps, the instability was still presented at initial times. Figure 4.7 also shows fluctuation at

initial times although the number of time steps is increased by 10 times larger than in the previous example.

Loss of accuracy at initial times seems inevitable consequence of using the Ei-function solution.

From this example, it is observed that reservoir pressure decreased as the radius of investigation increased,

but compared to the Self-Similarity solution this method does not show pressures below the initial reservoir

pressure Pi. This means that the Ei-Function solution is not sensitive to the reservoir parameters in the

equation.


28

Figure 4.6 3D plot of unstable Ei-Function solution.

Figure 4.7 3D plot of unstable Ei-Function solution, increasing number of time steps.

It can be concluded that the Ei-function is unsuitable for our purpose, because it presents unwanted

oscillations at initial times and distances close to the wellbore, where the pressure gradient undergoes a rapid

change. Therefore the general numerical solution needs to be evaluated to solve the ordinary differential

equation that describes the radial flow in porous media.


29

4.8.3 Numerical Prediction

Figure 4.8 shows the pressure distribution on the near-wellbore reservoir, predicted using the numerical solution

to the diffusivity equation. This model provides a more consistent characterization of the pressure distribution

through the reservoir with a high degree of accuracy. It shows a stable solution for short times and short

distances from the wellbore. Also the numerically predicted pressure distribution gradually builds-up with

time and decrease with distance from the wellbore. No pressure fluctuations with time were observed, so this

method is therefore useful for achieving a consistent result, by preventing the generation of spurious

oscillations in the solution. Furthermore the initial pressure condition is maintained along the whole

simulation.

Figure 4.8 Pressure in the reservoir using the numerical model.

Figure 4.9 shows the results of the numerical solution in a 3D plot of the pressure versus time and radius. We

can see from the plot how smooth the pressure curve builds-up until it reaches a constant pressure. No

physical inaccuracies or fluctuations are observed.

Also it shows stability to general variation in reservoir radius and injection times. Furthermore this numerical

result is in agreement with the physical representation of the behaviour of the pressure in the reservoir.

Figure 4.10 shows the results of the numerical and analytical solution using a total injection time of 500 sec

and the same injection rate of 30cc/sec. The final pressure achieved by both models differs by 3 % only.

However the pressure profiles are different. Due to stronger non-linearity, the slope of the analytical

approximation tends to over predict the pressure build-up and hence result in a larger slope.

But there are still tradeoffs using this method. In particular, the more detailed the mathematical model is, the

slower the computer can calculate a solution. Therefore, we must seek a balance between the accuracy of the

computation and the time we want to spend on simulation. Finally, because the numerical model is reliable

and robust, it will be implemented in the final Mini-Fracture simulator.


30

Figure 4.9 3D plot of numerical model simulation.

Figure 4.10 Comparison between analytical and numerical solution.


31

5 Rocks Mechanics Modelling

This chapter is devoted to the geomechanical modelling of the near-wellbore rock failure mechanics. First the

basics of rocks mechanics terminology will be recalled and the basic fundamentals of stress analysis in

underground formations will be reviewed. Moreover, the workflow process to determine the breakdown

pressure magnitude starting with Wireline logs will be introduced. Each step in the workflow is explained

including the necessary equations required to calculate the stresses around the wellbore. The effect of highly

inclined wellbore in rock mechanics will be explained in detailed. Finally, the tensile failure criterion is

introduced and the necessary pressure required to breakdown the formation is calculated.

As in the previous chapters, these concepts will be modelled in Matlab and evaluated with real field data to

calculate the fracture initiation pressure. Once the rock mechanics model is introduced in the simulator, the

breakdown pressure will set the objective for the MDT tool to create a fracture.

5.1 Terminology and Rock Mechanics Concepts

Rock mechanics is concerned with the application of Newtonian mechanics (i.e. statics and dynamics) to

study rocks in the ground. In particular, it is concerned with how the rock behaves in response to disturbances

and alterations caused by excavation, changes in stress, fluid flow, temperature changes, erosion, burial and

other phenomena (Economides et al, 1990).

5.1.1 Basic Concepts

Stress is defined as the force per unit area on a surface and is notated by the symbol σ.

( )

( )

Force F

Area A (5.1)

Two types of stresses i.e. Normal and Shear stress can act on a surface. If the force acts perpendicular (i.e.

normal) to the face or plane being considered, it is defined as a normal stress and is notated by the symbol σ.

On the other hand, if the force acts parallel to the face or plane being considered, it is defined as a shear stress

and is denoted by the symbol τ. Both stresses are important since normal stress acting on a rigid body can

lead to the generation of shear stresses in another plane. By convention, the stress with greatest magnitude

(major principal stress) is denoted by σ1, the minimum magnitude stress (minor principal stress) is denoted by

σ3, and the intermediate magnitude stress (intermediate principal stress) is denoted by σ2 (Fjaer, et al., 2008).

Another important concept in rock mechanics is strain. When a body is subjected to a stress field, the relative

position of points within is altered (the body deforms). If these new positions of the points are such that their

initial and final locations cannot be made to correspond to a translation and/or rotation, the body is strained

(Carnegie et al., 2002). If the undergoing deformation is recovered after the stress is removed, it is defined as

an Elastic Strain otherwise it is defined as a Plastic Strain and is denoted by the symbol ε.

*l l

l

(5.2)

where l is the original length and *l is the resultant length after the stress is applied.


32

To characterize rocks properties, two proportionality coefficients are defined. The first coefficient is the

Young’s Modulus which is a coefficient of proportionality between the normal stress applied to a body and

the strain related to it when the body undergoes elastic deformation. It is denoted by the symbol E (Fjaer, et

al., 2008).

( )

( )

NormalStressE

Strain

(5.3)

The second coefficient is the Poisson Ratio and is defined as the ratio of lateral expansion to longitudinal

contraction. It is denoted by the symbol υ.

2

(5.4)

where 2

*d d

d

been d the diameter before expansion and d* the new diameter.

5.1.2 Stresses in Undisturbed Ground

From a practical rock mechanics point of view, the present state of a rock and mechanical properties are very

important and hence of interest in this study. All the geological activities and events such as sedimentary

subsidence, sea level change and tectonic forces will affect not only current rocks mechanics properties, but

also in-situ stresses and pore pressures of the rock (Jaeger, et al 2008).

Underground rocks are subject to three principal stresses that are unequal and can come from different

sources. The three principal undisturbed underground stresses are the vertical or overburden stress σv which

generally has the greatest magnitude, the maximum horizontal stress σH which is the intermediate in-situ

stress and minimum horizontal stress σh which generally has the lowest magnitude (Jaeger, et al 2008). Figure

5.2 shows a representation of the three undisturbed ground principal stresses acting over a solid body.

Figure 5.1 Stressed in undisturbed ground (Zacharia, 2007).


33

5.1.3 Pore Pressure

The pore spaces of sedimentary rocks usually contain fluids such water, oil and gas. The pressure exerted by

these fluids is call pore pressure (Jaeger, et al 2008). One refers to the pore pressure at depth D as normal if it

is given by the weight of the fluid column above and is defined as:

0

( )

D

p fP z gdz (5.5)

where ρf is the density of the material g is the acceleration of gravity and z is the height of the fluid column.

The pore fluid density in case of Brine with sea water salinity is in the range 1.03-1.07 g/cc, so the pore

pressure increase with depth is roughly 10MPa /Km.

5.1.4 Effective Stress in a Rock

Generally the effect of the vertical and horizontal stresses on a rock is reduced by the presence of the pore

pressure. This is because the pore pressure acts in all directions within the rock, and helps to support or

relieve some part of the applied stresses, which are transmitted through the rock via grain-contacts and

cementation. The remaining stresses that are transmitted and are not relieved from the rock by pore pressure

are defined as effective stresses and they are responsible for the deformation and failure of a rock (Fjaer, et

al., 2008).The effective stresses in a rock are thus given by:

Vertical Effective Stress: σ’v=σv-αPp (5.6)

Maximum Horizontal Effective Stress: σ’H=σH-αPp (5.7)

Minimum Horizontal Effective Stress: σ’h=σh-αPp (5.8)

The coefficient is the Biot Constant. This constant defines the efficiency with which internal pore pressure

offsets the externally applied vertical total stress. When decreases, the net inter-granular stress increases,

so that the pore pressure variations have less impact on net stress. A close approximation of is taken to be

equal to one (Fjaer, et al., 2008).

Now that the basic geo-mechanics terminology has been introduced, we proceed with the explanation of the

steps required to compute the breakdown pressure starting from wireline log data. These logs will be used as

an input for the simulation.

5.2 Estimation of Breakdown Pressure

To predict the pressure required to initiate a fracture in the formation, an extensive stress profile modelling is

developed. This analysis starts with the integration of data from wireline logs and laboratory core analysis to

calculate all necessary parameters required to compute the Tensile Failure criteria. This section discusses the

most elementary and well knows models for rock failure. Also note that the models used for rock failure

analysis are simplified descriptions of the behaviour of real rocks. Rock failure is a complex process which is

still not fully understood. Much of the framework used to handle the rock failure is based on convenient

mathematical description of observed behaviour rather than derivations from basic laws of physics.


34

To simplify the analysis further is assumed that rocks are homogeneous and isotropic and have a uniform

wellbore pressure profile. The workflow of the process developed in this project to determine the stress

profile also defined as 1D geomechanical model is provided below. These same calculations and workflow

are used as a base to create the geomechanical model in the Mini-Frac simulator.

Figure 5.2 Breakdown pressure calculation workflow.

( , )

( , )

v

h

H

, ,y x z

r

z


35

5.2.1 Input Data

Evaluation of in-situ rock mechanics behaviour requires relevant input data. Our main data sources are

wireline logs and core analysis. Wireline logs provide semi-empirical relationships between rock properties

and stress existing in the near-wellbore face. Sonic based measurements are perhaps the best known logging

technique for obtaining stresses. Knowledge of sonic velocities allows important rock moduli such as

dynamic Poisson’s ratio to be inferred and hence a rough estimation of horizontal stress can be calculated.

The most relevant wireline logs that need to be acquired for rock mechanics analysis are presented in Table

5.1

Table 5.1 Relevant Wireline logs description.

Wireline Log Description

Acoustic Log

The Primary output from acoustic logging tools is the sonic velocity and

waveforms, such as the compressional, shear and stonely waves.

Density Log

The density tool uses an active gamma ray source to measure the electron density of

the formation. By appropriate lithology corrections, the electron density is

converted to mass density. The density is needed to convert from acoustic velocities

to dynamic elastic moduli. Secondly the density integrated over the vertical depth of

the well is usually a good estimation of the vertical stress.

Pressure Log

Pore pressure can be measured directly with the MDT tool, creating a pressure

gradient and estimating the type of fluid that is present in the formation. Pore

pressure can also be estimated by the weight of the fluid column above it.

Porosity Logs

Neutron porosity measurement uses a neutron source to measure the hydrogen

index in a reservoir, which is directly related to porosity. The porosity values are

then used to calibrate the data obtained by core analysis.

Natural GR logs

The GR log is a recording/measurement of the natural radioactivity of the

formation. The GR values are used to calculate the volume of shale in the formation

and calibrate the data obtained by the core analysis.

The logs provide data recorded continuously versus depth, but some parameters need to be calibrated with

core analysis. Core provides a possibility of directly measure the rock strength parameters and static elastic

properties. There are different methods and techniques used to determine the in-situ stress from laboratory

testing of core samples. These techniques are mainly based upon performing a variety of strains experiments

such as differential strain curve analysis and inelastic strain recovery, on cores recovered from the well. The

interpretation of these results, allows an inference to be made on the stress directions and magnitude at the

locations from where the core was retrieved. The Interpretation is based on assumptions which cannot be

easily verified. For this reason it is advisable to correlate and calibrate this data with the mechanical

properties and parameters acquired from wireline logs (Fjaer et al., 2008). The most relevant mechanical

parameters retrieved from a core analysis are the unconfined compressive strength UCS, the static Young’s

modulus Efr, the static Poisson’s ratio υfr, and the angle of internal friction (Friction angle). The parameters

used in the simulator and its units are summarized in Tables 6.1 and 6.2.


36

5.2.2 Dynamic Young Modulus and Poisson Ratio

Dynamic rock elastic parameters (Young’s Modulus and Poisson’s Ratio) are determined using advanced

sonic measurements providing shear and compressional slowness. The density log is also needed for this

purpose. Dynamic Poisson’s ratio υ is derived from the following equation, where νp is the compressional

velocity and νs is the shear velocity.

2 2

2 2

2

2( )

p s

dyn

p s

(5.9)

Also the dynamic Poisson’s ratio can be expressed in terms of compressional delta time c and shear delta

time s

2

2

11

2

1

dyn

s

c

s

c

(5.10)

The dynamic Young’s modulus Edyn is determined using the following equation, where ρ is the formation

density:

2 2 2

2 2

(3 4 )s p s

dyn

p s

E

(5.11)

Also the dynamic Young’s Modulus can be expressed in terms of compressional delta time c and shear

delta time s

2

2(1 )dyn dyn

s

E

(5.12)

Shear modulus G and Bulk modulus K are also related to the density of the formation and they can be

calculated by the following equations:

2dyn

s

G

(5.13)

2 2

1 4

3dyn

s

Kc

(5.14)

Once the dynamic moduli are known, the next step is to relate them to the static parameters. The difference

between static Young’s modulus and the dynamic Young‘s modulus may be quite significant, in particular at

low stress level. The Dynamic modulus may be several times larger than the static modulus e.g. in a weak

sandstone (Chardac et al., 2005). For this reason, the dynamic moduli should be calibrated with the static

moduli obtained from the core data.


37

5.2.3 Calibration of Dynamic Parameters

Elastic properties computed from the logs are termed “dynamic” (νdyn, Edyn, Gdyn, Kdyn). Logging data is used

to estimate these parameters. Because wellbore deformation is a relatively slow process compared to high

frequency wave propagation, “static” data sets from laboratory core tests are used as geomechanical

references. Therefore “dynamic” elastic property logs are usually calibrated against single point “static” core

measurements.

A number of public domain and proprietary correlations are available for computation of static elastic moduli

from dynamic values (Fjaer et al., 2008). Similarly, since rock strength is estimated from empirical

relationships, it is advisable to calibrate the log-derived UCS against the static data obtained from core. In the

Mini-Frac simulator, static data is introduced as a main input to compute the geomechanical profile.

5.2.4 Vertical or Overburden Stress Estimation

It is very common and convenient in the oil industry to assume that the vertical stress is the principal stress.

This is reasonable at large depths within a homogeneous earth, in areas that had not been exposed to tectonic

activity or are relaxed. In normal conditions the underground formation is subject to the weight of underlying

formations. The vertical stress σv or overburden stress at the bottom of a homogeneous column of height z is

given by:

v gz (5.15)

Here is the density of the material and g is the acceleration of gravity. If the density varies with depth the

vertical stress at depth D becomes.

0

( )

D

v z gdz (5.16)

Note that the z-axis here is pointing vertically downwards with z=0 corresponding to the earth surface. The

average density in the overburden is between 1.8 and 2.2 g/cc. This density values can be calculated with the

help of density logs (Cantini et al., 2010).

5.2.5 Minimum Horizontal Stress

At this point the minimum stress can be determined by choosing the appropriate model. Two models were

implemented in the simulator. The first model states that the stress regime of the rock is under uniaxial strain

conditions. To estimate the state of stress that is generated under this regime, it is assumed that the rock is a

semi-infinite isotropic medium subjected to gravitational loading and no horizontal strain. With uniaxial

strain assumed, the other two principal stresses are equal and lie in the horizontal plane. If they are written in

terms of effective stress, they are a function of only the overburden (Engelder, 1993).

(5.17)

where 0k is the coefficient of earth pressure at rest and is defined as:

(5.18)

0' 'h vk

01

k


38

and σ’h is the minimum effective horizontal stress. The minimum stress can be re-arrange in terms of pore

pressure Pp, the overburden stress v , the Biot’s constant α, and the Poisson’s ratio υ.

(5.19)

However, stress predictions using these assumptions must be used with great caution as there is no horizontal

strain anywhere in the equations. Nevertheless, they are useful for understanding the state of stress in the

earth and can be used as a reference (Engelder, 1993).

The second method employed to calculate the minimum horizontal stress is called Poroeslastic method. With

this method, tectonic stresses and the notion of strain is introduced, which is a quantity added to or subtracted

from the horizontal strain components (Li et al., 2010). If incremental tectonic strains are applied to rock

formations, these strains add a stress component h in an elastic rock as follows:

2 21 1h h H

E E

(5.20)

where h and H are the tectonic strains in the direction of h and H respectively. Now the minimum

horizontal stress can be defined as:

2 2( )

1 1 1h v h H

E EPp Pp

(5.21)

5.2.6 Maximum Horizontal Stress

The maximum horizontal stress can be determined from the minimum horizontal stress, vertical stress, pore

pressure and Poisson’s ratio. Current maximum horizontal stress prediction from wellbore breakout is based

on the maximum tangential stress on the wellbore wall being equal to the rock uniaxial compressive strain.

We assume that the vertical, minimum and maximum horizontal stresses, define a specific relation when the

stresses in the formation are in equilibrium. Based on a generalized Hooke’s law with coupling the

equilibrium of stresses and pore pressure, the maximum horizontal stress can be solved using this relation.

2hH v

PpPp

(5.22)

The three principal stresses σH, σh and σv should satisfy Hooke’s law in order to keep the stress-strain

equilibrium. According to Hooke’s law, the maximum horizontal stress can be written in the following form

(Fjaer et al., 2008):

2 2( )

1 1 1H v h H

E EPp Pp

(5.23)

( )1

h v Pp Pp


39

5.2.7 Stresses around Boreholes and Failure Criteria.

In this section an overview of in-situ stresses assessment is presented. Furthermore, we will drive the

equations necessary to calculate the critical breakdown pressure required to induce a tensile failure in the

formation. This will be achieved by implementing Mohr-Coulomb rock failure criteria.

When a well is drilled into a formation, stressed solid material is removed. The wellbore wall is then

supported only by fluid pressure in the hole. As this fluid pressure generally does not match the in-situ

formation stress, there will be stress redistribution around the well. To examine these stresses, we need to

express them in cylindrical coordinates (Fjaer et al., 2008). The equations for stresses around a circular

opening were first published by Kirsch 1898, where the opening is assumed to be parallel to a principle stress

axis (Charlez, 1991). The circumferential (hoop) stress in the proximity of the borehole is generally the

major principal stress and is defined as:

2 2 2 2 4 4' ' ' '' (1 / ) ( ) / (1 3 / )cos 2

2 2

H h H hR r Pw Pp R r R r

(5.24)

The longitudinal stress z in the proximity of the borehole is generally the intermediate principal stress.

'z v Pp (5.25)

The radial stress r in the proximity of the borehole is generally the minor principal stress. This is generally

the wellbore pressure due to the fluid column at that point.

2 2 2 2 2 2 4 4' ' ' '' (1 / ) ( ) / (1 4 / 3 / )cos 2

2 2

H h H hr R r Pw Pp R r R r R r

(5.26)

Theses equations define the effective radial and hoop stresses as a function of the principal effective stresses,

the radial distance r (r = R at the wellbore wall), the direction angle , borehole fluid pressure Pw and the

pore pressure Pp . Figure 5.4 shows the stresses distribution around the wellbore wall.

Figure 5.3 Stress at borehole wall in the presence of far field stresses and pore pressure (Zacharia, 2007).


40

At the wellbore radius R, the effective principal stresses can be represented by:

(5.27)

(5.28)

(5.29)

Equation 5.27 shows that the tangential stress at the borehole wall varies between the maximum value,

max' , (3 ' ' ) ( )H h w pP P (5.30)

and the Minimum value

min' , (3 ' ' ) ( )h H w pP P (5.31)

Here the maximum value occurs in the direction of 'h and the minimum value in the direction of 'H as

show in Figure 5.5

Several authors argued that vertical hydraulic fractures will always propagate perpendicular to the orientation

of the least horizontal principal stress (Hubbert and Willis, 1957). In Figure 5.6, it is observed that tensile

failure will occur at the position where is smallest and hence fracturing will occur in the direction parallel

of maximum horizontal stress (Fjaer et al., 2008).

Figure 5.4 Minimum and maximum stress distribution at wellbore wall.

Implementing the maximum tensile strength criterion, a tensile fracture is assumed to form at the borehole

wall and propagate into the formation if any of these principal stresses becomes sufficiently tensile (negative)

i.e. 0T , where 0T is the tensile strength of the rock. This means that equation 5.31 can be rewritten as:

0 (3 ' ' ) ( )h H w pT P P (5.32)

' ( ' ' ) ( ) 2( ' ' ) cos 2

' ( )

'

H h w p H h

r w p

z v p

P P

P P

P


41

If we consider the application of Terzaghi’s effective stress concept where the effective stress is defined as

the total stress minus the pore pressure, the maximum horizontal principal stress is determined by:

03( ) ( )w h p H p pP P P P T (5.33)

This can be simplified and rewritten as

03w h H pP P T

The criterion for fracture will usually be met only when the wellbore fluid pressure reaches the value of Pw

referred to as the breakdown pressurefract

wP .This is the pressure that the MDT tool needs to apply to the

formation to be able to initiate a hydraulic fracture.

03fract

w h H pP P T (5.34)

As shows in Figure 5.6, once this pressure is reached, a fracture which abruptly propagates into the formation

is formed. Fluid from the borehole then rushes into the newly created fracture, causing an instantaneous drop

in the wellbore pressure.

Figure 5.5 Tensile failure and fracture propagation in vertical wells.

5.2.8 Non-vertical borehole stress analysis and failure criteria.

So far we have seen stresses acting in a well that has an axis in the direction of the vertical principal stress.

Now it will be construct the solution in an inclined borehole.

First the in-situ principal stresses , ,v h H associated with the co-ordinate system (x,y,z) as illustrated in

Figure 5.7 should be transform to another co-ordinate system (x’,y’,z’), to conveniently determine the stress

distribution around a borehole. In this new coordinate system, the z-axis is parallel to the borehole axis, the x-

axis is parallel to the lowermost radial direction of the borehole, and the y-axis is horizontal (Chenevert et al.,

1988).


42

Figure 5.6 Stress coordinate system for deviated borehole (Fjaer et al., 2008).

Using the new co-ordinate system (x’,y’,z’), the transform of the in-situ stress field on the borehole becomes:

(5.35)

(5.36)

(5.37)

(5.38)

(5.39)

(5.40)

where ' ' '' , ' , 'x y zz are the normal and parallel effective stresses on the new co-ordinate system.

' ' '' , ' 'xy yz xy are the shear stresses in the new co-ordinate system, is the well inclination, is the well

azimuth and is the angular position around the wellbore. The stresses at borehole wall are calculated with

the following relations:

(5.41)

(5.42)

(5.43)

Now that we had characterized the stresses around the wellbore in vertical and deviated boreholes, we need to

introduce the tensile failure criteria to calculate the necessary breakdown pressure required to create a

continuous fracture in the wellbore wall. As discussed in section 5.2.7 a tensile failure occurs when the least

effective principal stress exceed the rock tensile strength i.e. 0T this means that:

2 2 2 2

'

2 2

'

2 2 2 2

'

'

'

2 2

'

' ( 'cos 'sin )cos sin

' ( 'sin 'cos )

' ( 'cos 'sin )sin cos

' 0.5( ' ') sin(2 )cos

' 0.5( ' ') sin(2 )sin

' 0.5( 'cos 'sin )sin(2 )

x h H v

y h H

zz h H v

xy H h

yz H h

xy h H v

' ' ' ' '

' ' ' '

'

' ' ' 2( ' ' ) cos(2 ) 4 sin

' ' 2 ( ' ' ) cos 2 4 sin

r w

x y w p x y xy

z zz x y xy

P

P P


43

0 ' ' ' ' '' ' 2( ' ' )cos(2 ) 4 sinx y w p x y xyT P P (5.44)

However the angle around the wellbore is also a variable so we must find the critical value for this angle.

Differentiating the principal stress with respect to and setting the resultant equals to zero, the minimum

principal stress is found at 0 for the case of equal horizontal in-situ stress. Introducing this in the

previously defined equation we obtain:

0 ' ' ' '' ' 2( ' ' )x y w p x yT P P (5.45)

When the wellbore pressure is solve, it becomes

' ' 03 ' 'w y x pP P T (5.46)

If we consider the application of effective stress concept, the equation 5.46 is written as:

' ' 03w y x pP P T (5.47)

Which give us the breakdown pressure fract

DEV wP necessary to create a fracture in the formation on deviated

wells.

' ' 03fract

DEV w y x pP P T (5.48)

5.3 Simulation of rock mechanics model

After understanding each step in the workflow process required to calculate the breakdown pressure, a Matlab

geomechanical simulator was created to replicate this process and predict the pressure necessary to initiate a

fracture at the wellbore wall. As in the previous Matlab scripts real field data was introduced to validate its

predictive capability. Introducing wireline log and core data as variables, the Matlab geomechanical simulator

computes the parameters summarized in Table 5.2.

Table 5.2 Output of geomechanical simulator.

Parameter Unit

Vertical or overburden stress [psi]

Pore pressure [psi]

Tensile strength [psi]

Minimum and maximum horizontal stress [psi]

Volume of shale -

Breakdown pressure [psi]

Porosity [m3/m

3]

Formation density [g/cc]

Sonic compressional and shear velocity [ft/us]


44

The solution was validated against fully implicit commercial Mechanical Earth Model (MEM) software

provided by Schlumberger. In this commercial simulator the mechanical properties of the rock were derived

from open-hole logs and were calibrated with laboratory testing results after which a wellbore stability

analysis was done, in order to predict the stresses around the wellbore area. Finally, a failure analysis was

done base on Mohr-Coulomb failure criteria, and validated with data acquired during drilling excavation

process. Overall the MEM used to validate the solution fulfils the requirements needed as a corroboration

tool, which means that the results obtained from the Matlab script should be similar. This validation and the

detailed description of the Matlab geomechanical simulator are explained in the following chapters.

Here below finds an example of the output deliver by the Geo-Mechanical Simulator:

Figure 5.7 Example of Geo-Mechanical simulator output.


45

6 Mini-Fracture Simulator User Interface

This chapter explains in detail the hierarchical program of the Mini-Fracture Simulator flow structure

developed in this study. To create the simulator, each sub-system related to the entire process was modelled.

This involves a detailed modelling of each part separately at various levels and joining them together to

create the main structure.

The main program is composed of four modules. These are MDT Tool Performance Module, Wellbore Flow

Module, Near-Wellbore Reservoir Module and Geomechanical Module. For each module, the input

parameters, process, and output solution are described. In addition the link between modules and user

interface is illustrated. Finally, the outcome of the Mini-Fracture simulator is discussed.

Each module is integrated to the main core program by using higher-order functions to generate a final

solution. This main core program also works as an interface between the input data provided by the user and

the variables needed in each module.

The final output solution is also built in the main module. The output consists of various displays to evaluate

whether the MDT tool is able to breakdown the formation or not. The Mini-Fracture simulator structure with

each module and their specific parameters is shows in the diagram on Figure 6.1.

Figure 6.1 Mini-Fracture Simulator structure.

6.1 Input Data

The first step in the simulator consists on providing the state variables required in each module. All relevant

variables of the system are organized into two groups. Those which are considered as given and are not to be

manipulated (uncontrollable variable) and those which can be manipulated so that to come to a solution

(controllable variables).


46

6.1.1 Uncontrollable Variables

These input variables are mainly used to characterize the subsurface properties (i.e. petrophysical properties).

They are mainly obtained from wireline open-hole logs and laboratory data gathered from core analysis. In

this case a total of twenty two input variables are the uncontrollable variables. See Table 6.1

Table 6.1 Uncontrollable variables use in the Mini-Fracture simulator.

Input Variable Symbol Units

Borehole Depth Depth Meters [m]

True Vertical Depth TVDepth Meters [m]

Borehole Deviation from Vertical Dev Degrees [deg]

Borehole Azimuth Angle Azm Degrees [deg]

Formation Density Rhob Gram over cubic Centimetre [g/cm^3]

Gamma Ray Readings Gr American Petroleum Institute Units [API]

Sonic Compressional DeltaT Dtco Micro seconds over feet [us/ft]

Sonic Shear DeltaT Dtsm Micro seconds over feet [us/ft]

Porosity Poro Volume [m3/m3]

Permeability Perme Milli-Darcy [mD]

Unconfined Compressive Strength UCS Kilo Pascal [Kpa]

Static Young’s Modulus Dyme Giga pascal [Gpa]

Tensile Strength AT0 Kilo Pascal [Kpa]

Total Compressibility Factor Ceff 1/ Pascal [1/pa]

Fracture Fluid Viscosity Vis Centipoises [cP]

Borehole Diameter Dw Inches [in]

Formation Fluid density Fden Gram over cubic Centimetre [g/cm^3]

Initial Hydrostatic Pressure P_init Kilo Pascal [Kpa]

Strain in Shmin Direction Xs Milli-strain

Strain in Shmin Direction Ys Milli-strain

Leak off calibration Coeff. Cal1 Unitless [-]

Horizontal Stress Ratio Cal2 Unitless [-]

6.1.2 Controllable Variables

These variables are those that can be varied or controlled by the engineer who is using the simulator. They are

mainly related to the operational tool parameters and can be used to optimize the tool configuration. A total

of four parameters can be controlled by the user. See Table 6.2.

Table 6.2 Controllable variables use in the Mini-Fracture simulator.

Input Variable Symbol Unit

Interval Height H Meters [m]

Required depth for Mini-Fracture Depth_user Meters [m]

Type of Packer use Pack Unitless [-]

Type of Pump use Pump Unitless [-]

These parameters are entered into the simulator in two data sets. The first data set is entered by using an excel

file attach in the simulator package. Then the simulator reads this excel worksheet and converts the variables

to a matrix form that can be read by Matlab. An example of the excel input file is shown below.


47

Figure 6.2 Example input data worksheet.

The second sets of parameters are entered individually using the software interface. These parameters are

dynamic and mainly controlled by the user. They can be modified before run the simulation so as to visualize

how changes in tool configuration and tool positioning affect the result.

6.2 Software User Interface

Figure 6.3 shows the User Interface main window. This interface permits the operator to provide and manage

the input variables (controllable and uncontrollable). Moreover the graphical interface is efficient and user

friendly, which aids the engineer in making operational decisions. It guides the user throughout the

procedure, by following 4 steps to compute and display the results.

Figure 6.3 Mini-Fracture simulator, user-interface window.


48

6.2.1 Step # 1, Formation Parameters

In Step # 1, two windows emerged. The first window indicates the required variables related to the formation.

The second window shows a help message indicating how to calculate the initial pressure at the depth where

the MDT stress test is performed. See Figure 6.4.

Figure 6.4 Step # 1, Simulator formation parameters input windows.

6.2.2 Step # 2, Geomechanical Model

Figure 6.5 shows the process windows in Step# 2.The simulator requests the input of choice to compute the

geomechanical properties. The two options are uniaxial strain and poroelastic equation. Depending of the

selected model, a window indicates the required parameters. It also requests the interval in which the user

plans to display the geomechanical information.

Figure 6.5 Step # 2, Simulator geomechanical input windows.


49

6.2.3 Step # 3, Tool Parameters

Figure 6.6 shows the tool parameters required in Step# 3. The main window also provides an MDT tool

sketch in which the user can check the options available for pumps, packers and test intervals.

Figure 6.6 Step # 3 Simulator tool operational input windows.

6.2.4 Step # 4, Rheology Model

Next the simulator requests the input of choice for the rheological model to be used. Depending on the

selected rheological model, two windows will emerge indicating the parameters required to compute the

viscosity profile.

Figure 6.7 Step # 4 Simulator rheology parameters.


50

Once all the parameters are entered, the user can run the simulator by clicking the RUN bottom placed below

Step # 4. This allows each sub-routine to use the required parameter to compute a particular answer. The next

section details the workflow on each module and the link between modules to obtain the final result.

6.3 MDT Tool Performance Module

Figure 6.8 shows the detailed structure of the MDT tool performance module. This module intends to

quantify the flow rate delivered by the tool as a function of the pressure in the borehole. It also provides the

maximum allowed differential pressure information. Finally this model computes the maximum build-up

pressure that pumps can deliver. Depending on the type of pump selected, the model chooses a specific

routine that limits the maximum build-up pressure and the flow rate at which fluid is injected.

Figure 6.8 MDT Tool Performance Module structure.

The output of this function (pressure and flow rate) is used as an input in the Borehole Module and Near-

Wellbore Reservoir Module. Depending on the packer selected, the MDT tool module sets a maximum

differential pressure that is allowed in the interval. This value is used in the main simulator to indicate

whenever the pressure at the borehole exceeds the packer specification, showing a warning message on the

screen.

6.4 Wellbore Flow Module

The Wellbore Flow Module estimates the fluid pressure and velocity profile over the wellbore. This module

uses the initial flow rate and pressure from the MDT module as input parameters. It also employs the

parameters of borehole diameter and packer interval, introduced by the user. Using the analytical solution

explained in chapter 3, the model computes the pressure drop that occurs from the tool to the wellbore wall

and the fluid velocity over the interval. The output from this function is then used as an input in the Near-

Wellbore Reservoir Module.


51

Figure 6.9 Wellbore Flow Module structure.

6.5 Near-Wellbore Reservoir Module

This module estimates numerically the fluid flow in the reservoir in the near-wellbore area. It uses the initial

flow rate and initial borehole pressure from the Wellbore Flow Module as input parameters. It also uses fluid

viscosity and petrophysical properties (i.e. permeability, porosity and compressibility factor) introduced by

the user.

It also employs an interactive process with the MDT Tool Performance Module to compute new flow rates,

and viscosities as a function of pressure. Moreover, it solves the diffusivity equations numerically by means

of finite difference computational process. The output from this module is used in the main simulator to

compare the maximum breakdown pressure with the maximum build-up pressure deliver by the tool.

Figure 6.10 Near-Wellbore Reservoir Module structure.


52

6.6 Geomechanical Module

Figure 6.11 shows the structure of the Geomechanical module. It determines the breakdown pressure

necessary to create a tensile failure in the formation. It uses as input parameters, the wireline open-hole logs

and laboratory data previously introduced in the excel file.

Each step in the geomechanical earth model workflow is explained in Chapter 5. The output of the

Geomechanical module is used in the main module to compare the maximum build-up pressure with the

breakdown pressure, and display if a tensile failure can be achieved.

Figure 6.11 Geomechanical Module structure.

6.7 Main Module & Simulator Answer

Figure 6.12 shows the structure of the Main Module. It integrates the outputs delivered by the other modules,

and provides a feedback to the user indicating if the MDT tool is able to breakdown the formation. It allows

the user to modify parameters and make operational decisions.

Figure 6.12 Main Module structure.


53

Once all parameters are entered, the simulator will start the computation of the pressure build-up in the

reservoir. A dynamic window that shows how pressure increase for each time step, will emerge. This window

is accompanied by a progress bar that helps the user to visualize how long the computation will take.

The main User Interface window will display the pressure build-up profile along the reservoir and the

breakdown pressure for a particular depth. The pressure profile is displayed up to a radial distance of 10

meters. Here the user can monitor if the maximum pressure build-up, reached the breakdown pressure. In

addition, it illustrates how the pressure at the near-wellbore area increases with time.

Figure 6.13 Main User Interface Window Results Display.

The results section displays several outcomes in separate windows. The user can select between six different

displays.

Display Geomechanics.

Interval Geomechanics.

3D Plot.

Pressure Radial Influence.

Viscosity profile.

Display Logs.


54

The “Display Geomechanics” output consists on the most important geomechanical properties of the rock

presented in four continuous graphs. Together with “Display Logs” this outcome assists the user to check the

basic lithology of the formation and also investigate the depths at which a fracture is more likely to occur.

The second check box “Interval Geomechanics” displays the same geomechanic graphs previously explained

but also focus on the stress test interval selected by the user.

For a better visualization of the results, the simulator displays a 3D plot of the pressure versus time and

distance. It also displays a colour map of pressure profile distribution near the wellbore area. Here the user

can rotate and zoom the graphs to get a better understanding of the pressure profile.

The viscosity profile can also be displayed. It depends on the rheology model selected by the user in Step # 4.

It compares the Herschel-Bulkley model versus the Newtonian model. The final result displayed consists of

some of the most important logs properties used to compute the geomechanical model. These are the Density

log, Porosity log, Sonic Compression and Shear Delta Time.

If the pressure difference between the wellbore and the interval is higher than the maximum allowed

differential pressure, a window which indicates that the packer will leak through the seal will appear. See

Figure 6.5

Figure 6.5 Simulator warning message.

Finally if the maximum build-up pressure at the wellbore wall is higher than the breakdowns pressure, a

popup window indicates that a tensile failure can be attained, if on the other hand the build-up pressure is

lower that the breakdown pressures a popup window indicates that tensile failure will not be attained.

Figure 6.16 Simulator final result windows.


55

7 Results

To validate the simulator’s predictive capability, two data sets from two different wells are used as a

reference. See Tables 7.1 and 7.2

The data sets are composed of geomechanical and Mini-Fracture stress test results, in which several

attempted tests did not create a tensile failure. (The input parameters used for the simulation are not presented

in this document in order to maintain the confidentiality agreement for data release).

The first step in the corroboration consisted of introducing the physical properties of the formation required to

estimate the breakdown pressure for each test. These results were then compared with the results obtained

from the Mechanical Earth Model previously validated by Schlumberger. The second corroboration step is

based on estimating the MDT maximum build-up pressure, and comparing the results with the maximum

pressure achieved in the Mini-Fracture stress test. Finally the validation will be completed if the simulator

predictions and the stress test results attempted on the field match, indicating whether the tool can create a

tensile failure or not.

Table 7.1 Stress test results Well # 1. Table 7.2 Stress test results Well # 2.

Test Depth Lithology Failure Created

2.1 x439.5 Shale NO

2.2 x245.5 Shale YES

2.3 x757.5 Shale NO

2.4 x756.5 Shale NO

2.5 x690.5 Shale NO

2.6 x525.5 Sandstone YES



7.1 Breakdown Pressure Estimation

Based on wireline logs and laboratory data, the breakdown pressure for both wells was computed using the

simulator. The results from these simulations were compared with the results from the Mechanical Earth

Model validated by Schlumberger (SLB- MEM). See Figure 7.1.

The graphs in Figure 7.1 show an accurate match between the Mini-Fracture simulator results and the

breakdown pressure profile obtained from the SLB-MEM. Nevertheless it is observed that in some specific

depths, the deviation between results can be as large as 20 %. This large difference is attributed to the fact

that the SLB-MEM applies correction for depletion in its calculations, and the strain values used to calculate

the minimum and maximum horizontal stresses were adjusted for different depth intervals which influences

the output estimation.

Test Depth Lithology Failure Created

1.1 x744.5 Sandstone NO

1.2 x277.5 Shale NO


1.4 x382.5 Shale YES



1.7 x665.5 Shale NO


56

Well # 1 Well # 2

Figure 7.1 Breakdown pressure computed using the Mini-Fracture simulator and SLB-MEM.

Figure 7.2 and 7.3 shows the frequency distribution of the deviation between results using the SLB-MEM and

Mini-Fracture simulator in Well # 1 and Well # 2.

Figure 7.2 Difference in results for Well # 1, (frequency distribution).


57

Figure 7.3 Difference in results for Well # 2, (frequency distribution).

In Well # 2 the difference between results is larger. Here we observed that about 70 % of the results are

below six percent difference, but there are zones that have a difference of about 20 %. This is because in Well

# 2, the methodology used to estimate the stresses around the wellbore done by Schlumberger Geomechanics

team, was based on calibration and adjustment of the data using several values of strain coefficients for

different intervals. Even though the results obtained from Well # 2 shows larger differences, the estimation of

the breakdown pressure is still in a valid range for our purpose.

Table 7.3 demonstrates that in Well # 1, both simulation methods delivered a similar results on the intervals

were the stress test was performed, with a maximum difference of about 6% on test 1.3. For Well # 2, the

discrepancy is larger on the intervals where the lithology is characterized by shale. Nevertheless, the

breakdown pressure estimated from the Mini-Fracture simulator is higher than the SLB-MEM estimation.

This can be interpreted as an over estimation of the results in which case the MDT tool has to achieve higher

build-up pressure in order to create a tensile failure.

Table 7.3 Breakdown pressure estimation using the Mini-Fracture simulator and SLB-MEM.

Well Test

N0

Depth [m] Lithology SLB-MEM

Breakdown

Pressure [Psi]

Mini-Frac Sim

Breakdown

Pressure [Psi]

Result Difference

[%]

1 1.1 x744.5 Sandstone 11912 11804 0.9

1 1.2 x277.5 Shale 10778 10990 1.9

1 1.3 x417.5 Sandstone 11496 10869 5.4

1 1.4 x382.5 Shale 8156 8035 1.4

1 1.5 x345.0 Sandstone 6152 6214 1.5

1 1.6 x704.5 Sandstone 9603 9231 3.8

1 1.7 x665.5 Shale 8298 8031 3.2

2 2.1 x439.5 Shale 7457 8620 15.5

2 2.2 x245.5 Shale 5342 6549 22.2

2 2.3 x757.5 Shale 9496 11126 17.1

2 2.4 x756.5 Shale 9182 10782 17.4

2 2.5 x689.5 Shale 7003 8566 22.3

2 2.6 x523.5 Sandstone 7964 7543 5.2

2 2.7 x347.0 Sandstone 7061 6693 5.2

2 2.8 x136.5 Sandstone 6722 6397 4.8


58

7.2 Maximum MDT Build-up Pressure Estimation

Using the same MDT operational parameters (i.e. pump type, packer type, flow rate, interval depth) that were

used on the real stress test job, the maximum build-up pressure was estimated for each individual test in both

wells. These estimations were compared with the maximum pressure achieved in the field.

Table 7.4 shows the readings for maximum build-up pressure achieved by the MDT tool on the field, and the

maximum pressure that was estimated using the Mini-Fracture simulator.

The difference in results between the real data and the simulated estimation is lower than 7 % for all tests,

which indicates that the Mini-Fracture simulator is capable of predicting the maximum, build-up pressure

with high accuracy.

However, in those cases where tensile failure was achieved, the maximum build-up pressure could not be

compared because the breakdown pressure is lower than the maximum pressure that can be attained.

Table 7.4 Maximum build-up pressure comparison.

Well Test

N0

Depth [m] Lithology Field MDT Max.

Pressure [Psi]

Mini-Fracture Sim

MDT Max.

Pressure [Psi]

Result

Difference

[%]

1 1.1 x744.5 Sandstone 9541 8914 6.5

1 1.2 x277.5 Shale 8260 8331 0.8

1 1.3 x417.5 Sandstone 9031 8686 3.8

1 1.4 x382.5 Shale 8964 8685 [-]

1 1.5 x345.0 Sandstone 8923 8718 2.2

1 1.6 x704.5 Sandstone 8550 9287 [-]

1 1.7 x665.5 Shale 9446 9312 1.4

2 2.1 x439.5 Shale 7341 6973 5.0

2 2.2 x245.5 Shale 6346 6823 [-]

2 2.3 x757.5 Shale 7900 7966 0.8

2 2.4 x756.5 Shale 7967 7991 0.3

2 2.5 x689.5 Shale 7916 7838 0.9

2 2.6 x523.5 Sandstone 6777 6763 [-]

2 2.7 x347.0 Sandstone 4839 7007 [-]

2 2.8 x136.5 Sandstone 4817 5889 [-]

7.3 Mini-Fracture Test Results

For further validation, the simulator was run for each single test. The maximum build-up pressure was plotted

against the breakdown pressure. These results were then compared with the outcome of the real stress test

carried out on the field.

Below we find the results for each test. They are presented in Tables and graphically. Tables 7.5 and 7.6

show the maximum packer differential pressure, the predicted breakdown pressure and the predicted extra

pressure needed to create a tensile failure during the test in which a fracture was not initiated.


59

For Well # 1, five out of the seven test preformed were predicted correctly. For the two wrong predictions the

breakdown pressure simulated was lowered than the expected, which indicates that the values were

underestimated. It was also observed that in those cases where the fracture was not initialized, the MDT tool

and the simulator gave similar results on the maximum pressure that can be achieved before the packer

differential pressure limits is exceeded.

Table 7.5 Results comparison between simulated and real stress test for Well # 1.

Test N0 Lithology Tensile

Failure

Predicted

Tensile

Failure

Occurred

Simulated Max.

Packer pressure

Difference (psi)

Real Max

Packer pressure

Difference (psi)

Predicted

Extra pressure

to fracture

(psi)

1.1 Sandstone NO NO 4590 5017 2890

1.2 Shale NO NO 4595 4524 2658

1.3 Sandstone NO NO 4721 5066 2183

1.4 Shale YES YES 4780 5059 [-]

1.5 Sandstone YES NO 4876 5059 [-]

1.6 Sandstone YES YES 4856 4114 [-]

1.7 Shale YES NO 4945 5039 [-]

For Well # 2, six out of the eight test preformed were correctly predicted. For the two wrong predictions, the

breakdown pressure occurred at values lower than was expected, which indicates that the values were

overestimated or pre-existing micro fractures were presented in the formation reducing the strength of the

rock close to the wellbore. It was also observed that in those cases where the fracture did not initialize, the

MDT tool and the simulation, delivered similar results on the maximum pressure that can be achieve before

the packer differential pressure limit is exceeded.

Table 7.6 Results comparison between simulated and real stress test for Well # 2.

Test N0 Lithology Tensile

Failure

Predicted

Tensile

Failure

Occurred

Simulated Max.

Packer pressure

Difference (psi)

Real Max

Packer pressure

Difference (psi)

Predicted

Extra pressure

to fracture

(psi)

2.1 Shale NO NO 4523 4891 1825

2.2 Shale YES YES 4717 4239 [-]

2.3 Shale NO NO 4972 4906 3159

2.4 Shale NO NO 4999 4975 2790

2.5 Shale NO NO 4914 4993 718

2.6 Sandstone NO YES 4131 4145 780

2.7 Sandstone YES YES 4589 2481 [-]

2.8 Sandstone NO YES 3941 2868 507

Find below the graphical comparison of the results from each individual tests.


60

Figure 7.4 and 7.5 shows a comparison between the results of the simulated and real stress test 1.1. The field

data does not exhibit a breakdown signature of the seven injection cycles at maximum build-up pressure. The

simulator also predicted that no fracture can be achieved in this interval unless an extra pressure value of

2890 psi is applied by the MDT tool.

Figure 7.4 Real field pressure build-up and flow rate versus time for stress test 1.1

Legend: #1: Packers Inflation, #2: Leak off, #3 to 7: Injection cycles, #8: Deflate packers.

Figure 7.5 Simulated pressure build-up and breakdown pressure versus time for stress test 1.1.


61

Figure 7.6 and 7.7 shows a comparison between the result of the simulated and the real stress test 1.2. The

field data does not exhibit a breakdown signature at maximum build-up pressure of the eight injection cycles.

The simulator also predicted that no fracture can be achieved in this interval unless an extra pressure value of

2659 psi is applied by the MDT tool.





62

Figure 7.8 and 7.9 shows a comparison between the results of the simulated and the real stress test

1.3. The field data does not exhibit a breakdown signature at the maximum build-up pressure of the

eleven injection cycles. The simulator also predicted that no fracture can be achieved in this interval

unless an extra 2183 psi pressure is applied by the MDT tool.


Legend: #1: Packers Inflation, #2: Leak off, #3 to 11: Injection cycles, #12: Packers Deflation.



63

Figure 7.10 and 7.11 shows a comparison between the results of the simulated and real stress test 1.4. The

field data exhibits a signature of tensile failure and fracture propagation. The simulator also predicted that

fracture will be created around a pressure of 8035 psi. In this case, the fracture was created at a pressure of

8964 psi which is 10 % higher than the prediction, resulting in an underestimation of the breakdown pressure.

It is also observed that the fracture was initialized, reaching the packer differential pressure limits at 5059 psi.


Legend: #1: Packers Inflation, #2: Leak off, #3: Breakdown, #4 to 7: Fracture Propagation,

#8: Packers Deflation.



64

Figure 7.12 and 7.13 shows a comparison between the results of the simulated and the real stress test 1.5. The

field data does not exhibit a breakdown signature at maximum build-up pressure for the ten injection cycles.

On the other hand, the simulator predicted that a tensile failure will occur at a pressure of 6215 psi which is

much lower than the maximum pressure applied on the wellbore. It is also observed that the pressure at the

interval reached the packer differential limits at 5059 psi.





65


field data exhibits a signature of tensile failure and fracture propagation. The simulator also predicted that

fracture will be created around a pressure of 9231 psi. In this case, the fracture was created at a pressure of

8550 psi which is lower than the prediction, resulting in an overestimation of the breakdown pressure. It is

also observed that the fracture was initialized below the packer differential pressure limits at 4114 psi.


Legend: #1: Packers Inflation, #2: Leak off, #3 to 6: Cycles 1 to 4 (injection and falloff),




66

Figure 7.16 and 7.17 shows a comparison between the results of the simulated and the real stress test

1.7. The field data does not exhibit a breakdown signature, likely due to shale plasticity deforming

rather than fracturing. On the other hand, the simulator predicted that a tensile failure will occur at a

pressure of 8031 psi which is lower than the maximum pressure applied on the wellbore. It is also

observed that the pressure at the interval reached the packer differential pressure limits at 5039 psi.





67

Figure 7.18 and 7.19 shows a comparison between the results of the simulated and real stress test 2.1. The

field data does not exhibit a breakdown signature. The simulator also predicted that no fracture can be

achieved in this interval unless an extra pressure of 1825 psi is applied by the MDT tool. The packer

differential pressure limits was almost reached close to the 5000 psi. The maximum simulated build-up

pressure was 250 psi lower than the actual pressure achieved by the tool, but still much lower than the

pressure required to break the formation.


Legend: #1: Packers Inflation, #2: Leak off, #3 to 6: Injection cycles, #7 to 9: Injection and drawdown cycles,

# 10: Packers Deflation.



68


field data exhibits a signature of tensile failure creation and fracture propagation. The simulator also predicted

that fracture will be created around a pressure of 6823 psi. In this case, the fracture was created at a pressure

of 6349 psi which is a value reasonably close to the estimated value. It is also observed that the fracture was

initialized below the packer differential pressure limits at 4234 psi. This indicates that there is still room for

the tool to build-up more pressure, which will result in a maximum build-up pressure really close to the

estimated value of 6823 psi.


Legend: #1: Packers Inflation, #2: Leak off, #3 to 7: Cycle 1-5 (injection and falloff), #8: Packers deflation.



69




differential pressure limits was reached close to the 5000 psi. The maximum simulated build-up pressure was

66 psi higher than the actual pressure achieved by the tool, but still much lower than the pressure required to

break the formation.





70




differential pressure limits was reached close to the 5000 psi. The maximum simulated build-up pressure was

close to the actual pressure achieved by the tool, but still much lower than the pressure required for breaking

the formation.





71


field data does not exhibit a breakdown signature which is likely due to shale plasticity and deforming rather

than fracturing. The simulator also predicted that no fracture can be achieved in this interval unless an extra

pressure of 718 psi is applied by the MDT tool. The packer differential pressure limits was reached close to

the 5000 psi. The maximum simulated build-up pressure was almost the same as the actual pressure achieved

by the tool, but still much lower than the pressure required to break the formation.





72


field data exhibits a signature of fracture propagation but not breakdown initialization. The breakdown

signature is not clear, as the pressure does not fall drastically after reaching the maximum build-up. This

shows that the fracture might have developed from a pre-existing defect (drilling induced fracture or existing

fracture). The simulator predicts that a fracture will be created at a pressure value of about 7543 psi. In this

case the fracture was created at a pressure of 6777 psi which is lower than the prediction, resulting in an

overestimation of the breakdown pressure. Nevertheless due to the fact that probably the fracture was

developed before the test was carried out, the breakdown pressure should be higher than 6777 psi. It is also

observed that the fracture was initialized below the packer differential pressure limits at 4145 psi.


Legend: #1: Packers Inflation, #2: Leak off, #3 to 6: Injections cycles,




73


field data exhibits a signature of fracture propagation but not initialization. The breakdown signature is not

clear, as the pressure did not fall drastically after reaching the maximum build-up. This shows that the

fracture might have developed from a pre-existing defect (like drilling induced fracture or pre-existing

fracture). The simulator also predicted that a fracture will be created at a pressure around 6693 psi. In this

case the fracture was created at a pressure of 4839 psi which is lower than the prediction, resulting in an

overestimation of the breakdown pressure. Nevertheless, due to the fact that probably the fracture was

developed before the test was carried out the breakdown pressure should be higher than 4839 psi for a

formation with no pre-existing fractures. It is also observed that the fracture was initialized below the packer

differential pressure limits at 2481 psi.


Legend: #1: Packers Inflation, #2: Leak off, #3 to 6: Injection Cycles,




74

Figure 7.32 and 7.33 shows a comparison between the results of the simulated and the real stress test 2.8.The

field data exhibits a signature of fracture initialization and fracture propagation. On the other hand, the

simulator predicted that no fracture can be achieved in this interval unless an extra pressure of 507 psi is

applied by the MDT tool. In this case, the fracture was created at a pressure of 4817 psi which is lower that

the prediction, resulting in an overestimation of the breakdown pressure. It is also observed that the fracture

was initialized below the packer differential limits at 2868 psi.


Legend: #1: Packers Inflation, #2: Leak off, #3 to 6: Injection Cycles,




75

8 Discussion

This chapter discusses the comparison of the results obtained by the Mini-Fracture simulator and real field

data. The correct predictions and the causes of discrepancies in the incorrect predictions are also analysed. In

addition, drawbacks and limitations of predicting tensile failure with the Mini-Fracture simulator will be

studied.

Figure 8.1 shows the Mini-Fracture simulator validation summary. The simulator predicted the creation of a

tensile failure correctly in 74 % of the cases. From this 74 % correct prediction, in 64% of the tests a fracture

was not initialized, and in 36 % of the tests the breakdown pressure was achieved creating a fracture. In the

other hand 26% of the simulations, predicted incorrect results. In 50% of the erroneous predictions formation

breakdown was achieved, whereas in the other 50% the fracture was not initialized.

The validation summary shows that 7 out of 15 simulations correctly predicted that a tensile failure will not

occur, 4 out of 15 simulations correctly predicted that a tensile failure will occur, 2 out of 15 simulations

incorrectly predicted that a tensile failure will not occur and 2 out of 15 simulations incorrectly predicted that

a tensile failure will occur. Table 8.1 shows in detail the validation and comparison of the simulated results

versus real field data for both wells.

Figure 8.1 Validation summary of Mini-Fracture simulation predictions against real field data.


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Table 8.1 Detailed simulator versus real field stress test predictions.

8.1 Correct Predictions

As previously stated, with the Mini-Fracture simulator, 74 % of the predictions were correctly validated.

Analysing the data, first the type of lithology in which the test was carried out was identified. From the nine

unsuccessful tests, six were performed in shale. The Poisson’s ratio for shale and very shaly sandstones are

much higher than for clean sands, indicating that these rocks exhibit a more elastic, and plastic behaviour.

They will have the tendency to suffer plastic deformation rather than brittle deformation and fracturing, while

requiring a higher build-up pressure in order to create a fracture (Charlez, 1997).

Figure 8.2 shows the difference between the simulated maximum build-up pressure and the breakdown

pressure for each test. It has been clearly identified that in the unsuccessful tests extra pressure has to be

supplied by the MDT in order to create a fracture (2000-3000 psi extra pressure). In the other hand for those

tests in which the fracture was created the difference between breakdown pressure and maximum build-up

pressure is 50–500 psi, which indicates that the tool has been reaching its maximum build-up pressure

capacity when the fracture was initialized.

If the unsuccessful test are simulated with a different type of pump (High and X-high Pressure Pump), it is

clearly noticeable that higher build-up pressures can be achieved for the same interval. Therefore, the

breakdown pressure is possible to achieve. Figure 8.3 shows the comparison of the simulation in test 1.2

changing the pump used in the MDT configuration.

Moreover it has been noticed, that using a pump with higher pressure capacity requires longer pumping times,

due to the limited flow rate. This is a drawback on stress test jobs as longer pumping times increases the

chances of the MDT tool to get stuck in the well.

Well

Test

Number

Tensile

Failure

Predicted

Tensile

Failure

Occur

1 1.1 NO NO

1 1.2 NO NO

1 1.3 NO NO

1 1.4 YES YES

1 1.5 YES NO

1 1.6 YES YES

1 1.7 YES NO

2 2.1 NO NO

2 2.2 YES YES

2 2.3 NO NO

2 2.4 NO NO

2 2.5 NO NO

2 2.6 NO YES

2 2.7 YES YES

2 2.8 NO YES


77

Figure 8.2 Pressure difference between the simulated maximum build-up pressure and the breakdown

pressure for each test.

Figure 8.3 Simulation of test 1.2 using different types of pump.

The main concern when changing the standard pump for a pump with higher pressure capacity is the

transgression of the packer differential pressure limits. Figure 8.4 shows the difference between the test

interval and the wellbore pressure for each test. It is observed that in 8 out of the 9 unsuccessful tests the

packer differential limits was reached. Once the limit is reached, fluid will start leaking-off from the interval

to the wellbore and the fracture won’t be initialized. This demonstrates that the biggest limitation in the Mini-

Frac job is the packer sealing element. With a maximum of 5000 psi for allowed differential pressure, the

packer element SIP-A3A-5 and IPCF-PC-700 are the most recommended to be used. Nevertheless, their

limitations are still far below the required specifications, if Schlumberger aims for 100% of fracture

propagation.


78

Figure 8.4 Difference in pressure between interval and borehole for each test.

8.2 Incorrect Predictions

There are many parameters that influence the interaction between fluid flow and reservoir stress state. Some

of these parameters include permeability, porosity, compressibility factor, fluid viscosity, tensile strength,

Young’s modulus and Poisson’s ratio. The interaction between these parameters and how they affect the

result of the simulator is complex. Some may have a significant effect while others may exhibit a negligible

effect. This complex interaction suggests that a parametric study is a step toward a better understanding of the

sensitivity of the simulator output and what may be the cause of its incorrect predictions.

The purpose of this section is to perform a parametric sensitivity analysis that helps to understand how the

uncertainty in the simulator output is apportioned to the sources of uncertainty in the simulator input. The

procedure was based on conducting parametric analysis on seven variables influencing the formation

breakdown and pressure build-up. Each of the parameters being studied is varied from run to run whilst all

the other parameters are held constant. Table 8.2 shows the parameters and the value variation used for the

sensitivity analysis.

Table 8.2 Values used for the sensitivity analysis.

Parameter Units Lower Value Upper Value

Permeability [mD] <1 >100

Porosity [%] <1 >10

Compressibility Factor [1/Pa]

-50 % of nominal value

+ 50 % of nominal value

Viscosity [cP]

Tensile Strength [Pa]

Young’s Modulus [Pa]

Poisson’s Ratio [-]


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8.2.1 Permeability Sensitivity Analysis

The permeability value was changed from a value lower than 1mD to a value higher than 100mD. Figure 8.5

shows the reservoir build-up pressure as a function of time varying the permeability. As expected the

maximum build-up pressure decreased when the permeability increased. A maximum difference value of

12% between pressure build-up was observed. For a given permeability the build-up pressure increased

during early times until stable conditions were reached.

This is a logical result because in high permeable formation, the leak off rate will be higher which implies

that the MDT tool will have to pump at a faster rate to compensate this leak off and achieve the same build-

up pressure that is achieved in low permeable areas.

Figure 8.5 Permeability sensitivity analysis.

8.2.2 Porosity Sensitivity Analysis

The porosity of a zone can be estimated either from a single porosity log (sonic, density, neutron, or NMR) or

a combination of porosity logs, to correct for variable lithology effects in complex reservoirs. When sonic

porosities are compared with neutron and density porosities, difference will be present. The neutron and

density logs responds to pores of all sizes however the sonic log is largely insensitive to either fractures or

vugs (Habbib et al., 2000). Therefore the distinction on the porosity method used to compute the results is

important. For the simulator neutron and density logs were used.

Figure 8.6 shows the results of changing the porosity from a value lower that 1% to a porosity value higher

than 10 %. It can be appreciated that a change in porosity leads to equal maximum build-up pressure for all

values, which will be achieved after much longer pumping times (4 times larger). The pore space in highly

porous reservoir rocks allows them to hold more fluids. Therefore, the MDT tool will require more time to

build-up pressure in a reservoir with more pore space.


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Figure 8.6 Porosity sensitivity analysis.

8.2.3 Compressibility Factor Sensitivity Analysis

Figure 8.7 shows the pressure build-up as a function of time for various compressibility factors. Changing the

total compressibility factor (fluid + rock) will have similar response to the previous case in which the

maximum build-up pressure is the same, but the time to reach that maximum will depend on both the

compressibility of the fluid and rock. Nevertheless for this parameter the influence is much smaller than the

changes in porosity.

Figure 8.7 Compressibility factor sensitivity analysis.


81

8.2.4 Fluid Viscosity Sensitivity Analysis

Figure 8.8 shows the pressure build-up as a function of time for various fluid viscosities. Changing the fluid

viscosity will have an impact on the maximum build-up pressure that can be achieved. However it is observed

that by changing the value 50% higher than the original value, the increase in pressure is almost unnoticed. In

the other hand by decreasing the viscosity value by 50% the reduction in pressure is more visible. Either way

the change between readings is less than 3%.

Figure 8.8 Fluid viscosity sensitivity analysis.

For this analysis the most extreme values from the available database have been chosen, although future

investigations could also consider less extreme values. In the analysis it was observed that the permeability is

the most significant uncontrollable parameter affecting the maximum build-up pressure that can be achieved

in the formation. Therefore, an accurate description of the permeability in the well is required prior to

performing a stress test. Excellent permeability results are obtained with nuclear magnetic resonance (NMR)

logging that, by itself, provides information about both porosity and pore-size distribution. These parameters

are used to successfully derive continuous permeability logs (Tarek, 2001).

The fluid viscosity influences the final result but not as much as the permeability. Furthermore, it is observed

that changes in porosity and compressibility factor will not have an impact on the maximum build-up

pressure but it will affect the pumping time that is required to achieve the maximum build-up pressure.

The uncertainty related to breakdown pressure is more difficult to predict, because a consistent description of

rock mechanical attributes of the formation requires collection and integration of many variables from various

sources in the model. Variables as maximum and minimum horizontal strain, and formation fluid density may

be hard to predict and will depend to a certain degree on the experience of the interpreter and the amount of

information gathered from the field.


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8.2.5 Poisson’s Ratio Young’s Modulus Sensitivity Analysis

Figure 8.9 and 8.10 shows the build-up pressure as a function of pump time, including the breakdown

pressure for different Young’s modulus and Poisson’s ratio. The result shows that by varying the Young’s

modulus and Poisson’s ratio by 50% it will influence the breakdown pressure by 22% and 17% respectively.

The mathematical explanation behind this behaviour is that by increasing the Young’s modulus or Poisson’s

ratio and keeping constant the pore pressure and rock strength, the minimum and maximum horizontal

stresses also increase and applying the failure criteria on equation 5.34 will compute a higher breakdown

pressure. Nevertheless, is important to point out that this parametrical approach may miss possible

interactions between the factors being studied. Therefore, more advanced statistical techniques are

recommended to systematically analyse the interactions of important parameters and their effects on the

results.

Figure 8.9 Young’s modulus sensitivity analysis.

Figure 8.10 Poisson’s ratio sensitivity analysis.


83

8.2.6 Tensile Strength Sensitivity Analysis

Figure 8.11 shows the build-up pressure as a function of time and the breakdown pressure for various tensile

strength values. By changing the tensile strength by 50 % the breakdown pressure only changed 6%. This

indicates that the most important factors affecting the breakdown pressure calculation are the Poisson’s ratio

and the Young’s modulus.

Figure 8.11 Tensile strength sensitivity analysis.

If fractures are generated prior to performing the stress test, the tectonic forces will be reduced and the

breakdown pressure will be much lower than the simulator predictions (Fjaer, et al., 2008). Examples of this

are test 2.6 and 2.8 in which the simulator predicted that the fracture would not be initialized but it actually

did happen at a much lower pressure than expected. To confirm this, test 2.6 and 2.8 were simulated without

considering the tensile strength. The result shows that the breakdown pressure in both cases is lower than the

maximum build-up pressure. This confirms a predictable tensile failure in both cases. To avoid incorrect

predictions it is recommended to obtain a 360 deg Image log before run the stress test, in order to indicate

areas of fractures induction by the drilling process.

Figure 8.12 Test 2.6 and 2.8 results, using tensile strength equals to zero.


84

8.3 Simulator Limitations and Assumptions

Despite of the good predictable simulator capability, the user must bear in mind that the model has underlying

limitations and is based in assumptions that not necessarily represents accurately subsurface conditions in all

cases.

Summarizing, the assumptions used throughout the model and development of the simulator are:

Laminar flow.

Fluid in permanent regime.

Flow in direction of r coordinate r ( , )V V r z .

Flow is isothermal.

Gravity accts just in z direction (gθ=gz=0).

Parameters don’t change in the θ direction.

Infinite acting reservoir.

The reservoir is a uniform pressure Pi when injection begins.

The well, with a wellbore radius of rw is centre in a cylindrical reservoir of radius re.

No flow across the outer boundary re.

Negligence of anisotropy is considered as the first limitation. To create the pressure profile in the reservoir,

the model assumes constant properties in all radial directions. However, geological formations with distinct

layers of sedimentary material can exhibit different rock properties. Therefore, calculation of fluid flow in the

reservoir may take changes between vertical and horizontal properties into account. Otherwise the results

may be subjected to inaccuracies (Dake, 1978).

Furthermore, the model neglects the effect of formation damage or skin factor, which can be considered the

second limitation. This means that any distribution of damage along the wellbore was not incorporated in the

flow model. Permeability changes in the vicinity of the wellbore result from formation damage (permeability

decrease) (Tarek, 2001). This can have an impact on the output of the maximum build-up pressure, which

depends on the permeability as an input and is a sensitive parameter to variations as explained in section

8.2.1.

The simulator assumes a single phase flow for a single fluid in the reservoir, and no acceleration or

deceleration of fluid particles through pore space is included. However, flow through hydrocarbon reservoirs

is predominantly multi-phase and multi-component flow. As a result concepts such as capillary pressure,

saturation and relative permeability are not included in the model (Dake, 1978).

The geomechanical model computation and breakdown pressure calculation are based on a simple approach,

which are continues through the well. The utilization of different strain coefficients for different zones has

neither been included in the model. Moreover, the reservoir pore pressure used to calculate the horizontal

stresses, are dependent of the formation fluid density introduced by the user, and it does not take into

consideration different pressure gradients for different fluids.

Finally, as formation density readings are not recorded until surface, the user must extrapolate and estimate

the density reading using a curve fitting scheme for shallow depths. This can lead to inaccurate results on the

vertical stress computation and therefore on the final output of the simulator.

http://en.wikipedia.org/wiki/Geological

http://en.wikipedia.org/wiki/Sedimentary


85

9 Conclusion and Recommendation

9.1 Conclusion

This project demonstrates that combining the appropriate constitutive relations that reflect the coupling

among the tool operational performance with wellbore flow, reservoir and geomechanics modelling a Mini-

Fracture simulator can be developed. The Mini-Fracture simulator accomplishes its goal to predict precisely

the possibility of tensile failure in 74 % of the stress test that were conducted in the field.

It has been observed that by using simple mathematical models with analytical solutions, engineers can

provide basic performance predictions. However, for the more advanced and complicated predictions the

analytical approach might be insufficient. Instead numerical simulation models have to be applied. It is

shown that the numerical approach gives more steady results when an attempt was made to estimate the

pressure profile in the reservoir, while fluid injection. Compared with the Ei and self-similarity solution the

numerical method is more robust and showed no restriction on the values used to simulate a Mini-Fracture

operation. The Numerical solution showed stability with respect to variation in reservoir radius and injection

times.

The simulator showed that the pressure drop over the wellbore interval is relatively small (in the order of X-10 psi) and it can be neglected as a cause for the MDT tool to fail to reach the breakdown pressure. The

pressure drop gradient from the tool to the wellbore is proportional to the flow rate and inversely proportional

to the interval height, which means that the lowest pressure drops are achieved when the distance between

packers is at its maximum.

It is concluded that higher build-up pressures can be achieved for the same interval if a pump with a higher

pump capacity is used on the MDT configuration. This means that the main limitation of the MDT tool to

attempt to overcome higher breakdown pressures are the seal (Packers) elements, which are limited to a

maximum differential pressure of 5000 psi. This seems to be the main reason of unsuccessful stress tests,

which jeopardize the application of the Mini-Frac technique.

The simulator emphasized the importance of having a complete geomechanical study, including geological

structure and stress estimation in order to compute the most precisely breakdown pressure estimation, prior

performing a stress test. It has also been shown that it is less likely to create a tensile failure in shaly intervals,

due to their elastic, deforming plastic behaviour rather than fracturing.

It is demonstrated that the Mini-Fracture simulator is sensitive to geomechanical, and petrophysical

parameters and the integration of data from various sources into the model will more consistently describe the

prediction of a tensile failure. Results indicated that the most important variables affecting breakdown

pressure and maximum build-up pressure are the Young’s modulus and formation permeability. Moreover is

showed that uncontrollable variables as pre-existing fractures can be predicted and will influence the results.

Finally it can be concluded that regardless of the significant limitations and basic assumptions used to

develop the Mini-Frac simulator, this tool can be employed by the fields engineer to estimate the chances for

a certain MDT tool configuration to achieve the breakdown pressure and create a tensile failure.


86

9.2 Recommendations

This project focused on creating a basic simulator that helps to understand the causes of failure in Mini-

Fracture stress test. Nevertheless, there are many improvements that can be implemented to the simulator

model that can be subject of further studies. First of all, the simulator environment could be improved. These

are just some of the improvements that could be made:

Integrate the concept of anisotropy in which properties of the formation are directionally

dependent.

Include the consequence of formation damage, where permeability changes do to skin factor are

include.

Incorporate the effect of multi-phase flow and include its relevant concepts (i.e. Capillary

pressure, saturation, relative permeability).

Upgrade the geomechanical model to include different strain coefficients for different lithologies.

Upgrade the radial flow model to include grid refinement near the wellbore wall to get more

accurate solutions.

In terms of operational procedures to perform a Mini-Frac stress test it is recommended to use the packer

elements SIP-A3A-5 and IPCF-PC-700 as they hold the maximum packer differential pressure possible.

Schlumberger is well known for their research and development centers in which new technologies are been

developed and tested successfully. As the main limitation to reach higher breakdown pressure seems to be the

sealing elements, it is also recommended that futures research and developments on new elements design may

be done to produce new packer elements that can hold maximum differential pressures higher than 5000 psi.

Finally to help the engineers with the design and strategy of the stress test, it is recommended that the most

complete geomechanical study is done prior to perform the job. Inaccuracies on rock properties will lead to

erroneous prediction of the breakdown pressure.


87

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Tarek Ahmed, P. D., P.E., "Reservoir Engineering Handbook", 2001,Wildwood, MA, Butterworth.

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90

11 Appendix

11.1 Derivation of the Wellbore Flow Model

Starting with the Navier-Stokes equation and Continuity equation in the cylindrical coordinate r we have:

2 2 2

2 2 2 2

1 1 2r r r r r rr z r r

V V VV V V V V VpV V g rV

t r r r z r r r r r r z

(11.1)

1 1( ) ( ) 0rrV V

t r r r

(11.2)

Applying the assumption and approximations explained on chapter 3 to the Navier-Stokes equation and

continuity equation we obtain:

2

2

r rr

V VpV

r r z

(11.3)

( ) 0rrVr

(11.4)

The derivation of equation 11.4 holds:

r rV V

r r

(11.5)

To solve the system we define the function ϕ as:

rrV (11.6)

From these equations it can be assumed that the product rVr on equation 11.4 is not dependent of the

coordinate r, and since it can just be depend of coordinates z and r this means that ϕ is a function of

coordinate z, ϕ = ϕ(z).

Substituting equation 11.5 and equation 11.6 into 11.3 the result will be a differential equation of ϕ depending

just of z:

2 2

3 2

1dp d

r dr r dz

(11.7)

In this equation the variables can be separated since p is a function of r and ϕ is only a function of z. By

integration of equation 11.7 with respect to r over the interval Rt to Rw we obtain the following second order

differential equation for ϕ(z).


91

22

( )2 2 2

1 1( ) 0

2

wz t w

t w t

R dP P Ln

R R R dz

(11.8)

Where Pt is the pressure of the fluid at the tool radius Rt and Pw is the pressure at the wellbore radius Rw.

Summarizing the boundary conditions we get:

( ) ( )

( ) ( )

( )

( )

0 0

0 0

r z H Z H

r z H Z H

r Rt

r Rw

V

V

P Pt

P Pw

Now we can introduce the Reynol’s number by the following expression:

RtV HRe

(11.9)

where RtV is the average velocity at the tool radius. In order to facilitate the analysis of the problem the

equations are wrote in the dimensionless forms. In Table 11.1 is presented the dimensionless parameters that

are going to be used to solve the equation:

Table 11.1Dimensionless Parameters.

*

w

rr

R

* zz

H

* rr

Rt

VV

V

*

w Rt w Rt

r Vr

R V R V

*

/

w t

Rt

p pp

V H

* t

w

RR

R

The solution can be expressed in terms of Reynolds Number and geometrical parameters. Substituting the

dimensionless parameters into the equation 11.3 we obtain:

* 2 **

*

* * *2Re wr r

r

RV VpV

r r H z

(11.10)

Similarly we can introduce the dimensionless parameters into the equations 11.5 and 11.6

* * *

rr V (11.11)

* *

* *

r rV V

r r

(11.12)


92

Substituting these two equations into equation 11.10 we obtain:

*2 * 2 *

*3 * * *2

1Re tRp d

r r H r dz

(11.13)

Integrating equation 11.13 in function of r we obtain a similar solution to the equation 11.8 but in

dimensionless form:

2 *

*2 *

*2 * *2

Re 1 11 0

2

wRw

R dp Ln

R H R dz

(11.14)

From equation 11.14 we can analyze that the solution is a non-linear differential equation in the velocity field

( , )rV r z . To find a solution to this differential equation we will consider that the Reynold’s numbers are

really low Re 0 which simplify the equation 11.14 to the form:

2 *

* *

*2

wRw

R dp LnR

H dz

(11.15)

Integrating two times equation 11.15 in function of z we obtain a velocity profile in dimensionless form:

* * * *2

*

( )( , ) 1

12 ( )

Rw

w

p HVr r z z

Rr Ln

(11.16)

Expressing equation in the original parameters we obtain the velocity profile of the fluid in the wellbore:

22( )

( , ) 11

2 ( )

t w

w

t

p p H zVr r z

R Hr Ln

R

(11.17)

Considering the change in volume over the coordinate r we can obtain the average velocity by integrating:

1

4 4r r

A

Q QV V dA

A rH rH

(11.18)

where Q is the flow rate. Re arranging equation 11.18 and substituting rV from equation 11.17 we obtain the

pressure profile over the interval between Rt and Rw:

3

3

4t w

t

Q rp p Ln

H R

(11.19)


93

11.2 Derivation of the Analytical Self-Similarity Solution

First the co-ordinates are transformed by the Boltzmann transformation.

2

4

effc r

k t

(11.20)

Applying the derivatives respect to t to the Boltzmann transformation we get.

2

24

effc r

t k t

(11.21)

and rewriting it

2

24

effc r

t t k t

(11.22)

Applying the derivatives respect to r to the Boltzmann transformation.

2

effc r

r k t

(11.23)

and rewriting it

2

effc r

r r k t

(11.24)

The change of pressure with time can be described by:

eff

p q

t Vc

(11.25)

Whit the volume

2

eV h r (11.26)

Giving

2

e eff

p q

t h r c

(11.27)

Substituting equation 11.27 in the radial diffusivity equation gives:

2

1 1eff

e

cp p p qr r

r r r k t r r r k hr

(11.28)


94

Substituting the Boltzmann transformation and its derivatives into the diffusivity equation, and rewriting the

terms:

0p p

(11.29)

Where p is the pressure in Boltzmann coordinates.

Rewriting the equation 11.29 we get

p p p

(11.30)

Defining

(11.31)

the solution of the equation 11.30 is given by

(11.32)

This means that

2

ln( ') ln( )

exp( ) exp( )'

p C

Cp C

(11.33)

We express now the boundary conditions in terms of the Boltzmann transformation

0

0 0

lim2

lim lim 22 2

r

eff

r

p qr

r kh

c r p p qr

k t kh

(11.34)

This Boundary condition is valid for t>0 and r 0 which is equal 0,

2

exppC

(11.35)

Substituting in equation 11.34

'pp

' '' ' '1

' 1

'

p pp p p

p

p


95

(11.36)

(11.37)

Having as a result

(11.38)

Now the derivative can be integrated

(11.39)

Solving the Integration on the left side gives.

exp exp

4 4

o oi i

X X

q B q Bp p d p p d

kh kh

(11.40)

Solving the integration on the right side of the equation

ln '4

i

qp p x

kh

(11.41)

with 2

4

effc rx

kt

(11.42)

and ' =exp( ). Been define as Euler constant =0.5772

2 2

0 0

2 2

explim 2 lim 2 exp

2

22 4

qC C

kh

q qC C

kh kh

2

exp exp

4

p p qC

kh

exp exp

4 4i

p X

o o

p

q B q Bdp d dp d

kh kh


96

11.3 Derivation of the Numerical Solution

For the centre cell i=2,…,nr-1. The discretise form of the diffusivity equation is written as:

2 2 1 1 11/2 1/2 1/2 1/22 2

n n n nn n i i i i

eff i i i i i i

p p p pk kc r r p p t r t r

r r

(11.43)

This equation describes the flow rates at time nt by using the notation pin

Now the pressure terms are evaluated at time t=[n+1]t at the right hand side of the equation.

1 1 1 1

2 2 1 1 11/2 1/2 1/2 1/22 2

n n n nn n i i i i

eff i i i i i i

p p p pk kc r r p p tr tr

r r

(11.44)

This equation describes the pressure of the non-boundary grid-blocks.

Now the equation has to be rearranging in such a way that can be express in terms of vectors and matrices:

1 2 2 1 1 2 21/2 1/2 1/2 1/21 1/2 1/2 1 1/2 1/22 2 2n n n ni i i i

i eff i i i i i i i

r r r rk k kt p c r r t p t p r r p

r r r

(11.45)

Or its equivalent:

1 1 1

, 1 1 , , 1 1

n n n n

i i i i i i i i i i i i it a p t a d p t a p d p e

(11.46)

With the coefficients:

(11.47)

(11.48)

(11.49)

(11.50)

(11.51)

The first grid block is next to the well and the flow rate influx is specified, discretising the equation in the

form:

2 2 1 11/2 1/2 1/2

1

2n n i ieff i i i i i well

i i

p pkc r r p p t r tq

r r

(11.52)

Re-arranging the terms we get.

1/2, 1

1/2 1/2,

1/2, 1

2 2

1/2 1/2

2

2

2

0

ii i

i ii i

ii i

n

i eff i i i

i

r ka

r

r r ka

r

r ka

r

d c r r p

e


97

2 2 1 1 2 21/2 1/21/2 1 1/22 2n n ni i

eff i w i i eff i w i

r rk kc r r t p t p c r r p tq

r r

(11.53)

Or its equivalent:

1 1 1

, 1 1 , , 1 1

n n n n


(11.54)


(11.55)

(11.56)

(11.57)

(11.58)

(11.59)

For the last gridblock the outlet boundary condition is used, in which the flow rate at the boundary is equal to

zero.

2 2 1 11/2 1/2 1/2

1

0 2n n i ieff i i i i i

i i

p pkc r r p p r t

r r

(11.60)

Re-arranging the terms we get.

1 2 2 1 2 21/2 1/21 1/2 1/22 2n n ni i

i eff i i eff i i

r rk kt p c R r t p c R r p

r r

(11.61)

Or its equivalent:

1 1 1

, 1 1 , , 1 1

n n n n


(11.62)


(11.63)

(11.64)

(11.65)

(11.66)

(11.67)

Now the equations 11.54, 11.61and 11.62 can be written in the matrix form, which reads

, 1

1/2,

1/2, 1

2 2

1/2 1/2

/

2

2

/

i i

ii i

ii i

i eff i i

i

a N A

r ka

r

r ka

r

d c r r

e q h

1/2, 1

1/2,

, 1

2 2

1/2 1/2

2

2

/

0

ii i

ii i

i i

i eff i i

i

r ka

r

r ka

r

a N A

d c r r

e


98

1n nt D A p Dp e (11.68)

Solving 1n

p for 0n

11n nt p D A Dp e

(11.69)

Where P is the pressure vector, D and A are column vector and e is the column vector accounting for the

boundary conditions. With the entries given by

(11.70)

(11.71)

(11.72)

For (n=0) the pressure is equal to the initial pressure vector set by the initial condition:

0

initpp (11.73)

11.4 Instructions to Install and use the Mini-Frac Simulator

To install and use the Mini-Frac Simulator the user has to follow the steps listed below.

1. Copy the folder MF_SIM in the D:\ dive. (Don’t change the place or name of the folder).

2. The Folder MF_SIM contains the following files:

Input_Data.xls.

MF_SIM.exe.

MF_SIM_pkg.exe.

Readme file.

3. If the MCR is not installed in the machine, run MCRInstaller, located in the MF_SIM_pkg. MCR

allows you to run Matlab scripts without the need of Matlab software to be installed.

4. If the MCR is already installed in the target machine, run the executable MF_SIM. This will deploy

the simulator interface.

5. Fill up the Input_Data Excel file with the respective parameters on the columns starting from row 7.

The values on the rows 4 to 6 are use for estimation of formation density and vertical stress. They are

calculated from the Shallow density fit sheet included in the excel file.

6. Save the excel file in D:\MF_SIM\Input_Data.xls

7. Run the executable MF_SIM.exe

8. Follow the steps #1 to # 4 and press the RUN bottom at the end of step # 4

9. See the results.

, ,i i i i

ii i

i i

a

d

e

A

D

e

Documents

AES/PE/12-14 Novel Simulator for Wireline Mini-Fracture