hydraulic fracture mechanics

HYDRAULIC FRACTURE MECHANICS

Peter Valkó Michael J. Economides

r Hydraulic Fracture Mechanics

Hydraulic Fracture Mechanics

Peter Valkó and Michael J. Economides Texas A & M University, College Station, USA

JOHN WILEY & SONS Chichester • New York • Brisbane • Toronto • Singapore

Copyright © 1995 by John Wiley & Sons Ltd, Baffins Lane, Chichester, Wesl Sussex PO19 IUD, England

Nacional 01243 779777 lnternational (+44) 1243 779777 ReprintedOctober 1996, May 1997 Ali rights reserved.

CONTENTS

No part o~ this book may be reproduced by any means, or_ transmitted, _or translated into a machine language w1thout the wntten permission of the publisher.

Other Wtley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA Jagaranda Wiley Ltd, 33 Park Road, Milion, Queensland 4064, Australia

Preface List of Notation

1

John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W lll, Canada

Hydraulically lnduced Fractures in the Petroleum and Related Industries 1.1 1 .2

John Wiley & Sons (SEA) Pte Ltd, 37 Jalan Pemimpin #05-04, Block B, Union Induslrial Building, Singapore 2057

1.3 1.4 1.5

·1 .6

1. 7

2

Linear Elasticity, Fracture Shapes and lnduced Stresses 2.1

British Library Cataloguing in Publica/ion Data A catalogue record for this book is available from the British Library ISBN O 471 95664 3 Typeset in IO½ll2½ Times by Laser Words, Madras, India Printed and bound in Great Britain by Bookcraft (Bath) Ltd

Fractures in Well Stimulation Fluid Flow Through Porous Media 1.2.1 The Near-well Zone Flow from a Fracturad Well Hydraulic Fracture Design Treatment Execution 1.5.1 Fracturing Fluids 1.5.2 Proppants Data Acquisition and Evaluation for Hydraulic Fracturing 1.6.1 Well Lag Measurements 1.6.2 Core Measurements 1.6.3 Well Testing Mechanics in Hydraulic Fracturing References

2.2

2.3

Force and Deformation 2.1.1 Stress 2.1 .2 Strain Material Properties 2.2.1 Linear Elastic Material 2.2.2 Material Behavior Beyond Perfect Elasticity Plane Elasticity 2.3.1 Plane Stress 2.3.2 Stresses Relative to an Oblique Une (Force Balance 1) 2.3.3 Equilibrium Relations (Force Balance 11) 2.3.4 Plane Strain 2.3.5 Boundary Conditions

xi xiii

1 1 2 4 5 7

11 11 13

14 14 15 15 15 16

19 19 19 21 23 23

26 27 27 28 30 30 32

vi

Contents

2.4

Pressurized Crack 2.4.1 Solution of the Une Crack Problem 2.4.2 Constant Pressure 2.4.3 Polynomia/ Pressure Distribution 2.4.4 "Zippern Cracks 2.4.5 "Zipper" Crack with Polynomial Pressure Distribution 2.5 Stress Concentration and Stress lntensity Factor 2.5.1 Stress lntensity Factor, Symmetric Loading 2.5.2 Stress lntensity Factor, non-symmetric Loading 2.6 Fracture Shape in the Presence of Far-field Stress. The Concept of Net Pressure 2.7 Circular Crack 2.8 Volume and Strain Energy 2.9 Computational Methods References

Contents

1 32 32 34 35 37 40 41 42

5.2

5.3

5.4

43

43 45 47 49 50

6

Non-laminar Flow and Solids Transport 6.1

3

Stresses in Formations 3.1 Basic Concepts 3.2 Slresses al Deplh 3.3 Near-wellbore Stresses 3.4 Stress Concentrations for an Arbitrarily Oriented Well 3.5 Vertical Well Breakdown Pressure 3.6 Breakdown Pressure for an Arbitrarily Oriented Well 3.7 Umiting Case: Horizontal Well 3.7.1 Arbitrarily Oriented Horizontal Well 3.8 Permeability and Stress 3.8.1 Stress-sensitive Permeability 3.9 Measurement of Stresses 3.9.1 Small lnterval Fracture lnjection Tests 3.9.2 Acoustic Measurements 3.9.3 Determination of the Closure Pressure 3.9.4 Core Stress Measurements 3.9.5 Critique and Applicabillty of Techniques Reterences

53 53 55 59 63 65 66 69 70 71 72 73 74 75 76

6.2

7

Fracture Geometry 4.1

4.2 4.3

The Perkins and Kern and Khristianovich and Zheltov Geometries 4.1.1 The Consequences of the Plane Strain Assumption Fracture lnitiation vs. Propagation Direction 4.2.1 Fractures in Horizontal Wells Fracture Profiles in Multi-layered Formations References

7.1

77

79 80

83 83 86 88 90 92 95

7.3

8

Rheology and Laminar Flow 5.1

Basic Concepts 5.1.1 Material Behavior and Constitutive Equations 5.1.2 Force Balance

97 97 98 103

Foam Rheology 7.1.1 Quality Based Correlations 7.1.2 Volume Equalized Constitutive Equations 7.1.3 Volume Equalized Power Law 7.1.4 Turbulent Flow of Foam Accounting far Mechanical Energy 7.2.1 Basic Concepts 7.2.2 lncompressible Flow 7.2.3 Foam Flow Rheomelry 7.3.1 Pipe Viscometry 7.3.2 Slip Correction References

Material Balance 8.1

5

Non-laminar Flow 6.1.1 Newtonian Fluid 6.1.2 General Fluid 6.1.3 Drag Reduction 6.1.4 Turbulent Flow in Other Geometries Solids Transport 6.2.1 Settling of an Individual Sphere 6.2.2 Effect of Shear Rate lnduced by Flow 6.2.3 Effect of Slurry Concentration 6.2.4 Wall Effects 6.2.5 Agglomeration Effects References

Advanced Tapies of Rheology and Fluid Mechanics

7.2

4

Sial Flow 5.2. 1 Derivation of the Basic Relations 5.2.2 Equivalent Newtonian Viscosity Flow in Circular Tube 5.3.1 Basic Relations 5.3.2 Flow Curve 5.3.3 Equivalent Newtonian Viscosrty far Tube Flow Flow in Other Cross Sections 5.4.1 Flow in Annutus 5.4.2 Flow in Elliptic Cross Section 5.4.3 Limiting Ellipsoid Cross Section References

8.2

The Conservation of Mass and lts Relation to Fracture Dimensions Fluid Leakoff and Spurt Loss as Material Properties 8.2.1 Carter Equation 1 8.2.2 Formal Material Balance. The Opening Time Distribution Factor

vii

105 105 111 112 112 115 119 122 122 123 124 128

131 131 131 132 134 137 138 139 141 142 143 145 145

147 147 148 148 151 152 153 153 154 154 156 156 157 162

165 165 169 169 171

viii

Contents

8.3 8.4

8.5 8.6 8. 7

9

The Constan! Width Approximation (Carter Equation 11) The Power Law Approximation to Surface Growth 8.4.1 The Consequences of the Power Law Assumption 8.4.2 The Combination of the Power Law Assumption with lnterpolation Numerical Material Balance Differential Material Balance Leakoff as Flow in the Porous Medium 8.7.1 Filter-cake Pressure Drop 8.7.2 Pressure Drop in the Reservoir 8.7.3 Leakoff Rate from Combining the Resistances (Ehlig-Economides et al. [61) References

Coupling of Elasticity, Flow and Material Balance 9.1

Width Equations of the Early 2D Models 9.1.1 Perkins-Kern Width Equation 9.1.2 Geertsma-de Klerk Width Equation 9.1.3 Radial Width Equation 9.2 Algebraic (2D) Models as Used in Design 9.2.1 PKN-C 9.2.2 KGD-C 9.2.3 PKN-N and KGD-N 9.2.4 PKN-a and KGD-a 9.2.5 Radial Model 9.2.6 Non-Newtonlan Behavior 9.3 Numerical Material Balance (NMB) with Width Growth 9.4 Differential 2D Models 9.4.1 Nordgren Equation 9.4.2 Differential Horizontal Plane Strain Model 9.5 Models With Detailed Leakoff Description 9.6 Pressure Decline Analysis 9.6.1 Nolte's Pressure Decline Analysis (Power Law Assumption) 9.6.2 The No-spurt-loss Assumption (Shlyapobersky method) 9.6.3 Material Balance and Propagation Pressure Estimates of the Spurt Loss 9.6.4 Resolving Contradictions 9.6.5 Pressure Decline Analysis With Detailed Leakoff Description {Mayerhofer et al. Technique) References

1O

Contents

178 179 181 183 184 185 187 187

189 189 192 195 196

263

11

Fracture Height Growth (3D and P-3D Geometries)

267

11.1

269

196 199 200 201 202 202

245 245 245

246 246 247 247

247 249 252 256 258

11 .2 11.3

204

Equilibrium Fracture Height 11.1.1 Reverse Application of the Net-pressure Concept 11 .1.2 Different Systems of Notation 11.1.3 Basic Equations 11.1 .4 The Effect of Hydrostatic Pressure Three-dimensional Models 11.2.1 Surface Integral Method 11.2.2 The Stress lntensity Factor Paradox Pseudo-three-dimensional Models References

269 270 272 276 278

279 281 283 284

205

206 209 210 211

Appendix: Comparison Study of Hydraulic Fracturing Models: Input Data and Results

212

lndex

217 218

227 230 232

235

10.1

237 238 241

10.2

Retardad Fracture Propagation 10.3.1 Fluid Lag 10.3.2 Tip Dilatancy 10.3.3 Apparent Fracture Toughness 10.3.4 Process Zone Concept 10.3.5 The Reopening Paradox 10.4 Continuum Damage Mechanics in Hydraulic Fracturing 10.4.1 lip Propagation Velocity from CDM 10.4.2 CDM-NK Model 10.4.3 CDM-PKN Design Model 10.5 Pressure Decline Analysis and Trp Retardation 10.5.1 Resolving Contradictions with Continuum Damage Mechanics References

189

Fracture Propagation Fracture Mechanics 10.1.1 Griffith's Analysis of Crack Stability 10.1.2 Mott's Theory for the Rate of Crack Growth Classical Crack Propagation Criterion far Hydraulic Fracturing 10.2.1 Fracture Toughness Criterion 10.2.2 The lnjection Rate Dependence Paradox

10.3

172 174 174

ix

242 242

243

References

287 294

295

PREFACE This book addresses the theoretical background of one of the most widespread activities in hydrocarbon wells, that of hydraulic fracturing. It provides a treatment of basic phenomena including elasticity, stress distribution, fluid flow, and the dynamics of the rupture process from the point of view of the influence of those phenomena on the created fracture. Currently used design and analysis techniques are derived and improved using a comprehensive and unified approach. Numerical pxamples are elaborated to illustrate important concepts. Toe material grew out of university and industrial courses that have been taught

at Mining University of Leoben, Texas A&M University and several locations throughout the world. During these courses we have recognized that currently available monographs, often written by a great number of co-authors reflect diverse views, systems of notations and units. One of our main goals was to establish a common language that eases the way workers in the field can get acquainted with the material and experts of different background can communicate with each other. Our gratitude goes to our coworkers and students who have contributed a great List of Notation y y YCDM

y

y,.. Y~-N

º' º• ry

"

ry'

n, T/maD K K

K

. It is in such a reservoir that fluids are stored. Typical pore diameters range from 10- 7 m to 10-4 m, and reservoir porosities range frorn about 0.10 to (typical) 0.25 for sandstones to (extraordinarily high) 0.4 for sorne carbonate formations. While the porosity is important in defining the oil- or gas-in-place for a petroleurn producing reservoir or the storativity of an injection target, a second quantity, the permeability, k, describing the ability of fluids to flow in the reservoir, is essential. Toe permeability relates the pressure gradient, b. p, which is the driving force in the reservoir with the macroscopic fluid velocity, u.

u ex kt;.p.

(1.1)

This is the well known Darcy's law which in radial coordinates yields the following expression for the volurnetric flow rate, q: 2,rrkh dp q=---, µ dr

a2 p 1 ap ,pµc, ap ar2 +-; ar = -k-at'

(1.3)

where e, is the total system compressibility and t is the time. An analogous expression for gas (cornpressible) fl.ow ernploys the real-gas pseudopressure, m(p ), defined by Al-Hussainy and Ramey [2] as m(p)

=

p

J PO

and, thus,

2p -dp,

(1.4)

µZ

1 am(p) ,pµc, am(p) - -2+ - - - = - - - - - . Or rar k Bt a2m(p)

Toe non-petroleum reader is referred to References [3]-[6] and references therein for the developments and solutions to Eqs. 1.3 and 1.5 which are standard in petroleum, geothermal and groundwater engineering. Of interest are the constant-rate and the constant-pressure-at-the-well solutions. Toe general forro of the constant-rate solution is q=

2rrkht;.p

Toree different types of flow rnechanisms can be distinguished: transient, or in:finite-acting behavior, steady-state with constant outer boundary pressure, Pe, and pseudosteady-state, denoting a no-flow outer boundary condition. Table 1.1 contains the expressions for the driving pressure gradient b.p and the dimensionless pressure function, p D, for the three flow mechanisms. Analogous expressions can be written for compressible (gas) flow using b.m(p) instead of 6.p (see Dake [3]; Economides and Ehlig-Economides [6]). Interestingly, for transient rate production at constant PwJ, the solution yields

q

=

2rrkh(p; - Pwf)

1 µ-

(1.7)

,

qD

and the PD for constant rate is very nearly equil to the 1/qv for constant pressure production for almos! al] times (see Earlougher [7]). The relationship between q and Pw f and the antecedent engineering activities for their optimum adjustment are the essential functions of petroleum production engineering (see Economides and Ehlig-Economides [6]). Table 1.1

Pressure gradients and dimensionless pressure functions for radial reservoir flow at the well flp

Transient (infinite acting reservoir)

Pn

p; - Pwf

' . ( - 4to 1 ) Po= -2,Ei and to

kt = ---_, 4>µ,c,r;,

Po= ½Ont0 +0.8091)

Semilogarithmic approximation at to > 100

= In!!.. r.

Steady state

Pe - Pwf

p0

Pseudosteady state

P- PwJ

0.472re p0 =in--

E 1 = Exponential integral p¡

Pe

p Pwf

(1.5)

(1.6)

µpD

(1.2)

where µ is the viscosity, r is the radial distance, and h is the reservoir thickness. Combination of the continuity equation, Darcy's law and an equation of state, describing incompressible fluid, yields the well known diffusivity equation

3

re

rw

= Initial reservoir pressure = Outer boundary constan! pressure = Average reservoir pressure = Flowing bottom-hole pressure = Outer boundary radius = Well radius.

r.

4

Hydrau/ically induced fractures

1.2. 1

Flow from a tractured we/1

The Near-we/1 Zone

high 9.87 x 10- 14 m 2 (100 md) and PwJ

=2 x

5

107 Pa. Use the steady-state expression

of Eq. 1.8.

Converging radial flow de facto exaggerates the impact of the near-well zone. It is clear from Eq. 1.6 that for (e.g. steady-state) flow, the driving pressure gradient is proportional to the logarithm of the radial distance. An alternative way to state this is that for a constant production rate, the same amount of pressure gradient is consumed in the first meter as in the next 10 m, the next 100 m, etc. Thus, by analogy, it should be obvious that alterations to the natural permeability in the near-well zone would be critica! to the well production or injection rate at constant ó. p. Permeability-altering phenomena occur frequently in almost ali well operations including drilling, well completions or "workovers". Reduction ofthe near-well reservoir permeability is common, is referred to as damage, and has been characterized by a dimensionless skin effect, s (see Van Everdingen and Hurst [8]) analogous to the film coefficient in heat transfer. This skin effect, implying a steady-state pressure drop, is added to the dimensionless pressure in Eq. 1.6, resulting in a change in the well production or injection rate: q=

2nkht;p µ(po +s)

.

(1.8)

Toe reader is referred to Chapter 5 of Economides et al. [6] for an extensive description of the various causes of near-well damage, certain mechanical contributions to the skin effect and quantification of its impact. Toe skin effect is determined through the pressure transient testing of a well. A large and positive value implies damage or a flow impediment due to a mechanical reason (e.g. s = 20), whereas s = O is for undisturbed permeability in a vertical well. Zero skin could imply damage in a deviated well. A negative skin implies stimulation where the near-well permeability is larger than the original reservoir value. The latter case can be accomplished through matrix stimulation, which includes a seríes of possible chemical treatments intended to remove near-well damage once its nature is identified (see Economides et al. [6]). Larger post-stimulation permeabilities are possible, although rare. This could happen if the formation itself reacts with the injected stimulation fluids (e.g. a hydrochloric acid, HCI, solution and a carbonate rock). Hydraulic fracturíng may be attempted in those cases where matrix stimulation cannot result in an economically satisfactory well production or injection rate. To understand the need for an altemative to matrix stimulation the following example is offered.

Example 1.1

Matrix Stimulation vs. Hydraulic Fracturing

Suppose that a well with rw = 0.1 m is drilled in a reservoir with re = 300 m, h = 20 m and Pe= 3.5 x 10 7 Pa. If the fluid viscosity f.L = 1 x 10-3 Pa- s anda well test has provided s = 10, investigate the incremental well flow rate before and after a completely successful matrix stimulation (i.e. with s = 10 and s = O, respectively) for a range of penneabilities from a low value equal to 9.87 x 10-18 m2 (0.01 md) to a

Solution For k

= 9.87 x

10- 14 m2 , Eq. 1.8 yields (2)rr(9.87

q=

X

10- 14 )(20)(3.5

X

[3ºº

107

l

(1 x 10- 3 ) ln-+s

0.1

-

2

X

10 7 )

0.186 8+s'

and for s = 10, q = 1.03 x 10- 2 m3 /s whereas for s = O, q = 2.32 x 10- 2 m3 /s. Both the incremental flow rate (1.29 x 10- 2 m3 /s = 7010 barrels/day) and the post-treatment rate itself (2.32 x 10- 2 m3 /s = 12 600 barrels/day) are very attractive, pointing towards matrix stimulation. Assuming that a minimum well production rate equal to 9 .2 x 10-5 m3/s (50 barrels/day) is required, then from Eq. 1.8, with s = O, the minimum reservoir permeability for which matrix stimulation is attractive would be k = 3.9 x 10- 16 m2 (0.4 md). In production engineering the attractiveness of the stimulation is subject to the costs of the treatment which must be balanced against the benefits of the incremental production rate of 5.1 x 10- 5 m 3/s (28 barrels/day). In this exercise, for permeabilities less than 3.9 x 10- 16 m 2 (or in sorne cases for much higher permeabilities if economic considerations indicate) hydraulic fracturing is likely to be the appropriate well stimulation operation. O

1.3

Flow from a Fractured Well

Once a hydraulic fracture is created in a well or, in the not uncommon case, where the well intersects a natural fracture, fluid will flow normal to the fracture face from or to the reservoir (production or injection) and then along the fracture path from or to the well. Far almos! all depths of interest (as will be expounded upon in detail in Chapter 2) a hydraulic fracture will be largely vertical. Gringarten and Ramey [9] have described the :flow performance of an infinite-conductivity fracture whereas Cinco-Ley and Samaniego [10] dealt with the finite-conductivity fracture case. The latter is a reasonable description of created hydraulic fractures. In the case of an infinite-conductivity fracture (the upper limit of high conductivity) flow of fluid is characteristically linear, i.e., from the reservoir into the fracture. Once the fluid enters the fracture, it is presumed to enter the wellbore instantaneously, relative to the time it would take without the fracture. For the finite-conductivity fracture a discernible linear flow develops within the fracture, in addition to the linear component from the reservoir into the fracture, hence the characteristic term bilinear flow (see Cinco-Ley and Samaniego [10]). Figure 1.1 is the Cinco-Ley and Samaniego [10] solution, as plotted by Agarwal et al. [11] for the transient flow of a finite-conductivity fractured well. On the ordinate is the dimensionless pressure, PD, on the abscissa is the dimensionless time, lD:ct and the parameter is the dimensionless fracture conductivity, F CD·

- - - - - - - - - - - - - - - - - - - - - - - - - - · --------

6

Hydraulically induced fractures

Hydraulic fracture desígn

10

3

Region of

~

Dimensionless fracture conductivity Feo

"'"' ~

0.1

Q

o.

Region of line&r flow

bilinear flow

\

2.5

\

::,

'€ ~

'-

!l.

"'"' .!!l

e:: o ·¡;; e::

1\ 2

E

~-

+

10-1

_ 2nkhT.,,[&n(p)] G Po

" E o

qTp.e ID,/

1.5

\

85

'

=....!E__ ,j,¡.,c,:,:}

Fa;,=~ kx,

10-2

0.5

10·5

104

10-2

10·'

10-1

10-1

These are defined for liquid (oil) as: 2rrkh(p, - Pw¡)

qµ,

kt

tvx¡

= ---,, µ,c,x

(1.9)

,

(1.10)

1

and Fcv

k¡w

= --,

(1.11)

kx¡

In Eqs. 1.9 to 1.11 variables are as defined in Eq. 1.6 and Table 1.1, except far the fracture half-length, x¡, the fracture permeability, k¡, and the propped fracture width, w. Toe values of the fracture half-length and fracture conductivity are the essential quantities for the prediction of fractured well performance. Toe Cinco-Ley and Samaniego (10] solution becomes indistinguishable from the Gringarten and Ramey [9] solution far F CD > 300. For practica! purposes they can be considered as the same far F CD > 70. Long-term fractured well performance results in pseudoradial flow, and the presence of a hydraulic fracture of half-length, x ¡ and conductivity, F CD, can be manifested by an equivalent skin effect, sf, which can be read from Figure 1.2.

Example 1.2

ºº

10'

1

º'

10'

Figure 1.2 Equivalent skin effect for pseudoradial flow into a fractured well [10]

Finite-conductivity fracture solution. Dimensionless pressure vs. dimensionless time [11]

PD=

1

Feo

Dimensionless Time, toxr Figure 1.1

7

Performance of a Fractnred vs, an Unfractnred Well

Suppose that the well in Example 1.1 with penneability k = 3.9 x 10- 16 ni2, and for which matrix stimulation has been deemed unattractive, is hydraulically fractured. A

typical fracture permeability, k¡, is 9.87 x 10-11 m2 (100000 md) and the propped fracture width is 5 x 10-3 m. Calculate the steady-state production rate if the fracture half-length is 300 m.

Solution From Eq.1.11 Feo =4.2 and, therefore, from Figure 1.2, s¡ +ln(x¡/rw) = 0.96. Substituting the values of x 1 and rw (= 0.1 m) this would lead tos¡= -7. Using Eq. 1.8, for steady-state production and s¡ = -7 results in q = 7.35 x 10-4 m 3 /s (400 barrels/day) which is an 8-fold increase over the best case that this well would produce with matrix stimulation (i.e. s = O). It is essential to note that once a well is hydraulically fractured the overwhelming portian of the total flow is through the fracture, bypassing the damage zone and, thus, any pretreatment radial. skin effect can be ignored. O

1.4

Hydraulic Fracture Design

Toe proper engineering approach to hydraulic fracture design is to maximize the post-treatment performance and ensuing benefits at the lowest treatment costs. Thus, an economic criterion such as the net present value (NPV) has been employed for this purpose: the optimum fracture size would coincide with the maximum NPV (see Meng and Brown (12]), A common hydraulic fracture design optimization procedure starts from a fracture size, usually denoted by, but not limited to, the fracture half-length.

8


A fracture-propagation model then describes the hydraulic fracture geometry defi-

nitely including the width and, with an appropriate model, the fracture height. This issue is addressed in detail in Chapters 9-11. Toe required fracturing fluid volume is then estimated through a material balance accounting far the created fracture volume and the fluid leakoff normal to the fracture faces. This calculation simultaneously provides the required injection time.

Chapter 8 contains fracture leakoff rnodels and the manner in which they are incorporated in the fracture-propagation material balance. There are severa! techniques to estimate the required mass of proppant materials. Toe calculation depends on the manner of proppant addition to the fracturing fluid slurry. A common method suggests a ramped proppant schedule (see Nolte [13]) with its onset depending on the leakoff characteristics. Thus, after the end of injection the mass of proppant leads to the propped fracture width assuming that the fracture length is either equal to the hydraulic length or it is truncated by sorne practical criterion, e.g. where the width is equal to three proppant diameters. Toe choice of proppant is critical sin.ce the fracture permeability at the expected in situ stress depends on the strength of the proppant (see Brown and Economides [14]). Thus, the propped width, w, the fracture permeability, k¡, the assumed fracture half-length, x¡ and the reservoir permeability, k are sufficient to allow the forecast of the post-treatment well performance using the model presented in Section 1.3. This prediction leads readily to the future incremental benefits which, when discounted to the present, constitute the net present value of the incremental revenue. Inherent to this design procedure is the estimation of the required fluid volume, proppant mass and time of injection. These are the main components of the treatment costs which, when subtracted from the present value of the incremental revenue, lead to the NPV, specific for the assumed fracture half-length. Toe procedure is then repeated with increments of the fracture half-length and for each the corresponding NPV is determined. Optimum x ¡ is the one corresponding to the maximum NPV. In an appropriate engineering design it is this treatment that should be executed. Typically indicated half-lengths may range from less than 100 m for a higher permeability reservoir to more than 500 m for a low-permeability forrnation. With the advent of the tip screen-out technique ("frac & pack"), especially in highpermeability, soft, formations, it is possible to create short fractures with unusually wide propped width. In this context a strictly technical optimization problem can be formulated: how to select the length and width if the propped fracture volume is given. Example 1.3 deals with this problem.

Example 1.3

Hydraulic fracture design

fracture.) Use a realistic fracture permeability, taking into account possible damage to the pr~ppant: k¡ = 1 x 10- 11 m2 . Assume that the created fracture height equals the formatton th1ckness. Use the Cinco and Samaniego graph, Figure 1.2, which assumes pseudoradial flow.

Solution The. same propped volume can be established creating a narrow, elongated, fracture or a w1de but short one. The production rate will depend on the decision according to Eq. 1.8, which for steady-state production rate takes the form

q=

rw

Th~ pse~dor~dial, steady-s~!e flow im~lied by this relationship should emerge relatively rap1dly m h1gher-permeab1hty format1ons, which are the normal candidates for "frac & pack" treatments. Obviously, our aim is to minimize the denominator. This can be accomplished using the Cinco and Samaniego graph, which is a plot of the function J 1, defined by /1(log10 Feo)= s¡

+ In x¡. rw

We will use the function /1, replotted in Figure 1.3 for convenience. Given the function f 1 the denominator of the production rate can be expressed as re X¡ In- -In-+ / 1 (log10 Fc0 rw rw

),

5

¿

111111

4 3

2

.,~

"r~

V

~

o ·1

Optimal Fracture Conductivity

Consider once again the reservoir and well data of Example 1.2 (k = 3.9 x 10- 16 m2, h = 20 m, re= 300 m, µ, = 1 x 10-3 Pa-s, Pe= 3.5 x 107 Pa and Pwf = 2 x 107 Pa). Determine the optimum fracture half-length, x¡, the optimum propped width, w, and the optimum steady-state production rate if the volume of the propped fracture, V¡ = 100 m3 , is given. (Note that V I is the volume of the two-wing

µ

r, ln- +s¡

,..

fllli

.,.,

.,,rf' V

.... ~

I'~

1

~

~'•

~

·2 10·'

10•

10 1

10'

10'

Feo

Figure 1.3

Functions for optímal fracture conductivity as used in Example 1.3

9

------ - - - - - - - -

10

Hydrau/ical/y induced fractures

Treatment executíon

which can be further simplified to give lnre - Inx1

The optimum production rate (assuming Pe 107 Pa) is

+ J 1 (log 10 F cD).

2Tí

From the above expression we can eliminate the half-length using the relationship, V 1 = 2whx1 , and the definition of the fracture conductivity, Eq. 1.11. As a result, we arrive at the following minimization problem:

q

X

3 •9

and is plotted as a straight line in Figure 1.3. Toe function J 3 , which we wish to minimize, is simply the sum of J 1 and J 2- As seen from Figure 1.3 it has a minimum at F CD.opr = 1.2 where f 3 ,opr = 1.45. Therefore, the following results hold: Toe optimum half-length is given by

rv;;:;

V2.4M·

X¡=

the optimum width is obtained from w

= ✓ 0.6V¡k hk¡

'

and the optimal steady state production rate is 2rrkht.p

q=

µ

ln r, - ln

/Yifj- +

.

100

w=

X

10-ll

- - - - - - - = 232 m, 2.4 X 20 X 3.9 X 10-l6 0.6

X

100 X 3.9 X 1020 X 10-ll

16

_ 16

20(3.5 X

X )

107 - 2 )0-J

X

=2 x

10 7 )

100

3

X

10-ll

- - - - - - ~ + 1.45

ln300- In

2

X

20

X

3.9

X

10-10

3

10- m /s(247 barrels/day).

In general it is necessary to check if the resulting half-length is less than re (otherwise x¡ has to be selected to be equal to re)- Similarly, one has to check if the resulting optimum width is re alis tic, i.e., it is greater than, say, three times the proppant diameter (otherwise a threshold value has to be selected as the optimum width.) In our example both conditions are satis:fied. Toe above example provides a deeper insight into the real meaning of dimensionless fracture conductivity. Toe reservo ir and the fracture can be considered as a system working in series. The reservoir can deliver more hydrocarbon if the fracture is longer but with a narrow fracture the resistance to flow may be significant inside the fracture itself. The optimum dimensionless fracture conductivity (F CD.opr = 1.2) corresponds to the best compromise between the requirements of the two subsystems. O

1.5

Treatment Execution

Hydraulic fracturing is a massive operation, frequently resulting in the injection of more than 2000 m3 of fracturing fluids, 5 x lü5 kg of proppants at bottomhole pressures that could be over 5 x 107 Pa (corresponding to wellhead pressures of 2 x 107 Pa) while employing as many as two dozen active ar standby pumping units each capable of delivering 1500 to 2000 hhp (1100 to 1500 kW). Analogous power may be available on specially designed stimulation vessels for offshore operations. Figure 1.4 is a schematic depiction of the execution operation. Fracturing fluids with appropriate addit.ives are blended with metered proppant and then injected through appropriate fracturing strings into the target forrnation. Below, there is a brief description of fracturing fluids, their expected performance, the additives that affect this performance and comrnon propping materials. Brown and Econornides [14] contains a much more detailed description along with large amounts of data required for the selection of fluids and proppants. Chapters 5 to 7 of this book describe the rheology and fluid mechanics of fracturing slurries.

1.45

Retuming to our numerical example the following results are readily calculated:

X¡=

10

107 Pa and PwJ

= - ----¡c'======éx~=--= 4.54 x

where the only unknown variable is F cv- Toe first two tenns are constant, and hence do not affect the location of the mínimum. Toe last two terms do not contain any problemspeci:fic data. Therefore, the optimum F CD is a given constant for any reservoir, well and proppant. (Moreover, the same optimum F cv would result for pseudosteady state production rate.) To find the optimum F CD we introduce two new functions: the first one, denoted by f 2, is defined by 2-303 f2(log 10 Fco) = - -log 10 Fco, 2

X

= 3.5 x

11

= 0.0ll m.

1.5. 1

Fracturing Fluids

Fracturing fluid properties are expected to facilitate fracture initiation (breakdown), fracture propagation and proppant transport while they minimize leakoff and longterm residual darnage to the proppant-pack permeability. Viscosity is, thus, the essential property and may be augrnented by additives during execution. It must be destroyed by other additives after the treatment.

12

Treatment execution


Figure 1.4 Toe fracturing operation. Fracturing fluids and proppants are blended and injected downhole at the target fonnation

The ideal fluid has low viscosity in the horizontal and vertical tubulars to reduce the friction pressure and, therefore, the required treating pressure. After the fluid enters the fracture, the viscosity should have a high value to cause a larger width and better proppant transport. In addition, the same agents that enhance viscosity may be used for the building of a filtercake on the fracture walls to reduce leakoff. After the treatment, the high viscosity is no longer needed but, instead, it is highly detrirnental to the flow of produced or injected fluids. Thus, it must be reduced considerably. These contradictory functions are essential elements in the fracturing fluid design. Fracturing fluids have been based on water, oil, mixed water and oil (emulsions), mixed water and gas or mixed oil and gas (foams). For water-based fluids, common polymer thickeners are hydroxyethyl cellulose (HEC) and hydroxypropyl guar (HPG) in quantities varying (in field units) from 20 lb/Mgal (2.4 kg/m 3 ) to 80 lb/Mgal (9.6 kg!m 3 ). At ambient conditions these polymer solutions may lead to viscosities up to 0.1 Pa • s (at expected shear rates in a fracture) but at reservoir temperatures (T = 65ºC to 115ºC or even highér) they

13

have considerably reduced viscosities (e.g. < 2 x 10- 2 Pa-s) which are insufficient for proppant transport. Required minimum viscosity in the fracture, where large shear rates at the tip may reduce the viscosity further, is considered to be 0.1 Pa • s (see Brown and Economides [14]). To increase the viscosity substantially, crosslinkers of the polymer chains have been employed. For temperatures below 115ºC borate crosslinkers are considered desirable. For higher temperatures, organometallic crosslinkers such as titanium and zirconium complexes are necessary. To meet the demand for lower viscosity in the tubulars and higher viscosity in the fracture, delayed crosslinkers have been used. These are triggered by activators that are sensitive to the high-shear values as the fluid passes through the perforations. To avoid oxidative degradation in the fracture, oxygen scavengers are often added to the fluid. A "40-lb borate-crosslinked gel" ( 40 lb/Mgal = 4.8 kg!m 3 ) at a reservoir temperature of 90ºC would still have an apparent viscosity of 0.2 Pa • s after 4 hours of injection-induced shear (see Brown and Economides [14]). Oil-based fluids have been used in water-sensitive formations with a phosphate ester as the gelling agent. These :fluids are losing their "market share" because of environmental and obvious safety considerations. Focus of research has been the development of non-intrusive, non-damaging waterbased fluids. A very common practice is the foaming of fracturing fluids with carbon dioxide or nitrogen. Foam qualities (gas volume fraction) from 50 to 90% have been used with 70 being very common. Toe purpose in using these fluids is to minimize filtrate damage and, more importantly, to facilita te the cleanup: fluid flowback after the treatment. After the injection stops the formidable task of breaking down the polymer emerges. Unbroken polymer chains result in a marked reduction in the permeability of the proppant pack. Thus, oxidizers or enzymes, and at times encapsulated breakers, are added to the fracturing fluid. The breaking action is critica! to the success of hydraulic fracturing and is the subject of active ongoing research.

1.5.2

Proppants

Toe hydraulic width created during the injection is reduced to zero after supplied fracturing pressure subsides to the closure pressure, unless propping materials are used. It is this residual propped width that can be used for the forecast of fractured well performance that was outlined in Section 1.3. Proppant size and proppant strength are the main criteria for selection. Toe general families of proppants are divided into low, intermediate and high strength. The demand for strength is directly related to the level of stress that the proppant will experience in the long term. Low-strength proppants are natural sands in typical sizes from 12/20 mesh to 20/40 mesh (average particle diameter is 2 x 10-4 m to 1 x 10-4 m). They are usually attractive at depths less than 2000 m because although they are the Ieast

----.·

14

--·

Hydraufical/y induced fractures Mechanics in hydraulic fracturíng

expensive proppants they undergo severe crushing resulting in substantial proppantpack permeability reduction. An NPV-based design procedure allows the balancing of these effects and is an invaluable aid in deciding on the appropriate proppant. Frequently, sands are coated with resins which allow the fragments to stay together and thus maintain a high fracture permeability at larger stress values. Synthetic, intermediate- and high-strength proppants are used at depths up to 3000 and 5000 m, respectively. Brown and Economides [14] contains an extensive coverage of proppant properties including their degradation from long-term exposure to stresses.

1.6

Data Acquisition and Evaluation for Hydraulic Fracturing

Field and laboratory measurements are often conducted before and after a hydraulic fracture treatment to predict and evaluate fracture geometry and conductivity. Data acquisition involves well logging, core laboratory investigations, well testing and fracture calibration injections. Seismic techniques, although expensive, can be used in critica! cases. The data acquisition has a cost and, thus, the selection of tests depends on the benefits from the knowledge of particular variables and the opportunity cost of their ignorance. Appropriate selection of data acquisition techniques is an essential part in the success of hydraulic fracture design. Although this book falls outside the scope of data acquisition and evaluation, below is an account of common techniques complete with appropriate references for further reading.

1.6.1

We/1 Log Measurements

Pretreatment log measurements are intended to obtain mechanical properties of the target and adjoining intervals and predict stress values and, especially, stress contrast. This would give indications for the fracture height migration (see Newberry et al. [15]; Ahmed et al. [16]). Borehole acoustic televiewers are used for the measurement of sonic travel time and amplitude (see Pasternak and Goodwill [17]; Plumb and Luthi [18]). ldentifying borehole ellipticity and the presence of vugs and natural fractures provide evidence of stress anisotropy and, thus, the expected hydraulic fracture azimuth. Mechanically fitted dipmeter logs with four and six arms are used to detect open-hole ellipticity and stress-related wellbore breakouts. These e:ffects have been correlated clearly with stress anisotropy [19-21]. Dipmeter logs with a dense array of microresistivity detectors provide wellbore images where natural fissures can be mapped. These devices are used in both vertical and horizontal wells [18,22,23]. More recently, a downhole extensiometer has been introduced by Lin and Ray [24] with two six-arm calipers and very sensitive pressure transducers to detect small

15

wellbore deformations, corresponding to stress anisotropy before the treatment and, potentially, the stress induced after a treatment.

1.6.2 Core Measurements The appearance and disappearance of fissures as the stress on a core is reduced or increased and the counting of these fissures has been used for the determination of stress anisotropy in oriented cores. Strain relaxation and its measurement with sensitive devices has been referred to as anelastic strain recovery (see Blanton [25]; Teufel [26]). Oriented cores are specially prepared and fitted with gauges which detect the relative displacement resulting from strain recovery. The reverse procedure is used for the di:fferential strain recovery analysis where cores are re-stressed and the relative differences in displacements are correlated with stress aniso trap y (see Strickland and Ren [27]).

1.6.3

We/1 Testing

Pressure transient testing is widely practiced by engineers dealing with porous media. Analysis of the pressure and rate data while the ?,'ell is flowing (drawdown) or shut in (buildup) or observed by another well (interference) allows the determination of important well and reservoir variables. These include the skin effect, reservoir permeability and permeability anisotropy, types and locations of boundaries and formation heterogeneities (such as two-porosity systems.) For wells that could be candidates for hydraulic fracturing, a pretreatment well test can revea! the reservoir permeability and skin effect allowing a decision for stimulation (matrix vs. fracturing vs no treatment at ali.) If fracturing is indicated the reservoir permeability is a critical variable for the design optimization (see Balen et al. [28]). A post-treatment well test, and assuming the reservoir permeability is known, can provide the fracture half-length and fracture conductivity. Such a determination is essential for the design evaluation. In Chapter 11 of Reference 6, modem well test analysis techniques are presented, complete with well-test design guidelines and types of tests that are presently practiced in the industry.

1. 7

Mechanics in Hydraulic Fracturing

Rock, fracture and fluid mechanics are critica! elements in the understanding and engineering design of hydraulic fracture treatments. Rock mechanical properties dictate the stress and stress distribution at depth (Chapter 3) and elastic properties control the created fracture geometry (Chapters 2 and 4). Contrast between the properties of adjoining layers controls the vertical fracture height migration (Chapter 11).

16


Fracture mechanics is an obvious field of study in this endeavor allowing for the interaction between the provided pressure and the resisting stresses. Tip propagation mechanisms and their effect on the observed net pressures are subjects of ongoing research and controversies (Chapters 10 and 11). The combination of rock, fracture and fluid mechanics results in the study of fracture propagation, the interaction and sensitivity between treatment variables and the formation to be fractured and the resulting hydraulic fracture morphology. These concepts are treated extensively in Chapters 9 to 11. They are also the central elements of this book.

References l.

2. 3. 4. 5. 6. 7. 8. 9.

10. 11.

12.

13. 14.

15.

16.

Anonymous, OPEC's Facts and Figures, Organization of Petroleum ~ Exporting Countries, Vienna, 1993. Al-Hussainy, R. and Ramey, H.J., Jr.: Applications of Real Gas Theory to Well Testing and Deliverability Forecasting, JPT, (May), 637-642, 1966. Dake, L.P., Fundamentals of Reservoir Engineering, Elsevier, Amsterdam, 1978. Craft, B.C. and Hawkins, M. (Revised by Terry, R.E.) Applied Petroleum Reservoir Engineering, 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1991. Amyx, J.W., Bass, D.M. aod Whiting, R.L.: Petroleum Reservoir Engineering: Physical Properties, McGrawHill, New York, 1960 Economides, M.J., Hill, AD. and Ehlig-Economides, C.A.: Petroleum Production Systems, Prentice Hall, Englewood Cliffs, N.J., 1994. Earlougher, R.C., Jr.: Advances in Well TestAnalysis, SPE, Dallas, TX, 1977. Van Everdingen, A.F. and Hurst, N.: Toe Application of the Laplace Transformation to Flow Problems in Reservoirs, Trans. AIME, 186 305- 324, 1949. Gringarten, A.C. and Ramey, AJ., Jr.: Unsteady State Pressure Distributions Created by a Well with a Single-Infinite Conductivity Vertical Fracture, SPEJ, (Aug.), 347-360, 1974. Cinco-Ley, H. an9, Samaniego, F.: Transient Pressure Analysis for Fractured Wells, JPT, 1749-1766, 1981. Agarwal, R.G., Carter, R.D. and Pollock, C.B.: Evaluation and Prediction of Performance of Low-Permeability Gas Wells Stimulated by Massive Hydraulic Fracturing, JPT (March), 362-372, 1979; Trans. AIME, 267. Meng, H.Z. and Brown, K.E.: Coupling of Production Forecasting, Fracture Geometry Requirements and Treatment Scheduling in the Optimum Hydraulic Fracture Design, SPE Paper 16435, 1987. Nolte, K.G.: Determination of Proppant and Fluid Schedules from Fracturing Pressure Decline, SPEPE, pp. 255-265, July 1986. Brown, J.E. and Economides, M.J.: Practica! Considerations in Fracture Treatment Design, in Economides, M.J.: Practica/ Companion to Reservoir Stimulation, Elsevier, Amsterdam, 1992. Newberry, B.M., Nelson, R.F. and Ahmed, U.: Prediction of Vertical Hydraulic Fracture Migration Using Compres3ibility and Shear Wave Slowness, Paper SPE/DOE 13895, 1985. Ahmed, U., Newberry, B.M. and Cannon, A.M.: Fracture Pressure Gradients Determination from WeII Logs, Paper SPE/DOE 13857, 1985.

References

17. 18. 19.

20. 21. 22. 23. 24. 25. 26.

27. 28.

17

Pastemak, E.S. and GoodwilI, G.D.: Application of Digital Borehole Televiewer Logging, Proc. 24th Annual SPlVLA, 1983. Plumb, R.A. and Luthi, S.M.: Application of Borehole Images to Geologic Modeling of an Eolian Reservoir, Paper SPE 15487, 1986. Brown, R.O., Forgotson, J.M. and Forgotson, J.M., Jr.: Predicting the Orientation of Hydraulically Created Fractures in the Cotton VaUey Formation of East Texas, Paper SPE 9269, 1980. Gough, D.I. and Bell, J.S.: Stress Orientations from Oil-Well Fractures in Alberta and Texas, Can. Jour. Earth Sci.., 18, 638-645, 1981. Zoback, M.O. and Zoback, M.L.: in Neotectonics, G.S.A., 1988. Svor, T.R. and Meehan, D.N.: Quantifying Horizontal Well Logs in Naturally Fractured Reservoirs - I, Paper SPE 22634, 1991. Meehan, D.N. and Svor, T.R.: Quantifying Horizontal Well Logs in Naturally Fractured Reservoirs - 11, Paper SPE 22792, 1991. Lin, P. and Ray, T.G.: A New Method to Determine In-Situ Stress Directions and In-Situ Formation Rock Properties During a Microfrac Test, Paper SPE 26600, 1993. Blanton, T.L.: The Relation Between Recovery Deformation and In-Situ Stress Magnitudes, Paper SPE 11624, 1983. Teufel, L.W.: Prediction of Hydraulic Fracture Azimuth from Anelastic Strain Recovery Measurements of Oriented Cores, Proc. 23rd U.S. National Rock Mechanics Symposium, 1982. Strickland, F. and Reo, N.: Predicting the In-Situ Stress of Deep Wells Using the Differential Strain Curve Analysis, Paper SPE 8954, 1980. Balen, R.M., Meng, H.-Z. and Economides, M.J.: Application of the Net Present Value (NPV) in the Optimization of Hydraulic Fractures, Paper SPE 18541, 1988.

2 LINEAR ELASTICITY, FRACTURE SHAPES AND INDUCED STRESSES A purely elastic body has a natural state to which the body returns if all the externa! forces are removed. An elastic deformation is therefore reversible: The work done on the body is saved as elastic energy which is totally recoverable. If deformations and their inducing forces (or forces and their inducing deformations) are connected by a linear relationship, this is linear elasticity. The appearance and propagation of a fracture means that the material has responded in an inherently non-elastic way and an irreversible change has occurred. At first glance, therefore, it seems that elastic theory (linear or even non-linear) might be of little use in fracture mechanics. Nevertheless, linear elasticity is a useful tool when studying fractures, because both the stresses and deformations (except for the fracture surface and perhaps the vicinity of the tip) may be still well described by elastic theory.

2.1

Force and Deformation

Forces considered in elastic theory (see Billington and Tate [I]; Fenner [2]) are distributed by nature. Surface forces are distributed along a surface and body forces along a volume. In both cases what really matters is the intensity, i.e. the force acting on a unit area of the surface or in a unit volume of the material. The action of the surrounding material on any volume element of it is transmitted by surface forces and thus, we concentrate on them. 2. 1. 1

Stress

The ratio of the force to the elementary surface area it is acting on is the force intensity called stress (or surface traction):

. (t:,F) , M

a= hm

llA.-o

measured in N/m2 or, briefly, Pa.

~

(2.1)

---------

20

Force and deformation

Linear e/asticity

The stress is a vector with magnitude and direction. An elementary ·surface is contained in a plane which can be rotated arbitrarily and hence there is an infinite set of stress vectors associated with a given point. The stress state is given if we provide an appropriate means to determine the stress corresponding to any arbitrarily selected plane direction. Stresses normal to the plane may be tensile or compressive, while those parallel to the plane are called shear. A normal stress is readily visualized based on everyday experience. To understand shear stress properly sorne abstraction is needed. Any stress can be decomposed into two orthogonal shear components and a tensile (or cornpressive) one. A common system of notation includes two suf:fixes: the first one refers to the direction of the stress while the second one denotes the direction of the outward normal to the plane on which it acts. A tensile stress (positive by convention) and a compressive stress (negative) have two identical suffixes. Shear stresses have different suffixes. To emphasize the difference, shear stresses are often denoted by r. If there is no danger of misinterpretation, the second suffix of a normal stress can be deleted (since it is identical to the first one.) Figure 2.1 shows an elementary cube whose edges are parallel to the Cartesian coordinate axes. There are nine stress components but they cannot be selected independently. Rotational equilibrium poses three constraints on them. Toe state of stress at a point is determined by six independent stresses: In Cartesian coordinares these

•zx

are Cf:w ayy, restrictions:

crzz, rxy,

21

ry:: and rzx. Toe remaining three components are given by the (2.2)

lf the six independent stresses are specified, the stress acting on any arbitrarily oriented (obligue) plane can be obtained by applying force balance. Toe word "obtained" means that we can calculare the three components of the stress vector. (The actual expressions will be given later.) Once the stress vector is known, we can decompose it into a normal and a shear component relative to the specified plane. Given the state of stress at a point, we may continuously change the orientation of the obligue plane while the magnitudes of the normal and shear .stresses are v~rying. It happens that there are three specific orientations where the shear stress vamshes and (at the same time) the normal stress has a local maximum. Toe three local rnaxima are called principal stresses. Toe three eigenvalues of the matrix (2.3)

denoted by u 1 2:: a 2 2:: a-3 give the magnitude of the principal stresses. The components of the corresponding eigenvectors are the direction cosines of the plane (with respect to which the maximum occurs) and hence cornponents of the direction vector of the principal stresses. Moreover, the eigenvectors are mutually orthogonal (a consequence of the symmetry of the matrix). In sorne applications the eigenvectors provide a natural coordinate system. In this coordinate system the matrix (2.3) will be diagonal. The eigenvalues of a diagonal matrix are the diagonal elements. If we know the directions of the principal stresses, then the only three additional data needed to specify the stress state are the diagonal elements of the matrix, i.e. 0'1, Uz and 0-3. In geologic applications often we may assume that one of the principal stresses is vertical. Toen one additional angle has to be given to specify the direction of the second principal stress in the horizontal plane. The third principal direction is also horizontal and orthogonal to the second one. Since such a direction is, unique the only additional data we need are the values 0-1, O'z and 0'3.

•yx

2. 1.2 Strain We can think of a deformation as a transition from a reference con:figuration into another one. Simple translation ar rotation of a rigid body are also deformations, but are of little interest in the present context. In elasticity theory the interest is with deformations, where the relative position of the points changes. For defining a suitable measure of the deformation let us consider two material points. If l is the original distance between the two points and l + b.l is the new distance, the engineering strain is defined by Figure 2.1

Stresses acting on one surface of an elementary cube

(2.4)

22

Linear etasticity

Material properties

Tensile strain corresponds to extension whereas compressive strain corresponds to contraction. By convention, strain associated with extension is negative and cornpressive strain is positive. However, in rock mechanics and especially in hydraulic fra~turing sometimes the opposite convention is more appropriate. Toe actual sign convention should be clear frorn the context. A shear strain is associated with plane Iayers sliding over each other. For small strains the angle of distortion (in radians) is a suitable measure of the shear strain. For the full definition of strain in the three-dimensional space, it is necessary to consider a point in the original configuration with coordinates x, y and z. After deformation, the new coordinates will be x + ux, y+ uy and z + Uz, respectively (see Figure 2.2). Toe quantities Ux, uy and Uz are the components of the displacement vector. With changing location of the original point the displacement may vary but smoothly. If we considera straight line starting from (x, y, z), parallel to the x. axis and of length 8x (where this length is short enough) then l

= 8x,

(2.5)

and l

+ tll = 8x +

aux -8x.

Hence, a suitable definition of the first component of the strain state, dance with Eq, 2.4 is Exx

aux = -. ax

Exx,

23

in accor-

(2.7)

Similar arguments lead to the definition of other strain components listed in Table 2.1. Again, six independent components (Exx, Eyy, Ezz, Exy, Eyz and Ezx) should be specified to give the state of strain at a given point.

2.2

Material Properties

Real materials have cornplex behavior when subjected to a stress :field. ldealized models help to understand the main features of the behavior. A perfectly elastic material stores the work done on it by externa! forces, and then it allows full recovery. How an elastic material responds with strain to a specific stress state (or vice versa) can be described by a constitutive equation. Of particular interest is the case where the constitutive equation is linear.

(2.6)

ax

2.2. 1 Linear Elastic Material )

(x'+V,..y'+u~)

For a linear elastic material the stress varies linearly with strain. Hooke's law states that under uniaxial compression the stress induced is proportional to the strain

I +ti/

(2.8)

where E is Young's modulus. As shown in Figure 2.3, the deformation in the x direction is accompanied by an additional deformation in the y direction. This "side

(X', y')

(x+e,,y+ey)•~'

(x.yJ\~/

Figure 2.2 Table 2.1 Second index First index

Displacement and strain Strain components,

cu

X

y

z

X

au, ax

, C"•ay + au,) ax

y

see xy

,C"'ax az -,C"' +au,) az ay

z

see xz

2

auy

ay

see yz

-2 - +au,) -

2

".w

v:-&xx

auz

a,

Figure 2.3

Uniaxial compression. Determination of Young's modulus and Poisson ratio

24

Linear elasticity

Material properties

effect" is given by

Table 2.2 Eyy

ª= = -VE,

(2.9) ).

where the Poisson ratio, v, is always positive and less than 0.5. In general, a static deformation test consists of (1) the preparation of a specimen of prescribed form, (2) the application of stress (or displacement) at sorne of the boundaries, (3) the measurement of the resulting displacement of the boundary surface ( or the resulting stress on the boundary surface). Toe uniaxial compression test illustrated on Figure 2.3 is suitable to determine Young's modulus and the Poisson ratio in one

experiment. Toe compressive stress and the strains are readily derived according to the expressions shown on Figure 2.3. Toe two material properties, E and v, are obtained from their definitions. Other simple tests give rise to other material properties. During the torsion of a circular bar around its axis, the shear stress and shear strain are related by (2.10)

where G is the shear modulus. Under hydrostatic compression the relative volume change is related to the hydrostatic pressure through the bulk compressibility, K. At this point an important question arises. Is there any relation between the observable material properties? In other words, how many independent properties are necessary to characterize the material already known to behave linearly? Starting with the generalized Hooke's law:

ª= ayy

ª"

Cu C21 C31

D v

=-

fz

= 2 x 1010 Pa = 20 GPa

= ~o,,-º0"'º04'5"'- = º· 2 ·

l

E

&

(a)

(e)

(b}

cr

cr

cr

0.2

•

From Table 2.2 we obtain G K

=

E ~-c--c 3(1 - 2v)

''

= 11 GPa,

(d)

Figure 2.4

and hence, Eq. 2.13 predicts

VL

3K+4G) = (3p

112

~

E

= -2 (!-+= 8.5 GPa, v)

9

9

X 11 X 10 +4 X 8.5 X 10 ) = (3----e-==--3 X 2700

&

(e)

~

&

Stress-strain relations of several types of materials

112

= 2900 ffi / S.

0

2.2.2 Material Behavior Beyond Perlect Elasticity A hypothetica1 linear elastic body "answers" with a continuously increasing strain to a linearly growing stress. No material can be loaded infinitely, because, eventually, it will fail. At this critica} value of the stress any further "strain" can be achieved easily because the material loses its ability to resist deformation. This is seen from the stress-strain curve (a) of Figure 2.4. Toe fact that the curve (up to the failure) is a straight line indicates that the material is linear elastic. The dashed line represents the brittle rupture. The stress-strain curve (b) corresponds to a plastic material in which strain occurs without any change in the stress. Toe work done in the plastic region is dissipated and the material flows. Curve (c) shows a material with an elastic and a plastic region. Curve (d) illustrates a material without a distinct elastic region. The stress necessary to start any deformation is called yield stress. Curve (e) corresponds to a material exhibiting typical nonlinear behavior, most likely due to continuous damage evolution. Toe limit of the elastic behavior depends on the type of loading. Many solids, failing at moderate tensile stresses, may carry much higher loading in the form of compressive stresses. In general, the failure will occur (or yielding will start) at speci:fic combinations of the three principal stresses. If we represent the state of the stress as a point in the three-dimensional stress space (taking the principal stresses as coordinates) then the stable and unstable states will be separated by a surface called

the envelope of stability or failure (yield) surface. A failure (or yield) criterion is the equation of this envelope. A detailed description of the material thus consists of an elasticity constitutive equation, a yield criterion and another constitutive equation valid in the post-yield region. Since yielding is inherently irreversible, a simple curve is not enough to represent the behavior in this region. In fact the actual behavior depends not only on the strain but on the history of the total loading process.

2.3

Plane Elasticity

Toe description of stress and strain in three dimensions is complicated. Fortunately, in many cases we can make simplifying assumptions to reduce the problem into a two-dimensional one.

2.3. 1 Plane Stress When one of the principal stresses is zero, the condition of plane stress is satis:fied. Toe plane stress condition is a good approximation, for instance, for a thin plate (Figure 2.5). If the sheet is in the x, y plane, load is allowed only in the same plane but the deformation is not restrained in the z direction. Toe following stresses are zero: (2.14)

28

Unear elastici'ty

Plane elasticity

uu=O

't¿:

Uyy

,.,.= '"' =O

,..,.=

X

rzy

=O

,,,

.

y

AII stresses remain on this plane

Figure 2.5

t

29

Material in the state of plane stress

.

It is useful to remember, however, that plane stress does not mean automatically the absence of strain in the z direction. (Toe plate may, for example "swell".)

X

'yx

Figure 2.6 Stresses acting within a plane

2.3.2

Stresses Relative to an Oblique Line (Force Balance /)

Now we retum to the problem of determining the normal and shear stress vectors relative to an oblique plane. Since we C,

= O,

X::::: O.

·•xy(X, O)

(2.31)

Toe :first condition states that the pressure acting on the line is compensated by the normal stress of the same magnitude. (The second argument of the unknown functions is y. Clearly, y= o- at the crack line.) Toe pressure is a function of the location x. It is supposed that the problem is symmetric with symmetry axes x = O (the pressure acts on both faces) and y= O (the function p(x) is an even function.) Since the above boundary conditions are written for the upper right quadrant only, the requirement for p(x) to be an even function is not restrictive in this half-plane, but it is understood that p(-x) is defined to be egua! to p(x). It is assumed that ali stresses disappear in infinity, i.e. the far-field stress state is zero. Non-zero far-field stresses will be treated in Section 2.6. Toe complete solution of the problem includes the construction of the stress state and the displacements

Pressurized line crack

at every location. For practical purposes, however, we are only interested in the displacement of the crack surface and the stress state at the tip and further along the crack axes. Mathematical solutions, based on the pioneering work of Muskhelishvili [9], have been accomplished by solving integral equations (England and Green [10], Green and Zema [11]) or applying integral transformation (Sneddon [12]). Toe solution procedure starts with the construction of a function g(~) according to ['

g(!;)

= }0

p(x)dx

(!;' _ x2)1/2'

o
(2.32)

Note that ~ is a dummy variable having the same dimension as x. Toe function g(~) has the same dimension as the pressure and it can be considered as a modified pressure summing up the effect of the pressure acting not only at the given location but at every other location. Once that function is known, the normal displacement of any point on the upper side of the crack line is given by u x O -

y( ' ) -

_...±._[ /;g(/;)d/; rrE' x (x' -

!;2)1/2'

O ::;x :Se,

(2.33)

where E 1 is the plane strain modulus, given in terms of the other properties in the last row of Table 2.2. By virtue of the second boundary condition, the displacement is zero outside the crack. Clearly, a similar displacement of the lower line occurs in the negative y direction, and hence the width of the crack is sirnply twice the value given by Eq. 2.33. Toe normal stress in the y direction is known inside the crack (it is the

34

Linear e/asticity

Pressurized crack

35

opposite of the pressure). Along the crack axes (but outside the open interval) the normal stress is given by 2 [

O"yy(x, 0)

=;

xg(c) (x' - c')'I'

g(O) - x [

tC§Jct, ]

O

(x' _ §')'!' ,

x > e,

(2.34)

The above equation is the one given by Sneddon (12, p. 318] with a modification. It is derived for the case of differentiable g(;). If the pressure is a continuous function of the location, then g(~) is differentiable. Fortunately, Eq. 2.34 gives correct results even for those cases where there are sorne distinct jurnps in the pressure. Along the line y = O the normal stress in the x direction equals that in the y direction: O".u(x, O) = ayy(x, 0), x > c. Along the same Iine the shear stress, rxy(x, 0), disappears (see e.g. Green and Zema (11, p. 276]). For severa! specific p(x) functions of practica! importance, the above integrals can be solved in closed form.

,

e

e

X

b

2.4.2 Constant Pressure e

If the pressure opening the crack is constant, p 0 , the g(g) function is given by

['

Po dx

Po"

gCsJ = lo (§' - x2 J112 = 2'

(2.35)

o XQ.

The maximum width is twice the normal displacement at x 8 = -,-(s1 +s2)xoln E rr

= O,

i.e.

(c+q

- -1) xo

l

x In

1 + cos

7rSz (

2(s¡

)

+ s2)

·( 2(s¡,rsz) + s2)

(2.49)

sm

Finally, the stress at the tip is

2x5x

x")112,

,:S:

(2.47)

2 2 2 1 [ q¡ + qz c x5 - 2q1q2xox + c2-x ] uy(x,O)=-,-(s1+s2) 4xoln-_--+xln 2 2 2 2 2 - 2x x 2 E rr q3 e x 0 + 2q1q2xox +ex 0 where(2.48) xi;)l/2,

X

+ s2)c sm . [ n-s2 = 8(s¡E'n2(s¡ + s2)

This condition was first derived by Khristianovich and Zheltov [3]. Toe corresponding displacement is obtained from Eq. 2.33. Again, two different expressions should be used, depending on the location x.

= (c2 _ qz = (c2 _

if if

(2.46)

= O, is satisfied if

xo =csm

q¡

x2)112, = {(xi;_ (x2 _ x5)lf2,

wo X> XQ

and the zipper crack equation, g(c)

q3

XQ·

6 x 10-, [ 20✓31 s+JIO0-x' ,r n J75 - x'

O

=o

2

2r 3r ) rre =-½(ax - ay) ( 1 + r; - ,.: sin(20).

a0

= (c,x + ay)-2(ax -

and r,e

= O.

O'y)cos(20),

(3.19)

(3.20)

60

Stresses in formations

Near~wellbore stresses

61

Considering only the directions parallel and perpendicular to the minimum horizontal stress directions, i.e. 0 = O and 0 = :n:/2, Eq. 3.19 simplifies further:

z

(3.21) and (3.22)

y X

Example 3.3

Calculation of the Stresses at the Borehole

A well has been drilled in a reservoir where the pore pressure is 2.0 x 107 Pa and the absolute minimum and maximum horizontal stresses are 2.4 x 107 Pa and 3.6 x 107 Pa, respectively. Calculate the effective stresses a9 at 0 = O and 0 = :rr /2. Assume that a= l.

Solution (a)

First, from Eq. 3.8 the effective far-field stresses are fYx

= 2.4 X

107

-

2.Q

uy

= 3.6 x

107

-

2.0 x 107

X

107

= 0.4 X

107 Pa,

and

z

= 1.6 x 107 Pa.

From Eqs. 3.21 and 3.22 the effective stresses at 0 = O and 0 (o-e),-o = (3)(1.6 x 107 )

-

= rr/2 are

0.4 x 107 = 4.4 x 107 Pa.

and (o-,),,.,,12

•re (b) Figure 3.4 Description and nomenclature for (a) Cartesian and (b) radial stress components

= (3)(0.4 x 107 ) -

1.6 x 107

-

-0.4 x 107 Pa. O

Suppose that in Example 3.3 the tensile strength of the rock was approximately equal to 4 x 106 Pa. The calculation shows the possibility for tensile failure to occur in a direction perpendicular to the minimum principal stress solely as the result of drilling the borehole. Such a value of tensile stress is in the range of typical tensile strengths of reservoir rocks. It should be noted that these induced stresses diminish rapidly to zero, away from the wellbore as shown in. Figure 3.5 (two-dimensional view) and Figure 3.6 (three-dimensional view). While the two-dimensional view of Figure 3.5 provides the stresses in one direction, Figure 3.6 provides their profiles around the well. As can be seen u0 (Figure 3.6a) anda, (Figure 3.6b) have severa! minima which would provide the "venue" for the second branch of a fracture. Once initiated from a point ora plane of the well the fracture starts propagating in a :first branch. Friction pressure resistance and other retardation effects could allow the fracture to find another point of minimurn stress concentration for a second fracture branch. Depending on the required path length (and thus the perforation phasing) it is conceivable that in certain cases a second branch may not develop. Toe near-well stress concentration

62

Stresses in formations Stress concentrations far an arbitrarily oriented we/1 12ox101:1

u,,= 3.3x10

affects greatly the near-well geometry and contact between the fracture and the well. Away from the well the far-field stresses take over.

7

Pa ""Y= Pa az= 3.3x107 Pa

•

3.3x10 7

100x10

63

r,..=0.1 m

ca • BOx101:1

a. sox101:1

e,

3.4

"•

Toe solution for the stresses and displacements because of an infinitely long circular hale in a homogeneous, isotropic, linearly elastic medium is given by the superposition of Kirsch's solution, the antiplane solution, and the solution for an intemally pressured hole (Figure 3.7, Deily and Owens [4], Bradley [5], Richardson [6]). To represent this solution mathematically, it is necessary to define the orientation of the borehole with respect to the in situ stresses. Define a as the angle between the ax(ah,min) directíon and the projection of the borehole axis onto the ax - ay plane, and f3 as the angle between the borehole axis and the a::: direction as shown in Figure 3.8. Toe rotation of the stresses from the in situ system of coordinates to the borehole local system of coordinates is obtained from the following:

:~

40x10 6

e,

20x10 6

o 0.1

10

Radial distance from weU,

Figure 3.5

100

1000

m

Stress concentration around and away from a wellbore

(a)

Stress Concentrations for an Arbitrarily Oriented Well

sin 2 f3

)

2

hw n

q

Umax

ctKE

i )'I•

2

= Uavg

and therefore the dependence of the wall stress on the nominal Newtonian she¡p- rate can be given as

Bingham plastic, power law and Newtonian formulas for slot flow

Bingham plastic

Umax

111

Slot flow

Rheology and laminar flow

110

óp

12µq

L

hw3

+ ~)w

µ,w=K

12uavg

( + ),_, ( )"-' 1

2n

-3n

6uavg

-w

- -2"-'(1+2n)" -nKwl-"un-I µ, eavg

3

xh-nw-'-4>

= Kjl'

r

)ª""' D

D(l+n)/n

8uavg

y,,,=D

z3n+2n-"K (

(Hw)/wl

= µ,}'

.

t:.: =

4 = - - ~ - 2 - - - -

Toe nominal Newtonian shear rate can be defined for any non-Newtonian fluid as (8uavg/D), but in general it differs from the true wall shear rate. Toe wall shear rate for a yield power law fluid depends on both n and

13

b, = 2(b;' + bt) b2=I+2bo+b6

+ Jr4c-b-,-+-6"b')_+_4~b~j-+~bri

lln

= 24µPq

t,.p

L bo

=

nD3ry and ,P as above, where 6µpUavg

D,,

1

2 . 6n 3 + + 6n + y - ~~-~-_c...~--'--'--~-~~ 2 3 2 w - 8n + 12n + 4n - (4n + 4n)pg

Uo=---.

Care should be taken if the resulting terminal velocity corresponds to a shear rate that is considerably less than the one at which the power law parameters were detennined. In such case the power law extrapolation overestimates the apparent viscosity. As a result, the predicted terminal velocity might be underestimated as demonstrated by Roodhart [18]. A convenient empirical approach to present the results of settling experiments was suggested by Shah [19]. He found that the plot of the dimensionless group

Jci-n

(6.18)

18µ

Toe more general relation equivalent to Eq. 6.18 in the Stokes regime but valid for any Reynolds number is 4gd't, p p

)1;2

(6.19)

( 3p¡Cv

Toe terminal velocity corresponds to idealized conditions (infinite fluid). Nevertheless, it serves as a suitable reference velocity for describing more realistic situations.

6.2.2

Effect of Shear Rate lnduced by Flow

For hydraulic fracturing applications we are interested in settling under the additional shear induced by the flow. The flow induced shear results in further changes of the drau force if the fluid is non-Newtonian. To account for the thinning effect Novotn; [14] suggested the use of Stokes' law with an apparent viscosity calculated at an "effective shear rate"

y= Non-Newtonian Fluid Results of the drag force acting on a moving sphere are available for power law fluids. Acharya [17] obtained F

= 3n:d;K

(:J

n f(n),

(6.20)

where the function f (n) is given by f(n)

= 3 (3n-3)/2 [33n

5

¡(=:r

+ (j,¡)2r/2

(6.25)

where y1 is the shear rate which would be generated by the fluid flow only. Using a parallel plate approximation the shear stress induced by the fluid flow can_ be obtained from the constitutive equation (see Table 5.2). Since Y1 depends on the d1stance from the wall, the effective shear rate is higher in the vicinity of the wall than in the center. Assuming power law behavior the equation becomes

(6.26) 4

3

2

- 63n - lln + 97n + 16n] . 4n 2 (n + l)(n + 2)(2n + 1) '

therefore the power law form of Eq. 6.18 is

(6.24)

(NR.e,p)2 versus the particle Reynolds number, NR.e.p i~ a unique ~urve for a given flow parameter, n', and presented the curves both graph1cally and m the form of simple correl~ting equations.

from which the terminal velocity is

uo =

= 24/(n).

(6.21)

We can salve (numerically) the above equation at any y and hence a profil~ of settli~g velocities can be determined. If only one representative value of the settlmg veloclty is needed, the above equation may be solved at y= w/4.

--------------··----·--------

142

Non-laminar flow and solids transport

Solids transport

143

Dallavalle equation (Eq. 6.16) is still valid in a slurry characterized by the void fraction of the solids. ef,, if both the Reynolds number and the drag coefficient is modified according to exp [ (u,;) uo :§"

0.00008

s

~

Co,,;

u0 )

= ( u,;

5ef, ] 3(1 - ef,)

2 (

l

1-ef,)

+ q,1/3

0.00006

(d ,u,;p¡) ,

(6.27)

µ,

4F) (d~n:

1

2 -1

(,PJU,;)

'

(6.28)

0.00004

where F is still given by Eq. 6.12, understanding that the moving velocity is now u¡f¡. Thus. a terminal velocity calculation should involve Eqs. 6.12. 6.16, 627 and 628.

0.00002

6.2.4

Wall Effects

Toe terminal velocity decreases ( relative to the unbound fluid case) of an individual sphere falling between walls can be described by the wall effect function

o

o

y

Wall

w/2

Uow

fw=-,

Centerline

Figure 6.4 Settling velocity far Example 6.4

The velocity field around a falling sphere is disturbed by the presence of walls and/or other particles, leading to additional effects.

A treatable form of f w derived theoretically by Faxen (see the detailed treatment by Happel and Brenner (21]) for a particle located ata distance y= w/4 from the wall is f

Example 6.4

Settling Velocity Profile in the Facture

(6.29)

Uf)

w

=l

- 0.652&v + 0.1475w2

-

0.13lw4

-

0.0644w5

+,, -

where

2d,

Construct a curve representing the terminal velocity of a 20/40 mesh bauxite proppant (dp = 6.4 x 10-4 m, Pp = 3700 kg/m 3 ) in a borate crosslinked polymer characterized by P¡ = 1010 kg/m 3 , n = 0.68, K 33.5 Pa. sn. Assume fracture width w = 0.008 m and average linear velocity Uavg = 33.5 m/s.

=

(6.30)

w=--

w

(6.31)

Although there is no consensus in the question how to take into account the concentration and wall effects simultaneously, it is reasonable to apply the wall correction only at the final stage of the calculation in a non-iterative manner.

Solution

Example 6.5 Toe solution of Eq. 6.26 can be obtained numerically far different distances frorn the wall. Toe results are plotted in Figure 6.4. Toe calculated terminal velocity at the centerline (Y= w/2) is only Uo 3.4 x 10-6 mis, an unrealistically low value. The terminal velocity at y= w/4 (u0 = 8.6 x 10-5 m/s) can be accepted as a reasonable representative value. O

=

Representative Settling Velocity Involving Slurry and Wall Effects

Repeat the calculations of Example 6.4 for a representative location in the fracture, y= w/4, taking into account slurry and wall effects. Assume that the slurry was obtained adding 10 lbm of proppant per gallan of fluid (10 ppga).

6.2.3 Effect of Slurry Concentration

Solution

The presence of other particles causes an additional decrease of the terminal settling velocity. According to the interesting observation of Barnea and Mizrahi (20] the

Adding 10 lbm bauxite (Pp = 3700 kg/m 3 ) to 1 gallon liquid results in fluid volume, V¡ = 1 gallen = 0.00379 m 3 , and partid e volume, V P = 10 lbm/ Pp = 10 x 0.456/

144

Non-laminar flow and solids transport

References

3700 = 0.00123 m3 • Therefore, the void fraction of solids is