Hydraulic Fracture Mechanics

----------=---- .._----_.

r-_-_._. ......_.-._._._----

JOHN WILEY & SONSChichester. NewYork. Brisbane. Toronto. Singapore

Peter Valko and Michael J. EconomidesTexasA & M University,College Station,USA

Hydraulic Fracture Mechanics

-"'-"-------"'-:---------------------~

28303032

191919212323262727

112457

111113141415151516

xixiii

2 Linear Elasticity, Fracture Shapes andInduced Stresses .2.1 Force and Deformation

2.1.1 Stress2.1.2 Strain

2.2 Material Properties2.2.1 Linear Elastic Material2.2.2 Material Behavior Beyond Perfect Elasticity

2.3 Plane Elasticity2.3.1 Plane Stress2.3.2 Stresses Relative to an Oblique Line

(Force Balance I)2.3.3 Equilibrium Relations (Force Balance II)2.3.4 Plane Strain2.3.5 Boundary Conditions

Hydraulically Induced Fractures in the Petroleumand Related Industries1.1 Fractures in Well Stimulation1.2 Fluid Flow Through Porous Media

1.2.1 The Near-well Zone1.3 Flow from a Fractured Well1.4 Hydraulic Fracture Design1.5 Treatment Execution

1.5.1 Fracturing Fluids1.5.2 Proppants

·1.6 Data Acquisition and Evaluation for Hydraulic Fracturing1.6.1 Well Log Measurements1.6.2 Core Measurements1.6.3 Well Testing

1.7 Mechanics in Hydraulic FracturingReferences

1

Prefacelist of Notation

CONTENTS

Typeset in 10~/12i TImes by Laser Words, Madras, IndiaPrinted and bound in Great Britain by Bookcraft (Bath) Ltd

ISBN 0 471 956643

A catalogue record for this book is available from the British library

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

Jagaranda Wiley Ltd, 33 Park Road, Milian,Queensland 4064, Australia

John Wiley & Sons, Inc., 605 Third Avenue,New York, NY 10158·0012, USA

Other WIley Editorial Offices

No part o~ this book may be reproduced by any means,or transmitted, or translated into a machine languagewithout the written permission of the publisher.

All rights reserved.

ReprintedOctober 1996, May 1997

National 01243 779777International (+44) 1243 779777

Copyright © 1995 by John Wiley & Sons Ltd,Baffins Lane, Chichester,West Sussex P019 IUD, England

vi Contents Contents vii

2.4 Pressurized Crack 32 5.2 Slot Flow 1052.4.1 Solution of the Line Crack Problem 32 5.2.1 Derivation of the Basic Relations 1052.4.2 Constant Pressure 34 5.2.2 Equivalent Newtonian Viscosity 1112.4.3 Polynomial Pressure Distribution 35 5.3 Flow in Circular Tube 1122.4.4 "Zipper" Cracks 37 5.3.1 Basic Relations 1122.4.5 "Zipper" Crack with Polynomial Pressure Distribution 40 5.3.2 Flow Curve 115

2.5 Stress Concentration and Stress Intensity Factor 41 5.3.3 Equivalent Newtonian Viscosity for Tube Flow 1192.5.1 Stress Intensity Factor, Symmetric Loading 42 5.4 Flow in Other Cross Sections 1222.5.2 Stress Intensity Factor, non-symmetric Loading 43 5.4.1 Flow in Annulus 122

2.6 Fracture Shape in the Presence of Far-field Stress. 5.4.2 Flow in Elliptic Cross Section 123The Concept of Net Pressure 43 5.4.3 Limiting Ellipsoid Cross Section 124

2.7 Circular Crack 45 References 1282.8 Volume and Strain Energy 472.9 Computational Methods 49

References 50 6 Non-laminar Flow and Solids Transport 1316.1 Non-laminar Flow 131

6.1.1 Newtonian Fluid 1313 Stresses in Formations 53 6.1.2 General Fluid 1323.1 Basic Concepts 53 6.1.3 Drag Reduction 1343.2 Stresses at Depth 55 6.1.4 Turbulent Flow in Other Geometries 137

6.2 Solids Transport 1383.3 Near-wellbore Stresses 593.4 Stress Concentrations for an Arbitrarily Oriented Well 6.2.1 Settling of an Individual Sphere 139

63 6.2.2 Effect of Shear Rate Induced by Flow 1413.5 Vertical Well Breakdown Pressure 65 6.2.3 Effect of Slurry Concentration 1423.6 Breakdown Pressure for an Arbitrarily Oriented Well 66 6.2.4 Wall Effects 1433.7 Limiting Case: Horizontal Well 69 6.2.5 Agglomeration Effects 1453.7.1 Arbitrarily Oriented Horizontal Well 70 References 1453.8 Permeability and Stress 713.8.1 Stress-sensitive Permeability 72

3.9 Measurement of Stresses 73 7 Advanced Topics of Rheology and Fluid Mechanics 1473.9.1 Small Interval Fracture Injection Tests 74

1473.9.2 Acoustic Measurements 75 7.1 Foam Rheology3.9.3 Determination of the Closure Pressure 76 7.1.1 Quality Based Correlations 1483.9.4 Core Stress Measurements 77 7.1.2 Volume Equalized Constitutive Equations 1483.9.5 Critique and Applicability of Techniques 79 7.1.3 Volume Equalized Power Law 151

References 80 7.1.4 Turbulent Flow of Foam 1527.2 Accounting for Mechanical Energy 153

7.2.1 Basic Concepts 1534 Fracture Geometry 83 7.2.2 Incompressible Flow 154

7.2.3 Foam Flow 1544.1 The Perkins and Kern and Khristianovich and 7.3 Rheometry 156

Zheltov Geometries 83 7.3.1 Pipe Viscometry 1564.1.1 The Consequences of the Plane Strain Assumption 86 7.3.2 Slip Correction 157

4.2 Fracture Initiation vs. Propagation Direction 88 References 1624.2.1 Fractures in Horizontal Wells 90

4.3 Fracture Profiles in Multi-layered Formations 92References 95 8 Material Balance 165

8.1 The Conservation of Mass and Its Relation to5 Rheology and Laminar Flow 97 Fracture Dimensions 165

8.2 Fluid Leakoff and Spurt Loss as Material Properties 1695.1 Basic Concepts 97 8.2.1 Carter Equation I 169

5.1.1 Material Behavior and Constitutive Equations 98 8.2.2 Formal Material Balance. The Opening Time5.1.2 Force Balance 103 Distribution Factor 171

-_--- ...._._-------

287294

295

267269269270272276278279281283284

258263

245245245246246247247247249252256

ix

Index

Appendix: Comparison Study of Hydraulic FracturingModels: Input Data and Results

References

11 Fracture Height Growth (3~ and P-3D Geometries)11.1 Equilibrium Fracture Height

11.1.1 Reverse Application of the Net-pressure Concept11.1.2 Different Systems of Notation11.1.3 Basic Equations11.1.4 The Effect of Hydrostatic Pressure

11.2 Three-dimensional Models11.2.1 Surface Integral Method11.2.2 The Stress Intensity Factor Paradox

11.3 Pseudo-three-dimensional ModelsReferences

10.3 Retarded Fracture Propagation10.3.1 Fluid Lag10.3.2 TIp Dilatancy10.3.3 Apparent Fracture Toughness10.3.4 Process Zone Concept10.3.5 The Reopening Paradox

10.4 Continuum Damage Mechanics in Hydraulic Fracturing10.4.1 TIp Propagation Velocity from COM10.4.2 CDM-NK Model10.4.3 CDM-PKN Design Model

10.5 Pressure Decline Analysis and TIp Retardation10.5.1 Resolving Contradictions with Continuum

Damage MechanicsReferences

Contents

242242243

235237238241

230232

218227

217

212

189189189192195196196199200201202202204205206209210211

187

178179181183184185187

172174174

........__ ....._--------

10 Fracture Propagation10.1 Fracture Mechanics

10.1.1 Griffith's Analysis of Crack Stability10.1.2 Mott's Theory for the Rate of Crack Growth

10.2 Classical Crack Propagation Criterion forHydraulic Fracturing10.2.1 Fracture Toughness Criterion10.2.2 The Injection Rate Dependence Paradox

9 Coupling of Elasticity, Flow and Material Balance9.1 Width Equations of the Early 20 Models

9.1.1 Perkins-Kern Width Equation9.1.2 Geertsma-de Klerk Width Equation9.1.3 Radial Width Equation

9.2 Algebraic (20) Models as Used in Design9.2.1 PKN-C9.2.2 KGD-C9.2.3 PKN-N and KGD-N9.2.4 PKN-a and KGD-a9.2.5 Radial Model9.2.6 Non-Newtonian Behavior

9.3 Numerical Material Balance (NMB) with Width Growth9.4 Differential 20 Models

9.4.1 Nordgren Equation9.4.2 Differential Horizontal Plane Strain Model

9.5 Models With Detailed Leakoff Description9.6 Pressure Decline Analysis

9.6.1 Nolte's Pressure Decline Analysis(Power Law Assumption)

9.6.2 The No-spurt-Ioss Assumption(Shlyapobersky method)

9.6.3 Material Balance and Propagation PressureEstimates of the Spurt Loss

9.6.4 Resolving Contradictions9.6.5 Pressure Decline Analysis With Detailed Leakoff

Description (Mayerhofer et al. Technique)References

8.3 The Constant Width Approximation (Carter Equation II)8.4 The Power Law Approximation to Surface Growth

8.4.1 The Consequences of the Power Law Assumption8.4.2 The Combination of the Power Law Assumption

with Interpolation8.5 Numerical Material Balance8.6 Differential Material Balance8.7 Leakoff as Flow in the Porous Medium

8.7.1 Filter-cake Pressure Drop8.7.2 Pressure Drop in the Reservoir8.7.3 Leakoff Rate from Combining the Resistances

(Ehlig-Economides et al. [6])References

Contentsviii

The Authors

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 basicphenomena including elasticity, stress distribution, fluid flow, and the dynamics ofthe rupture process from the point of view of the influence of those phenomenaon the created fracture. Currently used design and analysis techniques are derivedand improved using a comprehensive and unified approach. Numerical ~xamples areelaborated to illustrate important concepts.

The material grew out of university and industrial courses that have been taughtat Mining University of Leoben, Texas A&M University and several locationsthroughout the world. During these courses we have recognized that currentlyavailable monographs, often written bya great number of co-authors reflect diverseviews, systems of notations and units. One of our main goals was to establish acommon language that eases the way workers in the field can get acquainted withthe material and experts of different background can communicate with each other.

Our gratitude goes to our coworkers and students who have contributeda great deal to the final form of the book. The list below is far fromcomplete: T. Brugger, H. Buchsteiner, Zhongming Chen, C. Enzendorfer YongFan, T.P. Frick, M.J. Mayerhofer, H. Mosser, R. Oligney, W. Prassl, M. Prohaska,C.R. Rom, R.E. Schmid, 1. Smith, R. Seiler, W. Winkler, and M. Zettl.

Anybody interested in hydraulic fracturing is bound to be influenced by thepioneering work down in the fifties and sixties. The authors of this book have hadthe privilege to enjoy discussions with the developers of the first and compellingmodels including YP. Zheltov and T.K. Perkins, If. The hand-written remarks ofI. Geertsma are saved with particular honor. While the views expressed on thesepages have also been influenced by personalities such as M.P. Cleary, S.A. Holditch,M.K. Hubbert, K.G. Nolte, 1. Shlyapobersky and N.R. Warpinski, we take full

. responsibility for the content and format of presentation.We would like to express our gratitude to organizations for permitting us to

reproduce some of the figures and tables in the text: Society of Petroleum Engi­neers (Figure 1-1, Figure 1-2, Figure 3-12, Figure 3-14, Figure 4-7, Figure 11-4,and Tables AI-AS); American Institute of Physics, (Figure 7-5, Figure 7-6, andFigure 7-7).

PREFACE

---------------------------------~-~- ..~.----~-.....--._--

7area, m-Reidenbach et al. constant coefficient, kgQ,77 •m-L54 . s-o 77 (Ch, 7)cross sectional area for flow, m2

dimensionless fracture surface areafracture surface area at end of pumping, m2 (Ch, 8,9)fracture area per unit of bulk volume, m" (Ch.8)dimensionless fracture network area (Ch. 8)fracture area per unit of matrix volume, m" (Ch. 8)fracture surface area exposed to leakoff, m2 (Ch. 8)fracture surface areas at different time instants, m2

conductivity/porosity factor, m2/3 (Ch. 3)Kachanov parameter of damage accumulation rate, Pa-~s-l (Ch. 10)dimensionless Kachanov parameter (Ch. 10)drag coefficient, dimensionless (Ch. 7)modified drag coefficient, dimensionless (Ch. 7)element of the linear elasticity coefficient matrix, Paleakoff coefficient, m.s-I/2coefficient of pressure dependent leakoff, rn- S-I/2. Pa-1 (Ch, 11)diameter, well diameter, mdamage variable, dimensionless (Ch. 10)dissipation rate, J .S-1annulus smaller and larger diameter, m (Ch, 5)Young's modulus, Paeffective elastic modulus, Pa (Ch, 11)plane strain modulus, Paexponential integralcomplete elliptic integral of the second kindforce, Ndimensionless correction factor for proppant settlingdimensionless fracture conductivity (Ch. 1)dimensionless fracture conductivity, optimal (Ch. 1)shear modulus, Faauxiliary function for circular crackformation depth, m (Ch. 1,2)CDM width factor, dimensionless (Ch. 10)

EeE'EiE(k),E(m)FFcFCD

FCD,opt

GG(~)H[CDM

A'AcAD

A"Afe

AfbAfDAfmaALAn,AjBx,ByCCDCDCD,.,CijCL

CL,Q

DDDvD1,D2

E

A

LIST OF NOTATION

-------- ..~---- .-.~-------

energy, Jwork of inner pressure to move fracture faces apart, Jkinetic energy, Jstrain energy caused by net pressure, Jstrain energy caused by far-field stress, Jcompressibility factor of gas, dimensionlessexponent (Ch. 5)coefficient, m2. (Pa '5)-1 (Ch. 7)auxiliary variables, dimensionless (Ch. 5)auxiliary variable, m3/4 (Ch. 9)auxiliary variable, Pa .m-1/4 (Ch. 9)intercept, m3 (Ch, 8)intercept, Pa (Ch. 9,10)coefficient, m2 . Pa-2 • S-1 (Ch, 7)fracture width, m (Ch, 8)auxiliary variables, dimensionless (Ch. 5)half-length of two-dimensional line crack, mproportionality constant in the pressure vs. width relationship, Pa- m-1proportionality constant in the pressure vs. width relationship (KGD),Pa·m-1proportionality constant in the pressure vs. width relationship (PKN),Pa-rn"!proportionality constant in the pressure vs. width relationship (radial),Pa-rn"!total reservoir compressibility, Pa-1total fissure compressibility, Pa-!Nordgren coefficient of dimensionless length, mNordgren coefficient of dimensionless time, sNordgren coefficient of dimensionless width, mNordgren coefficient of dimensionless net pressure, Paparticle diameter, m (Ch. 7)Fanning friction factor, dimensionlessfactor of wall effect, dimensionless (Ch, 7)acceleration of gravity, m- S-2auxiliary function for line crackNolte's g-functionNolte's go-functionreservoir thickness, mproductive eight of fracture, m (Ch. 1)fracture height, mfracture height provided by the modeler, m (Ch. 8)matrix thickness, m (Ch, 8)perforated (target) height, minjection rate per one wing, m3 . s-!permeability, m2

argument of elliptic integral (Ch. 11)Matt's numeric factor (Ch. 10)

xvList of Notation

kkk

Clf

CI

C2

C3

C.

dpff.gg(~)g(6tD. Oi)go(a)hhhfhi.",»:hp

c,

bbbblbo, b!,b2

C

cf

Cf.KGD

aa

distortional creep compliance, Pa-1 (Ch. 3)dilatational creep compliance, Pa-I (Ch. 3)consistency index, Pa- s-·stiffness matrix, Pa .m (Ch, 11)generalized consistency index, Pa .s-"generalized consistency index for pipe flow, Pa- s-nstress intensity factor (mode I), Pa- m1/2

critical stress intensity factor, fracture toughness, Pa- m1/2

fracture toughness at bottom, Pa .m 1/2

fracture toughness at top, Pa- m1/2

nominal stress intensity factor, Pa- m1/2 (Ch, 10)geometry dependent volume equalized consistency index (pipe), Pa- s-n(Ch.7)volume equalized consistency index, Pa .s-n (Ch, 7)auxiliary coefficient, Pa-I (Ch. 7)auxiliary coefficient, dimensionless (Ch. 7)auxiliary coefficient, dimensionless (Ch. 7)rate of pressure increase, Pa- S-1 (Ch, 8)length, mlength of contact, m (Ch, 4)Deborah number, dimensionlessReynolds number, dimensionlessParticle Reynolds number, dimensionlesswall Reynolds number, dimensionlessgeneralized Reynolds number, dimensionlessradius of a circular crack, m (Ch. 2)distance between two points, m (Ch, 11)dimensionless filter-cake resistance (Ch, 8,9)estimate of fracture radius, from material balance, m (Ch. 9,10)estimate of fracture radius, from no-spurt-loss, m (Ch, 9,10)filter-cake resistance, m-! (Ch. 8,9)estimate of fracture radius, from propagation pressure, m (Ch, 9,10)stress vector, Pa (Ch. 2)spurt-loss coefficient, mestimate of spurt-loss coefficient from material balance, mestimate of spurt-loss coefficient from propagation pressure, mcomponents of the stress vector, Pa (Ch. 2)formation tensile strength, Pamatrix of areal elements, m2 (Ch. 11)volume, crack volume, m3

bulk volume, rrr'fracture volume at end of pumping, rrr'fracture volume, m3

volume of injected fluid, m3

leakoff volume, m3

leakoff volume at end of pumping, m3

matrix volume, m3 (Ch. 8)

List of Notation

Kv£KlK2K3K2LLNDe

NRe

NRe.p

NRe,w

N~(!RRRDRmb

Rnsp

RoRp,

SSpSp.mbSp.~,S" s,TTVVbV,VIViVL

VLeV"",

K'K~K/K/cK1C,boUOm

KIC.IOp

K1."

Kp.VE

xiv

flow rate outside the plug region, m3. S-1 (Ch. 5)radial distance, m (Ch. 1)distance from tip, m (Ch, 2)outer boundary radius, m (Ch, 1)wellbore radius, m (Ch. 1)skin effect, dimensionless (Ch. 1)equivalent skin effect, dimensionless (Ch. 1)repulsive distributed force, Pa (Ch. 2)cohesive distributed force, Pa (Ch. 2)time, scharacteristic time (Ch. 8)closure time, sdimensionless timedimensionless time for fracture + reservoir systemrelaxational time constant for the distortional creep, s (Ch. 3)relaxational time constant for the dilatational creep, s (Ch. 3)time at end of pumping, sdifferent time instants, S

fluid velocity, m .S-1dislacement, m (Ch. 2,10)average fluid velocity, m- S~1(Ch. 5)compressional wave slowness, m· s" (Ch. 3)fracture propagation rate (tip velocity), m- S-1 (Ch. 8-10)maximum fluid velocity, rn- S-1 (Ch. 5)shear wave slowness, m· S-1 (Ch. 3)terminal settling velocity, m· S-l (Ch, 7)terminal settling velocity, m .S-I (Ch. 7)terminal settling velocity with wall effect, rn S-1 (Ch. 7)slip velocity, m . S-1 (Ch. 2)displacement components, m (Ch. 2)longitudinal wave velocity, m- S-l (Ch. 2)leakoff velocity, m S-1 (Ch, 8)fracture width, m (Ch. 1)width of flow channel, m (Ch. 5)average fracture width at end of pumping, mestimate of average width at instant of shut-in, mwidth of longitudinal-to-transverse transition, m (Ch. 4)width of ideal transverse fracture, m (Ch. 4)fracture width at wellbore (KGD), rnmaximum fracture width at wellbore (PKN), mmaximum fracture width at wellbore, m (PKN)dimensionless fracture width at wellbore (NK)maximum width of a line crack, m (Ch, 2)maximum width of the elliptical cross section, mdimensionless maximum fracture width at wellbore (NK)fracture width at tip (Ch. 8-10)average fracture width, m

xviiList of Notation

WoWoWO,D

W~=x.f

W

Ww.O

Ww.O.PKN

W~'D

W,

W

Us

Umax

totctotDxf

[I

tzt,tj, In

U

U

s

qzrr

.-~ ~.-~---~-~~~------------------------

permeability of proppant pack in fracture, m"bulk formation permeability, m2 (Ch. 8)matrix permeability, m2 (Ch. 8)auxiliary variables in CDM-PKN model (Ch. 10)distance, mcharacteristic length of flow channel, maverage distance of microcracks. m (Ch. 10)dimensionless average distance of microcracks (Ch. 10)slope, m; . s-IIZ (Ch, 8)slope, Pa (Ch. 9,10)argument of elliptic integral (Ch. 5)mass flux, kg. S-1real gas pseudo pressure, Pa- S-1flow behavior index, dimensionlessgeneralized flow behavior index, dimensionlessnumber of time steps in numerical material balance methodpressure, Paclosure pressure, Papressure at the center of perforation, Pa (Ch. 11)dimensionless pressureouter boundary reservoir pressure, Pa (Ch, 1)average reservoir pressure, Pa (Ch. 1)average pressure in a GDK fracture, Pa (Ch. 9)initial pressure, Pa (Ch. 1)instantaneous shut-in pressure in stress determination test, Pa (Ch. 3)net pressure, Patip net pressure, Pa (Ch. 9)wellbore net pressure, Pa (Ch. 9,10)constant net pressure in crack, Pa (Ch. 2)fracture propagation pressure, Pa (Ch. 9,10)dimensionless reservoir pressure, Pawellbore flowing pressure, Pa (Ch. 1)wellbore instantaneous shut-in pressure, Pa (Ch. 9,10)wellbore propagation pressure, Pa (Ch. 9,10)constant pressure in line crack, Pa (Ch. 2)coefficient of polynomial pressure distribution in line crack, Pa- m-I(Ch. 2)coefficient of polynomial pressure distribution in line crack, Pa- m-z(Ch. 2)coefficient of polynomial pressure distribution in line crack, Pa .m-3(Ch.2)production rate, m3. -I (Ch. 1)flow rate in flow channel, tube, fracture, m3 . S-I

dimensionless production rate (Ch. 1)dimensionless flow rate into reservoir (Ch. 8,9)flow rates into reservoir at different times, m3 • S-1 (Ch, 8,9)plug flow rate, m3 .S-1 (Ch. 5)

List of Notation

.-._-_._....---._.__-..._----=-

P3

PoPI

pw.isi

Pw.pr

PPcPcpPDp,

IiPPo'Pis

PnPn,tip

Pn.w

pnO

PPT

Pres.DP",r

nn'

m(p)

mmmm

krkrDkmako. kl• k2, k3I

xvi

angle characterizing direction of horizontal well, rad (Ch. 3)auxiliary variable in BNS equation, dimensionless (Ch. 6)(CDM) geometry factor, dimensionless (Ch. 10)shear rate, S-1wall shear rate, S-Iwall shear rate (Newtonian fluid) or nominal Newtonian wall shear rate,S-I (Ch. 5)average wall shear rate, S-1 (Ch. 5)thickness of a slice of material, mstrain, dimensionlessroughness, dimensionless (Ch. 6)specific volume expansion ratio, dimensionless (Ch, 7)horizontal strain, dimensionlessvertical strain, dimensionlessfluid efficiency, dimensionless (or %)parameter related to poroelasticity and Poisson ratio, dimensionless (Ch. 3)viscosity parameter, Pa· sn' (Ch. 11)computed efficiency from modeler's data, dimensionless (Ch. 8)dimensionless matrix hydraulic diffusivity (Ch. 8)opening time distribution factor, dimensionlessKachanov exponent, dimensionless (Ch. 10)compressibility modulus, Pa (Ch. 2)opening time distribution factor from modeler's data, dimensionless(Ch.8)retardation time, s (Ch. 8)interporosity flow coefficient, dimensionless (Ch. 8)modified interporosity flow coefficient, dimensionless (Ch. 8)viscosity, Pa .sapparent viscosity, Pa- sequivalent Newtonian viscosity, Pa sfiltrate viscosity, Pa .s (Ch. 8)plastic viscosity, Pa s (Ch. 5)solvent viscosity, Pa- s (Ch. 5)wall viscosity, Pa- s (Ch. 5)low shear viscosity, Pa .s (Ch. 5)Poisson ratio, dimensionlessauxiliary variabledensity, kg- rn?fluid density, kgm-3 (Ch. 6)formation density, kg· m-3 (Ch. 3)normal stress, Pamaximum principal horizontal stress, Paminimum principal horizontal stress, Pamaximum stress, Paminimum stress, Panet stress, Pa (Ch 10)vertical stress, Pa

xixList of Notation

vf.-Lo

K

rJrJrJ'

eeeene•.

Y.·s

YYYCDM

YY.Y"·",,,

.._ ..... _._._---

average fracture width from GDK width equation, maverage fracture width from PKN width equation, mlateral coordinate, mdimensionless lateral coordinatedimensionless lateral coordinate at time instant jfracture half length, mapparent fracture half-length, m (Ch. 1)dimensionless fracture lengthfracture half-length at end of pumping, mfracture half-length at different time instants, mfracture length provided by the modeler, m (Ch, 8)maximum estimate of fracture length, mestimate of fracture length from material balance, mestimate of fracture length from propagation pressure, mestimate of fracture length from unretarded propagation , mlocation of jump of pressure in line crack, m (Ch, 2,9)coordinate, mdimensionless vertical ordinate for height containment (Ch. 11)location of bottom of perforation, dimensionless (Ch. 11)location of top of perforation, dimensionless (Ch, 11)vertical coordinate for height containment, m (Ch. 11)vertical coordinate for height containment, m (Ch, 11)coordinate, mfoam quality, dimensionless ratio (or %) (Ch. 7)upward height migration, m (Ch. 11)pressure drop, Papressure drop across filter-cake, Pa (Ch. 8)pressure drop across polymer-invaded zone, Pa (Ch. 8)pressure drop in the reservoir, Pa (Ch. 8)shut-in time, s (Ch. 9,10)after-growth time, S (Ch. 9,10)after-growth time, observed, s (Ch. 9,10)dimensionless shut-in time (Ch. 9,10)downward height migration, m (Ch. 11)density difference, kg . m-3 (Ch. 7)angle of oblique plane, rad (Ch, 2)poroelastic constant, dimensionlessexponent of fracture length growth, dimensionless (Ch. 9,10)angle characterizing direction of horizontal well, rad (Ch. 3)kinetic energy correction factor, dimensionless (Ch. 5,7)permeability anisotropy ratio, dimensionless (Ch. 3)auxiliary variable for Carter equation Il, dimensionless (Ch. 9)angle characterizing direction of horizontal well, rad (Ch. 3)slip coefficient (Mooney method), mS-1 .Pa-I (Ch.7)modified slip coefficient (Oldroyd-Jastrzebski method), m2 . 8-1 . Pa-1(Ch.7)geometry factor relating average to maximum, dimensionless

List of Notation

----_ .._-_ ..

Y

ClKE

f3f3f3f3f3c

Cl

Cl

Cl

rSh;tJ.ptJ.Pf:acetJ.PpiztJ.P,e,6t6t.6ta, otJ.to6td6pe

Yy

Ydv;YRYw

WGDK

WPKN

X

Xo

XOj

xfXfxfO

XI'X t.s- Xf.n

X/.mxl.max

Xf.mb

Xf,pr

XJ.III-

Xo

xviii

Subterranean porous media have been the source of valuable fluids such as ground­waters and petroleum, both liquid (oil) and natural gas. Oil and gas, combined, stillaccount for over 60% of all the energy needs of the world (with coal providing anadditional 30%.) The demand for hydrocarbons is likely to continue unabated at theseexceptionally high levels throughout the twenty-first century (see OPEC data [1]).

Porous formations have also been used for the injection of slurried wastes such ashazardous chemicals or radioactive byproducts. Certain special geologic structureshave been used for the seasonal storage and quick recovery of already processedpetroleum products and natural gases.

In all of these cases access to the geologic formations has been accomplishedwith drilled wells. Historically wells were first vertical with targets of progressivelyincreasing depth. Then, wells could be drilled deviated and, since the early 1980s theycan be started vertical and after a "buildup" angle they can be turned fully horizontalinto the target formation, with some horizontal lengths exceeding 2500 m (over8000 ft). Horizontal wells have become commonplace with continuously increasingestimates on their future share of all wells drilled.

Depths of formations of interest range from a few hundred meters to deeper than6000 m for natural gas formations. Typical oil reservoirs are usually between 2000and 3500 m.

Although exact estimations are difficult, it is widely believed that in the USAalone more than one million petroleum wells have been drilled in the history of theindustry (since 1859 and Col. Drake's well). A comparable number has been drilledin the former USSR. In the rest of the world the number is smaller.

1.1 Fractures in Well Stimulation

HYDRAULICALLY INDUCEDFRACTURES IN THEPETROLEUM ANDRELATED INDUSTRIES

1

(I)

UJ

UJ

rr

in situ minimum stress in target, upper and lower layer, respectively, Pa(Ch. 11)closure quality, Pa (Ch. 3)shear stress, Paopening time, s (Ch. 8-11)dimensionless opening timeyield stress, Pavolume equalized yield stress, Pawall stress, Paaverage wall stress, Pastress parameter, Pa (Ch. 5)porosity, dimensionlessratio of yield stress to wall stress, dimensionless (Ch. 5)exponent of width growth, dimensionless (Ch. 8)fissure porosity, dimensionless (Ch, 8)matrix porosity, dimensionless (Ch, 8)angle with the .r axis, rad (Ch. 11)interporosity constant (Ch, 8)ratio of inner and outer annulus diameter, dimensionless (Ch. 5)ratio of particle diameter to half width, dimensionless (Ch. 6)

17' effective stress, Pa17], 172, 0'3 principal stresses, Pa

xx List of Notation

Ei = Exponential integralPi = Initial reservoir pressurePe = Outer boundary constant pressurep =Average reservoir pressurepwf = Flowing bottom-hole pressureYe = Outer boundary radiusrw = Well radius.

D..P PD

Transient Pi - pwf

1 • ( 1)(infinite acting reservoir) Po= -2Ez - 4tokt

and to= ~-,-2-4>I-LC, w

Semilogarithmic Po = ~(IntD+0.8091)approximation at to > 100

Steady state P. - pwf PD= In!j_rwO.472rePseudosteady state p- Pwf po=[n--

rw

Table 1.1 Pressure gradients and dimensionless pressure functions forradial reservoir flow at the well

and the PD for constant rate is very nearly equal to the l/qD for constant pressureproduction for almost all times (see Earlougher [7]).

The relationship between q and Pwf and the antecedent engineering activitiesfor their optimum adjustment are the essential functions of petroleum productionengineering (see Economides and Ehlig-Economides [6]).

(1.7)

The non-petroleum reader is referred to References [3]-[6] and references thereinfor the developments and solutions to Eqs. 1.3 and 1.5 which are standard inpetroleum, geothermal and groundwater engineering.

Of interest are the constant-rate and the constant-pressure-at-the-well solutions.The general form of the constant-rate solution is

q = 2rckhD.p (1.6)fJ-PD

Three different types of flow mechanisms can be distinguished: transient, orinfinite-acting behavior, steady-state with constant outer boundary pressure, Pe, andpseudosteady-state, denoting a no-flow outer boundary condition.

Table 1.1 contains the expressions for the driving pressure gradient D.p and thedimensionless pressure function, pD, for the three flow mechanisms. Analogousexpressions can be written for compressible (gas) flow using D.m(p) instead of 6.p(see Dake [3]; Economides and Ehlig-Economides [6]).

Interestingly, for transient rate production at constant Pw r, the solution yields

2:Jrkh(pi - Pwf)q = 1 '

fJ--qD

3Fluid flow through porous media

(1.5)a2m(p) 1am(p) ¢fJ-Cr am(p)-- + --- = -----.or2 r ar k at

and, thus,

(1.4)

where c, is the total system compressibility and t is the time.An analogous expression for gas (compressible) flow employs the real-gas pseu­

dopressure, m(p), defined by Al-Hussainy and Ramey [2] as

l p 2pm(p) = -dp,

PO fJ-Z

(1.3)[pp 1ap ¢fJ-Cr ap-+--=---,or2 r ar k at

(1.2)2:Jrrkhdpq= ~fJ-~dr'

where fJ- 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

This is the well known Darcy's law which in radial coordinates yields the followingexpression for the volumetric flow rate, q:

(Ll)u ()(kD.p.

A porous medium is a geologic formation whose rock contains voids (pores). Theratio of the pore volume to bulk volume is defined as the porosity, ¢. It is in such areservoir that fluids are stored. Typical pore diameters range from 10~7 m to 10-4 m,and reservoir porosities range from about 0.10 to (typical) 0.25 for sandstones to(extraordinarily high) 0.4 for some carbonate formations.

While the porosity is important in defining the oil- or gas-in-place for a petroleumproducing reservoir or the storativity of an injection target, a second quantity, thepermeability, k, describing the ability of fluids to flow in the reservoir, is essential.The permeability relates the pressure gradient, D.p, which is the driving force in thereservoir with the macroscopic fluid velocity, u,

1.2 Fluid Flow through Porous Media

Of the producing wells drilled in North America since the 1950s about 70% ofgas wells and 50% of oil wells have been hydraulically fractured. The majority ofinjection wells have been fractured also (personal communication from SchlumbergerDowell and Halliburton companies, 1994). Similar percentages are expected in therest of the world, as those reservoirs mature (age).

Why is hydraulic fracturing such a common well "stimulation" procedure andhow is it practiced in the modern petroleum and other industries?

These issues are addressed in this chapter and form the rationalization for thestudy of hydraulic fracture mechanics.

Hydraulically induced fractures2

Once a hydraulic fracture is created in a well or, in the not uncommon case, wherethe well intersects a natural fracture, fluid will flow normal to the fracture face fromor to the reservoir (production or injection) and then along the fracture path from orto the welL

For almost 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 describedthe flow performance of an infinite-conductivity fracture whereas Cinco-Ley andSamaniego [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 thefracture, 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 Agarwalet al. [11] for the transient flow of a finite-conductivity fractured well. On theordinate is the dimensionless pressure, PD, on the abscissa is the dimensionlesstime, tD:cf and the parameter is the dimensionless fracture conductivity, FCD.

1.3 Flow from a Fractured Well

and for s = 10,q = 1.03 X 10-2 m3/s whereas for s = 0, 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-treatmentrate itself (2.32 x 10-2 mJ /s = 12600 barrels/day) are very attractive,pointing towardsmatrix 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. L8, with s = 0, the minimum reservoirpermeability for which matrix stimulation is attractive would be k = 3.9 X 10-16 m2

(0.4 rnd), In production engineeringthe attractivenessof the stimulation is subject tothe costs of the treatmentwhichmust be balancedagainst the benefitsof the incrementalproduction rate of 5.1 x 10-5 m3/s (28 barrels/day).

In this exercise, for perrneabilitiesless than 3.9 x 10-16 m2 (or in some cases formuch higher permeabilitiesif economicconsiderationsindicate) hydraulicfracturing islikely to be the appropriatewell stimulationoperation.0

For k = 9.87 X 10-14 m2, Eq, 1.8 yields

(2)Jr(9.87x 10-14)(20)(3.5 x 107 - 2 X 107) 0.186q= [300] =8+s'(1 x 10-3) In- +s0.1

Solution

high 9.87 x 10-14 m1(100 md) and Pwf = 2 X 107 Pa. Use the steady-stateexpressionof Eq. i.s.

5Flow from a fractured well

-----_ ....•......-._-----_. __ ..._-----

(1.8)

Example 1.1 Matrix Stimulation vs. Hydraulic Fracturing

Suppose that a well with 'w = 0.1 m is drilled in a reservoir with r = 300 m, h=20 m and Pe = 3.5 X 107 Pa. If the fluidviscosity f.L = 1 X 10-3 Pa s and a well testhas provided s = 10, investigate the incremental well flow rate before and after acompletely successful matrix stimulation(i.e. with s = 10 and s =0, respectively)fora range of permeabilitiesfrom a low value equal to 9.87 x 10-18 m2 (0.01 md) to a

example is offered.

2rrkhtlpq = J.i(PD +s)"

The reader is referred to Chapter 5 of Economides et al. [6] for an extensivedescription of the various causes of near-well damage, certain mechanical contribu­tions to the skin effect and quantification of its impact.

The skin effect is determined through the pressure transient testing of a well. Alarge and positive value implies damage or a flow impediment due to a mechanicalreason (e.g. s = 20), whereas s = 0 is for undisturbed permeability in a vertical welLZero 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 aseries of possible chemical treatments intended to remove near-well damage onceits nature is identified (see Economides et al. [6]). Larger post-stimulation perme­abilities are possible, although rare. This could happen if the formation itself reactswith the injected stimulation fluids (e.g. a hydrochloric acid, HCI, solution and acarbonate rock).

Hydraulic fracturing may be attempted in those cases where matrix stimulationcannot result in an economically satisfactory well production or injection rate.

To understand the need for an alternative to matrix stimulation the following

Converging radial flow de facto exaggerates the impact of the near-well zone. It isclear from Eq. 1.6 that for (e.g. steady-state) flow, the driving pressure gradient isproportional to the logarithm of the radial distance.

An alternative way to state this is that for a constant production rate, the sameamount 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 thenatural permeability in the near-well zone would be critical to the well productionor injection rate at constant tlp.

Permeability-altering phenomena occur frequently in almost all well operationsincluding drilling, well completions or "workovers", Reduction of the near-well reser­voir permeability is common, is referred to as damage, and has been characterizedby a dimensionless skin effect, s (see Van Everdingen and Hurst [8]) analogous tothe 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:

1.2.1 The Near-well Zone

Hydraulically induced fractures4

...._-_ -----------------------

The proper engineering approach to hydraulic fracture design is to maximize thepost-treatment performance and ensuing benefits at the lowest treatment costs. Thus,an economic criterion such as the net present value (NPV) has been employed forthis purpose: the optimum fracture size would coincide with the maximum NPV (seeMeng and Brown [12]).

A common hydraulic fracture design optimization procedure starts from a fracturesize, usually denoted by, but not limited to, the fracture half-length.

1.4 Hydraulic Fracture Design

From Eq. 1.11 FCD = 4.2 and, therefore, from Figure 1.2, sf + In(xI /rw) = 0.96.Substituting the values of XI and rw (= 0.1 m) this would lead to 51 = -7.

Using Eq. 1.8, for steady-state production and SI = -7 results in q = 7.35 X

10-4 m3/s (400 barrels/day) which is an 8-fold increase over the best case that thiswell would produce with matrix stimulation (i.e. s = 0).

It is essential to note that once a well is hydraulically fractured tbe overwhelmingportion of the total flow is through the fracture, bypassing the damage zone and, thus,any pretreatment radial skin effect can be ignored. 0

Solution

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

Figure 1.2 Equivalent skin effect for pseudoradial flow into a fractured well [101

10' 10'10'10010·'

\\

1\I"

t-

0.5

Hydraulic fracture design 7

----~-- ..-.- .....__ ---

3

2.5

~.... 2"::..CE.+",- 1.5

Example 1.2 Performance of a Fractured vs. an Unfractured Well

Suppose that the well in Example 1.1 with permeability k = 3.9 X 1O-l6 m-, and forwhich matrix stimulation has been deemed unattractive, is hydraulically fractured. A

(1.11)

(1.10)kt

tDx! = ---2'¢MCrXf

kfwand FCD = kxr '

In Eqs. 1.9 to 1.11 variables are as defined in Eq. 1.6 and Table 1.1, except forthe fracture half-length, x f' the fracture permeability, kf' and the propped fracturewidth, w.

The values of the fracture half-length and fracture conductivity are the essentialquantities for the prediction of fractured well performance.

The Cinco-Ley and Samaniego [10] solution becomes indistinguishable from theGringarten and Ramey [9] solution for FCD > 300. For practical purposes they canbe considered as the same for FCD > 70.

Long-term fractured well performance results in pseudoradial flow, and thepresence of a hydraulic fracture of half-length, xf and conductivity, FCD, can bemanifested by an equivalent skin effect, sf, which can be read from Figure 1.2.

(1.9)

Figure 1.1 Finite-conductivity fracture solution. Dimensionless pressure vs. dimensionlesstime [111

These are defined for liquid (oil) as:

2rrkh(Pi - Pwf)PD= ,

qj.£

Dimensionless Time, tDXf

FCD

Functionsfor optimalfractureconductivityas usedin Example1.3Figure 1.3

10'10"10'10·

·210-'

!v -1

"I'.. / ~.,.....

V~ JI'- '1/1--'

,...

5

4

3

2~~-e:

0

·1

r. xfIn - -In - + fl(lOglo FCD),

Tw rw

Wewill use th~functionf 1. replotted in Figure 1.3 for convenience.Given the functionf I the denominator of the production rate can be expressed as

Th~pse~dor~dial,steady-s~a~eflowim~liedby this relationshipshould emergerelativelyrapidly III higher-permeability formations, which are the normal candidates for "frac& pack" treatments.Obviously, our aim is to minimize the denominator. This can beaccomplished using the Cinco and Samaniegograph, which is a plot of the functionf), definedby

f I(Iogro F CD) = Sf + In Xf.r",

The.same proppedvolume can be establishedcreating a narrow, elongated, fractureora Widebut .short one. The production rate will depend on the decision according toEq. 1.8, which for steady-stateproduction rate takes the form

Solution

fracture.)Use a realistic fracture permeability, taking into account possible damage tothe pr~ppant~kj = 1 X 10-11 m2. Assume that the created fracture height equals theformation thickness. Use the Cinco and Samaniegograph, Figure 1.2, which assumespseudoradialflow.

9Hydraulic fracture design

Consider once again the reservoir and well data of Example 1.2 (k = 3.9 x10-16 m2, h = 20 m, r.= 300 m, J.l = 1 X 10-3 Pa- s, Pe = 3.5 X 107 Pa andpwf = 2 x 107 Pa). Determine the optimum fracture half-length, Xj, the optimumpropped width,w, and the optimumsteady-stateproduction rate if the volumeof thepropped fracture, V f = 100m3, is given. (Note that V f is the volumeof the two-wing

Example 1.3 Optimal Fracture Conductivity

A fracture-propagation model then describes the hydraulic fracture geometry defi­nitely including the width and, with an appropriate model, the fracture height. Thisissue is addressed in detail in Chapters 9-11.

The required fracturing fluid volume is then estimated through a material balanceaccounting for the created fracture volume and the fluid leakoff normal to the fracturefaces. This calculation simultaneously provides the required injection time.

Chapter 8 contains fracture leakoff models and the manner in which they areincorporated in the fracture-propagation material balance.

There are several techniques to estimate the required mass of proppant materials.The calculation depends on the manner of propp ant addition to the fracturing fluidslurry. A common method suggests a rampedproppant schedule (see Nolte (13]) withits onset depending on the leakoff characteristics. Thus, after the end of injectionthe mass of proppant leads to the propped fracture width assuming that the fracturelength is either equal to the hydraulic length or it is truncated by some practicalcriterion, e.g. where the width is equal to three proppant diameters. The choiceof proppant is critical since the fracture permeability at the expected in situ stressdepends on the strength of the proppant (see Brown and Economides [14]).

Thus, the propped width, w, the fracture permeability, kI> the assumed fracturehalf-length, XI and the reservoir permeability, k are sufficient to allow the forecast ofthe post-treatment well performance using the model presented in Section 1.3. Thisprediction leads readily to the future incremental benefits which, when discountedto 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 treatmentcosts which, when subtracted from the present value of the incremental revenue, leadto the NPV, specific for the assumed fracture half-length.

The procedure is then repeated with increments of the fracture half-length and foreach the corresponding NPV is determined. Optimum xI is the one correspondingto 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 formation.

With the advent of the tip screen-out technique ("frac & pack"), especially in high­permeability, soft, formations, it is possible to create short fractures with unusuallywide propped width. In this context a strictly technical optimization problem can beformulated: how to select the length and width if the propped fracture volume isgiven. Example 1.3 deals with this problem.

Hydraulically induced fractures8

~~~~;;;;;;;;;;;;;;~~~~===:--.--.--- .

Fracturing fluid properties are expected to facilitate fracture initiation (breakdown),fracture propagation and proppant transport while they minimize leakoff and long­term residual damage to the proppant-pack permeability.

Viscosity is, thus, the essential property and may be augmented by additives duringexecution. It must be destroyed by other additives after the treatment.

1.5.1 Fracturing Fluids

Hydraulic fracturing is a massive operation, frequently resulting in the injectionof more than 2000 m3 of fracturing fluids, 5 x lOS kg of proppants at bottomholepressures that could be over 5 x 107 Pa (corresponding to wellhead pressures of2 x 107 Pa) while employing as many as two dozen active or standby pumping unitseach capable of delivering 1500 to 2000 hhp (1100 to 1500 kW). Analogous powermay be available on specially designed stimulation vessels for offshore operations.

Figure 1.4 is a schematic depiction of the execution operation. Fracturing fluidswith appropriate additives are blended with metered proppant and then injectedthrough appropriate fracturing strings into the target formation. Below, there isa brief description of fracturing fluids, their expected performance, the additivesthat affect this performance and common propping materials. Brown and Econo­mides [14] contains a much more detailed description along with large amounts ofdata required for the selection of fluids and proppants. Chapters 5 to 7 of this bookdescribe the rheology and fluid mechanics of fracturing slurries.

1.5 Treatment Execution

In general it is necessary to check if the resulting half-length is less than r, (otherwisexf has to be selected to be equal to r.). Similarly, one has to check if the resultingoptimum width is realistic, 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 exampleboth conditions are satisfied.

The above example provides a deeper insight into the real meaning of dimensionlessfracture conductivity. The reservoir and the fracture can be considered as a systemworking in series. The reservoir can deliver more hydrocarbon if the fracture is longerbut with a narrow fracture the resistance to flow may be Significant inside the fractureitself. The optimum dimensionless fracture conductivity (F CD.opr = 1.2) corresponds tothe best compromise between the requirements of the two subsystems. 0

= 4.54 X 10-3 m3/s(247 barrels/day).

In 300 -In

2 3 9 1016 20(3.5 X 107 - 2 X 107)

]I' X ,x - X -'-------,:-;:---'-• 1 X 10-3q =~---r======~==---

100 x 10-11=---=-=-=-=--:-=-~+ 1.452 x 20 x 3.9 X 10-10

The optimum production rate (assuming P. = 3.5 X 107Pa and pwl = 2 x107 Pa) is

11Treatment execution

----_._ •._---- ..._---_.

w=

100X 10-11 = 232 m,2.4 x 20 x 3.9 X 10-16

0.6 x 100 x 3.9 X 10-16 = 0.011 m.20 x 10-11

XI =

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

21ikh~p_ f.L

q - [V;k;In r. - In V -!if + 1.45

and the optimal steady state production rate is

the optimum width is obtained from

rv;k;xf = V 2.4hk'

and is plotted as a straight line in Figure 1.3.The function f 3, which we wish to minimize, is simply the sum of f 1 and f 2· As

seen from Figure 1.3 it has a minimum at FCD.opr = 1.2 where f 3,opr = 1,45. Therefore,the following results hold:

The optimum half-length is given by

where the only unknown variable is FCD. The first two terms are constant, and hence donot affect the location of the minimum. The last two terms do not contain any problem­specific data. Therefore, the optimum FCD is a given constant for any reservoir, welland proppant. (Moreover, the same optimum F CD would result for pseudo steady stateproduction rate.)

To find the optimum FCD we introduce two new functions: the first one, denoted by12, is defined by

From the above expression we can eliminate the half-length using the relationship,V f = Zwhx]; and the definition of the fracture conductivity, Eq. 1.11. As a result, wearrive at the following minimization problem:

which can be further simplified to give

Hydraulically induced fractures10

----------- ..------, •__ . " 'u.,,_~~ . __ ~ _

The hydraulic width created during the injection is reduced to zero after suppliedfracturing pressure subsides to the closure pressure, unless propping materials areused. It is this residual propped width that can be used for the forecast of fracturedwell performance that was outlined in Section 1.3.

Proppant size and proppant strength are the main criteria for selection. The generalfamilies of proppants are divided into low, intermediate and high strength. Thedemand for strength is directly related to the level of stress that the proppant willexperience in the long term.

Low-strength proppants are natural sands in typical sizes from 12/20 mesh to20/40 mesh (average particle diameter is 2 x 10-4 m to 1 x 10-4 m). They areusually attractive at depths less than 2000 m because although they are the least

1.5.2 Proppants

have considerably reduced viscosities (e.g. < 2 x 10-2 Pa- s) which are insufficientfor proppant transport. Required minimum viscosity in the fracture, where large shearrates at the tip may reduce the viscosity further, is considered to be 0.1 Pa- s (seeBrown and Economides [14]).

To increase the viscosity substantially, crosslinkers of the polymer chains havebeen employed. For temperatures below 115°C borate crosslinkers are considereddesirable. For higher temperatures, organometallic crosslinkers such as titanium andzirconium complexes are necessary. To meet the demand for lower viscosity in thetubulars 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 thefluid passes through the perforations. To avoid oxidative degradation in the fracture,oxygen scavengers are often added to the fluid.

A "40-1b borate-crosslinked gel" (40 lb/Mgal = 4.8 kg/nr') at a reservoir temper­ature of 90°C would still have an apparent viscosity of 0.2 Pa . s after 4 hours ofinjection-induced shear (see Brown and Economides [14]).

Oil-based fluids have been used in water-sensitive formations with a phosphateester as the gelling agent. These fluids are losing their "market share" because ofenvironmental 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 dioxideor nitrogen. Foam qualities (gas volume fraction) from SO to 90% have been usedwith 70 being very common. The purpose in using these fluids is to minimize filtratedamage and, more importantly, to facilitate the cleanup: fluid fiowback after thetreatment.

After the injection stops the formidable task of breaking down the polymeremerges. Unbroken polymer chains result in a marked reduction in the permeability ofthe proppant pack. Thus, oxidizers or enzymes, and at times encapsulated breakers,are added to the fracturing fluid. The breaking action is critical to the success ofhydraulic fracturing and is the subject of active ongoing research.

13Treatment execution

The ideal fluid has low viscosity in the horizontal and vertical tubulars to reducethe friction pressure and, therefore, the required treating pressure. After the fluidenters the fracture, the viscosity should have a high value to cause a larger width andbetter proppant transport. In addition, the same agents that enhance viscosity may beused for the building of a filtercake on the fracture walls to reduce leakoff. After thetreatment, the high viscosity is no longer needed but, instead, it is highly detrimentalto 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

(REC) and hydroxypropyl guar (HPG) in quantities varying (in field units) from20 lb/Mgal (2.4 kg/nr') to 80 lb/Mgal (9.6 kg/nr'). At ambient conditions thesepolymer solutions may lead to viscosities up to 0.1 Pa- s (at expected shear ratesin a fracture) but at reservoir temperatures (T = 6S·C to l1SoC or even higher) they

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

Hydraulically induced fractures12

Rock, fracture and fluid mechanics are critical elements in the understanding andengineering 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 2and 4). Contrast between the properties of adjoining layers controls the verticalfracture height migration (Chapter 11).

1.7 Mechanicsin Hydraulic Fracturing

Pressure transient testing iswidely practiced by engineers dealing with porous media.Analysis of the pressure and rate data while the well is flowing (drawdown) or shutin (buildup) or observed by another well (interference) allows the determinationof important well and reservoir variables. These include the skin effect, reservoirpermeability and permeability anisotropy, types and locations of boundaries andformation heterogeneities (such as two-porosity systems.)

For wells that could be candidates for hydraulic fracturing, a pretreatment welltest can reveal the reservoir permeability and skin effect allowing a decision forstimulation (matrix vs. fracturing vs no treatment at all.) If fracturing is indicated thereservoir permeability is a critical variable for the design optimization (see Balenet at. [28]).

A post-treatment well test, and assuming the reservoir permeability is known, canprovide the fracture half-length and fracture conductivity. Such a determination isessential 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.

1.6.3 Well Testing

The appearance and disappearance of fissures as the stress on a core is reduced orincreased and the counting of these fissures has been used for the determination ofstress anisotropy in oriented cores.

Strain relaxation and its measurement with sensitive devices has been referred toas anelastic strain recovery (see Blanton [25]; Teufel [26]).

Oriented cores are specially prepared and fitted with gauges which detect therelative displacement resulting from strain recovery.

The reverse procedure is used for the differential strain recovery analysis wherecores are re-stressed and the relative differences in displacements are correlated withstress anisotropy (see Strickland and Ren [27]).

1.6.2 CoreMeasurements

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

15Mechanics in hydraulic fracturing

Pretreatment log measurements are intended to obtain mechanical properties ofthe target and adjoining intervals and predict stress values and, especially, stresscontrast. This would give indications for the fracture height migration (see Newberryet al. [15]; Ahmed et al. [16]).

Borehole acoustic televiewers are used for the measurement of sonic travel timeand amplitude (see Pasternak and Goodwill [17]; Plumb and Luthi [18]). Identifyingborehole ellipticity and the presence of vugs and natural fractures provide evidenceof stress anisotropy and, thus, the expected hydraulic fracture azimuth.

Mechanically fitted dipmeter logs with four and six arms are used to detectopen-hole ellipticity and stress-related wellbore breakouts. These effects have beencorrelated clearly with stress anisotropy [19-21].

Dipmeter logs with a dense array of microresistivity detectors provide wellboreimages where natural fissures can be mapped. These devices are used in both verticaland 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

1.6.1 WellLog Measurements

Field and laboratory measurements are often conducted before and after a hydraulicfracture treatment to predict and evaluate fracture geometry and conductivity.

Data acquisition involves well logging, core laboratory investigations, well testingand fracture calibration injections. Seismic techniques, although expensive, can beused in critical cases. The data acquisition has a cost and, thus, the selection oftests depends on the benefits from the knowledge of particular variables and theopportunity cost of their ignorance.

Appropriate selection of data acquisition techniques is an essential part in thesuccess 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 forfurther reading.

1.6 DataAcquisition and EvaluationforHydraulic Fracturing

expensive propp ants they undergo severe crushing resulting in substantial proppant­pack permeability reduction. An NPV-based design procedure allows the balancingof 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 togetherand thus maintain a high fracture permeability at larger stress values.

Synthetic, intermediate- and high-strength proppants are used at depths up to 3000and 5000 m, respectively.

Brown and Economides [14] contains an extensive coverage of proppant proper­ties including their degradation from long-term exposure to stresses.

Hydraulically induced fractures14

. --------_ .._-----------------------._ .•..----.-- ..---.- -----_ ....

26.

25.

24.

23.

21.22.

20.

17. Pasternak, E.S. and Goodwill, G.D.: Application of Digital Borehole TeleviewerLogging, Proc. 24th Annual SPWLA, 1983.

18. Plumb, R.A and Luthi, S.M.: Application of Borehole Images to Geologic Modeling ofan Eolian Reservoir, Paper SPE 15487, 1986.

19. Brown, R.O., Forgotson,l.M. and Forgotson, I.M., Jr.: Predicting the Orientation ofHydraulically Created Fractures in the Cotton Valley Formation of East Texas, PaperSPE 9269, 1980.Gough, D.I. and Bell, 1.S.: Stress Orientations from Oil-Well Fractures in Alberta andTexas, Can. Jour. Earth Sci., 18, 638-645, 1981.Zoback, M.D. and Zoback, M.L.: in Neotectonics, G.S.A, 1988.Svor, T.R. and Meehan, D.N.: Quantifying Horizontal Well Logs in Naturally FracturedReservoirs - I, Paper SPE 22634, 1991.Meehan, D.N. and Svor, T.R.: Quantifying Horizontal Well Logs in Naturally FracturedReservoirs - II,Paper SPE 22792, 1991.Lin, P. and Ray, T.G.: A New Method to Determine In-Situ Stress Directions and In-SituFormation 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 RecoveryMeasurements of Oriented Cores, Proc. 23rd U.S. National Rock Mechanics Symposium1982. '

27. Strickland, F. and Ren, N.: Predicting the In-Situ Stress of Deep Wells Using the Differ­ential Strain Curve Analysis, Paper SPE 8954, 1980.

28. 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.

17References

1. Anonymous, OPEC's Facts and Figures, Organization of Petroleum. ExportingCountries, Vienna, 1993.

2. Al-Hussainy, R. and Ramey, H.I., Ir.: Applications of Real Gas Theory to Well Testingand Deliverability Forecasting, JPT, (May), 637-642, 1966.

3. Dake, L.P., Fundamentals of Reservoir Engineering, Elsevier, Amsterdam, 1978.4. Craft, B.C. and Hawkins, M. (Revised by Terry, R.E.) Applied Petroleum Reservoir

Engineering, 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1991.5. Amyx, J.W., Bass, D.M. and Whiting, R.L.: Petroleum Reservoir Engineering; Physical

Properties, McGraw Hill, New York, 19606. Economides, M.J., Hill, AD. and Ehlig-Bconomides, C.A: Petroleum Production

Systems, Prentice Hall, Englewood Cliffs, N.J., 1994.7. Earlougher, R.C., Jr.: Advances in Well TestAnalysis, SPE, Dallas, TX, 1977.8. Van Everdingen, AF. and HUrst, N.: The Application of the Laplace Transformation to

Flow Problems in Reservoirs, Trans. AlME, 186305- 324, 1949.9. Gringarten, AC. and Ramey, A.J., IT.: Unsteady State Pressure Distributions Created

by a Well with a Single-Infinite Conductivity Vertical Fracture, SPEl, (Aug.), 347-360,1974.

10. Cinco-Ley, H. and Samaniego, F.: Transient Pressure Analysis for Fractured Wells, JPT,1749-1766,1981.

11. Agarwal, R.G., Carter, R.D. and Pollock, C.B.: Evaluation and Prediction ofPerformance of Low-Permeability Gas', Wells Stimulated by Massive HydraulicFracturing, JPT (March), 362-372, 1979; Trans. AlME, 267.

12. Meng, H.Z. and Brown, K.E.: Coupling of Production Forecasting, Fracture GeometryRequirements and Treatment Scheduling in the Optimum Hydraulic Fracture Design,SPE Paper 16435, 1987.

13. Nolte, K.G.: Determination of Proppant and Fluid Schedules from Fracturing PressureDecline, SPEPE, pp. 255-265, July 1986.

14. Brown, J.E. and Economides, M.J.: Practical Considerations in Fracture TreatmentDesign, in Economides, MJ.: Practical Companion to Reservoir Stimulation, Elsevier,Amsterdam, 1992.

15. Newberry, B.M., Nelson, R.F. and Ahmed, U.: Prediction of Vertical Hydraulic FractureMigration Using Compressibility and Shear Wave Slowness, Paper SPE/DOE 13895,1985.

16. Ahmed, U., Newberry, B.M. and Cannon, AM.: Fracture Pressure GradientsDetermination from Well Logs, Paper SPE/DOE 13857, 1985.

References

Fracture mechanics is an obvious field of study in this endeavor allowing for theinteraction between the provided pressure and the resisting stresses. Tip propagationmechanisms and their effect on the observed net pressures are subjects of ongoingresearch 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 theformation to be fractured and the resulting hydraulic fracture morphology. Theseconcepts are treated extensively in Chapters 9 to 11. They are also the centralelements of this book.

Hydraulically induced fractures16

-----, .'_-,_._------_-_ ----

measured in N/m2 or, briefly, Pa,

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

a = lim (D.F) , (2.1)IlA .... O M

2.1.1 Stress

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

2.1 Force and Deformation

A purely elastic body has a natural state to which the body returns if all the externalforces are removed. An elastic deformation is therefore reversible: The work doneon the body is saved as elastic energy which is totally recoverable. If deformationsand their inducing forces (or forces and their inducing deformations) are connectedby a linear relationship, this is linear elasticity. The appearance and propagation ofa fracture means that the material has responded in an inherently non-elastic wayand an irreversible change has occurred. At first glance, therefore, it seems thatelastic 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 boththe stresses and deformations (except for the fracture surface and perhaps the vicinityof the tip) may be still well described by elastic theory.

LINEAR ELASTICITY,FRACTURE SHAPESAND INDUCED STRESSES

2

------ -----------._-----------_-----_------._------ __ -- -----

(2.4)t:.le=-.

I

We can think of a deformation as a transition from a reference configuration intoanother one. Simple translation or rotation of a rigid body are also deformations, butare of little interest in the present context. In elasticity theory the interest is withdeformations, where the relative position of the points changes.

For defining a suitable measure of the deformation let us consider two materialpoints. If I is the original distance between the two points and I + Sl is the newdistance, the engineering strain is defined by

2.1.2 Strain

denoted by CT1 :::: CT2 :::: 0'3 give the magnitude of the principal stresses. The compo­nents of the corresponding eigenvectors are the direction cosines of the plane (withrespect to which the maximum occurs) and hence components of the direction vectorof the principal stresses. Moreover, the eigenvectors are mutually orthogonal (aconsequence of the symmetry of the matrix).

In some applications the eigenvectors provide a natural coordinate system. In thiscoordinate system the matrix (2.3) will be diagonal. The eigenvalues of a diagonalmatrix 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 diagonalelements of the matrix, i.e, O'h CT2 and 173.

In geologic applications often we may assume that one of the principal stressesis vertical. Then one additional angle has to be given to specify the direction of thesecond principal stress in the horizontal plane. The third principal direction is alsohorizontal and orthogonal to the second one. Since such a direction is, unique theonly additional data we need are the values 0'1, 0'2 and 0'3·

(2.3)

If the six independent stresses are specified, the stress acting on any arbitrarilyoriented (oblique) plane can be obtained by applying force balance. The word"obtained" means that we can calculate the three components of the stress vector.(The actual expressions will be given later.) Once the stress vector is known, we candecompose 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 theoblique plane while the magnitudes of the normal and shear .stresses are v~rying. Ithappens that there are three specific orientations where the shear stress vanishes and(at the same time) the normal stress has a local maximum. The three local maximaare called principal stresses. The three eigenvalues of the matrix

(2.2)

are CT;m CTyy, Uu., .xy, ryz and .zx. The remaining three components are given by therestrictions:

21Force and deformation

Figure 2.1 Stresses acting on one surface of an elementary cube

The stress is a vector with magnitude and direction. An elementary 'surface iscontained in a plane which can be rotated arbitrarily and hence there is an infiniteset of stress vectors associated with a given point The stress state is given if weprovide an appropriate means to determine the stress corresponding to any arbitrarilyselected plane direction.

Stresses normal to the plane may be tensile or compressive, while those parallelto the plane are called shear. A normal stress is readily visualized based on everydayexperience. To understand shear stress properly some abstraction is needed. Anystress can be decomposed into two orthogonal shear components and a tensile (orcompressive) one. A common system of notation includes two suffixes: the first onerefers to the direction of the stress while the second one denotes the direction of theoutward 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 havedifferent suffixes. To emphasize the difference, shear stresses are often denoted byr. If there is no danger of misinterpretation, the second suffix of a normal stress canbe deleted (since it is identical to the first one.)

Figure 2.1 shows an elementary cube whose edges are parallel to the Cartesiancoordinate axes. There are nine stress components but they cannot be selected inde­pendently. Rotational equilibrium poses three constraints on them. The state of stressat a point is determined by six independent stresses: In Cartesian coordinates these

Linear elasticity20

---_ ....._--_ .._.....•--_ ....

Figure 2.3 Uniaxial compression. Determination of Young's modulus and Poisson ratio

ev=-cyyxx

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

(2.8)

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

2.2.1 Linear Elastic Material

Real materials have complex behavior when subjected to a stress field. Idealizedmodels help to understand the main features of the behavior. A perfectly elasticmaterial stores the work done on it by external 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 wherethe constitutive equation is linear.

2.2 Material Properties

Similar arguments lead to the definition of other strain components listed in Table 2.1.Again, six independent components (en, eyy, eZZ'exy, e}Z and ezx) should be spec­

ified to give the state of strain at a given point.

(2.7)

Hence, a suitable definition of the first component of the strain state, ex" in accor­dance with Eq. 2.4 is

23Material properties

-__ ------

Secondindex x y zFirst index

aux 1 (aux +auy) I Cu: au,)x - -+-ax 2 oy ax 2 ax oz

y see xy au}'I Cu}' au,)- -+-ay 2 az oy

see .rzau,z see yz8z

Table 2.1 Strain components, Cij

Figure 2.2 Displacement and strain

(x',y)

(X' + tt.. y'+ U~)

(2.6)aux1+tll = ox+-ox.ax

and

(2.5)I = ox,

Tensile strain corresponds to extension whereas compressive strain corresponds tocontraction. By convention, strain associated with extension is negative and compres­sive strain is positive. However, in rock mechanics and especially in hydraulicfracturing sometimes the opposite convention is more appropriate. The actual signconvention should be clear from the context. A shear strain is associated with planelayers 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 necessaryto consider a point in the original configuration with coordinates x, y and z. Afterdeformation, the new coordinates will be x + ux, y + uy and z + u-, respectively (seeFigure 2.2). The quantities u», uy and Uz are the components of the displacementvector. With changing location of the original point the displacement may vary butsmoothly. If we consider a straight line starting from (x, y, z), parallel to the x_ axisand of length ox (where this length is short enough) then

Unear elasticity22

._------------_---

A cylindrical sandstone specimen (density, p = 2700 kg/m-) is loaded by a compressiveforce, F = 0.8 X 106 N. The height (I = 20 em) decreases by 4 mm and the diam­eter (D = 5 em) increases by 0.2 mm. Determine the elastic constants and predict thelongitudinal wave propagation velocity.

Example 2.1 Determining Elastic Properties from a Uniaxial Test

Since longitudinal expansion and contraction involve volume change and shearstrain at the same time, it is not surprising that both the compressibility and the shearmodulus playa role in the final expression.

The great advantage of dynamic tests is that in situ measurements (without cuttingout a specimen and, hence, destroying the material) are available. To characterizethe material, the velocity of two different types of waves has to be determined.

(2.13)

where A- and G are often referred as the Lame constants. As seen, only two indepen­dent material constants are necessary compared to the original 21.

Thus, for an isotropic material the elastic constants E, G, v, A, K are related bysimple algebraic relations and any two of them determine the other ones. In thefracturing literature mostly E, G and v are used. Table 2.2 shows how the otherconstants can be related. Some authors prefer to introduce additional combinations.One of the combinations, the plane strain modulus, E', is particularly useful infracture mechanics, hence it is included in Table 2.2.

Static tests do not provide the only possibility to measure material properties.Dynamic tests consist of periodically changing the load on the surface of the materialand observing various characteristics of the forced elastic waves. The propagationvelocity of a longitudinal wave in the interior is, e.g., related to the density and theelastic constants according to Billington and Tate [1]:

Ev 2Gv G(E - 2G)A --

(l + v)(l - Zv) 1 - 2v 3G-EE

G2(1+ v)

E 2G(1 + v)

E 2G GEK

3(1 - 2v) 3(1+ v)(l - 2v) 3(3G -E)E-2G

v ---2G

E'E 2G 4G2-- --1- v2 I-I! 4G-E

Table 2.2 Interrelations of the elastic constants of an isotropicmaterial

25Material properties

..---.--- .. -------

A+2G A A- D 0 DA A+2G A- 0 0 0). A ).+2G 0 0 00 0 0 G 0 0 (2.12)

0 0 D D G 00 0 0 0 0 G

where C, the stiffness matrix, consists of 36 material constants, the situation appearscumbersome. Fortunately, the symmetry requirement, Cij = Cji, decreases thisnumber to 21, "which is now generally accepted to be the number of independentelastic constants (Billington and Tate [1])".

Determining 21 material properties is still very difficult. Assuming some addi­tional invariance properties, however, may further reduce the number of indepen­dent material constants. By far the most effective assumption is isotropy. For anisotropic material the properties are independent of direction. The stiffness matrix isof the form

a;u Cll Cl2 C13 C14 C15 C16 e;uayy C2l C22 C23 C24 Czs C26 EyyaZZ C3l C32 C33 C34 C35 C36 ezz (2.11)

C4l C42 C43 C44 C45 C46xaxy Exy

axz C5l CS2 C53 C54 C55 CS6 exzayZ C61 C62 C63 C64 C65 C66 8;;z

where G is the shear modulus. Under hydrostatic compression the relative volumechange is related to the hydrostatic pressure through the bulk compressibility, K.

At this point an important question arises. Is there any relation between theobservable material properties? In other words, how many independent propertiesare necessary to characterize the material already known to behave linearly? Startingwith the generalized Hooke's law:

(2.10)

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 some of the bound­aries, (3) the measurement of the resulting displacement of the boundary surface (orthe resulting stress on the boundary surface). The uniaxial compression test illustratedon Figure 2.3 is suitable to determine Young's modulus and the Poisson ratio in oneexperiment. The compressive stress and the strains are readily derived accordingto the expressions shown on Figure 2.3. The two material properties, E and v, areobtained from their definitions.

Other simple tests give rise to other material properties. During the torsion of acircular bar around its axis, the shear stress and shear strain are related by

(2.9)a=

eyy = -V-,E

effect" is given by

Linear elasticity24

~--------- '---""------"-----" '" ---

(2.14)

When one of the principal stresses is zero, the condition of plane stress is satisfied.The 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 sameplane but the deformation is not restrained in the z direction. The following stressesare zero:

2.3.1 PlaneStress

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

2.3 Plane Elasticity

the envelope of stability or failure (yield) surface, A failure (or yield) criterion is theequation of this envelope.

A detailed description of the material thus consists of an elasticity constitutiveequation, a yield criterion and another constitutive equation valid in the post-yieldregion. Since yielding is inherently irreversible, a simple curve is not enough torepresent the behavior in this region. In fact the actual behavior depends not onlyon the strain but on the history of the total loading process.

Figure 2.4 Stress-strain relations of several typesof materials

(c)

(J~

0,, ~-

(e) S

(b)(a)

27Plane elasticity

---_ ..'.._---- ...._---_ ..._-----

A hypothetical linear elastic body "answers" with a continuously increasing strain toa linearly growing stress. No material can be loaded infinitely, because, eventually,it will fail. At this critical value of the stress any further "strain" can be achievedeasily because the material loses its ability to resist deformation. This is seen fromthe stress-strain curve (a) of Figure 2.4. The fact that the curve (up to the failure) is astraight line indicates that the material is linear elastic. The dashed line represents thebrittle rupture. The stress-strain curve (b) corresponds to a plastic material in whichstrain occurs without any change in the stress. The work done in the plastic regionis dissipated and the material flows. Curve (c) shows a material with an elastic anda plastic region. Curve (d) illustrates a material without a distinct elastic region. Thestress necessary to start any deformation is called yield stress. Curve (e) correspondsto a material exhibiting typical nonlinear behavior, most likely due to continuousdamage evolution.

The 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 ofcompressive stresses. In general, the failure will occur (or yielding will start) atspecific combinations of the three principal stresses. Ifwe represent the state of thestress as a point in the three-dimensional stress space (taking the principal stresses ascoordinates) then the stable and unstable states will be separated by a surface called

2.2.2 Material Behavior Beyond PerfectElasticity

VL = (3K +4G) 1/2 _ (3 x 11X 109+4 x 8.5 X 109) 1/2 _3p - 3 x 2700 - 2900 m/s. 0

and hence, Eq. 2.13 predicts

K = 3(1 _ 2v) = 11 GPa,

EG = 2(1+ v) = 8.5 GPa,

E

From Table 2.2 we obtain

so 0.2 x 10-3

v = - fl = --;2'g'ii'g04;;<5,- = 0.2.

I 0.2

F 0.8 X 106_ D2;r/4 _ 0.052 x rr/4 10

E - ---;rr- - 0.004 = 2 x 10 Pa = 20 GPa

T aT

The test gives Young's modulus and the Poisson ratio directly:

Solution

Unear elasticity26

(2.21)(7= Sx cos e+ Sy sin e.Since (7 is the projection of S, we have

(2.18)

(2.20)Sy = <"yx cos e + Uyy sin fJ.

Figure 2.7 Stresses acting on an oblique line

x

cA

Sy

y

Figure 2.6 Stresses acting within a plane

x

where LAB,LAC and LBC are the lengths of the corresponding line segments. Since

LAB . LACcase =- and smB =-,LBe LBe

(2.17) and similarly, we obtain

posing restriction on the lengths of the components.Consider the balance of forces in the x direction:

(2.19)

Equation 2.17 can be rewritten as

Sx =Uxx case + <"xy sin e,

(2.15)

(2.16)

Now we return to the problem of determining the normal and shear stress vectorsrelative to an oblique plane. Since we deal with two dimensions (i.e, the stressesin the x, y plane) the oblique plane is represented by a straight line. Assume thatwe know the angle e and that we have a description of the stress state, i.e, the twonormal stresses (7=, Uyy and the shear stress <"xy. The definition of the stresses isshown on Figure 2.6 with the help of a square placed parallel to the axis.

In Figure 2.7 the straight line cuts off a comer of the same square. The directionvector s and the normal vector n of the straight line are determined uniquely bythe angle e. We are interested in determining the stress vector S, i.e. calculating itstwo components: Sx and Sy. Once we know the stress vector S, we also want todecompose it into a normal component (7 and a shear component r with respect tothe straight line characterized by the angle e.Applying the Pythagorean theorem wecan write

2.3.2 Stresses Relative to an Oblique Line (Force Balance I)

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

Figure 2.5 Material in the state of plane stress

y

--_ ...._._-_.

28 Unear elasticity Plane elasticity 29

ZtL au=0 Uyy tr,tz'= tz:w =0 tty...x 'YZ= 'zy =0

These relations are automatically satisfied if we consider the displacements ratherthan the strains as the unknown variables to be determined.

Plane strain is a reasonable approximation in simplified description of hydraulicfracturing. The main question is how to select the plane. Two possibilities arise andthis has given rise to two different approaches of fracture modeling. The state ofplane strain was assumed in horizontal plane by Khristianovitch and Zheltov [3,4)and by Geertsma and de Klerk [5], while plane strain in vertical plane (normal tothe direction of fracture propagation) was assumed by Perkins and Kern [6] andNordgren [7). Often in the hydraulic fracturing literature the term "KGD geometry"is used interchangeably with horizontal plane strain assumption and "PKN geometry"is used as a substitute for postulating plane strain in the vertical plane.

The selected plane (in which the plane strain condition is applied) should beorthogonal to the largest extent. For a long fracture (hundreds o~ me~e~sof length)with limited height (tens of meters) and small width (measurable Inmillimeters) onecan assume the state of plane strain in every vertical plane orthogonal to the lengthaxes. For a short fracture (a few meters of length) with considerable height (tens ofmeters) and small width (millimeters) one can assu~e the= of ~lane strain ~nevery horizontal plane. In Chapter 4 the two geometnes are discussed IIImore detail.

(2.30)

with all the other stress and strain components being zero.Remembering that the strains have been derived from displacements and in plane

strain only the displacements Ux and uy are not zero, it is not surprising that the threestrain components are not independent. Indeed, their variation is constrained by thecompatibility relation:

(2.29)

(2.28)1eyy = E[ayy - v(O"zz + axx)],

2(1 + v)exy = E 'XY'

Figure 2.8 Material in the state of plane strain

31Plane elasticity

--_ _--_ __ ._--------------------_ .. -.----

(2.27)

The strain is, thus, two-dimensional, with no displacement in the z direction. (Thenormal stress in the z direction is, however, not necessarily zero.) The relationbetween strain and stress for the plane strain system (without body forces and temper­ature variation) is

(2.26)

Another special case reducing the three-dimensional problem into a two-dimensionalone is the case of plane strain. Consider a (practically) infinite body of uniform crosssection lying parallel to the z axis (Figure 2.8). External forces applied are parallelto the x, y plane, i.e. they are "infinitely repeated" in every cross section. Intuitivelyit is obvious that the state of strain is independent of the coordinate z, i.e. it also"repeats itself'. In addition, the following strains are zero:

2.3.4 PlaneStrain

As we see, the equilibrium relations give two equations for the three independentstress components.

(2.25)

(2.24)aO"xx + a,yx = 0,ax ay

arxy + Myy = o.ax ay

Up to now we have studied relations valid at a given point. Another force balanceputs restrictions on the variation of the stress components with the location. Theseare the well-known equilibrium relations. For the case of plane stress (without bodyforces) we have

2.3.3 Equilibrium Relations (ForceBalance /I)

Equations 2.22 and 2.23 allow us to compute the normal and shear stresses relativeto a line characterized by the angle B. In Chapter 3 they will be used extensively.

(2.23)r = (O"xx - axx) sin a cos a - rXy(cos2a - sin2 B)

= ~(O"xx - O"y)") sin(2B) - rxy cos(2a).

Similarly, the shear stress is given by

(2.22)

Substituting Eqs. 2.19 and 2.20 into Eqs. 2.21, the final form of the normal stress is

Linear elasticity30

where E' is the plane strain modulus, given in terms of the other properties in thelast row of Table 2.2.

By virtue of the second boundary condition, the displacement is zero outsidethe crack. Clearly, a similar displacement of the lower line occurs in the negativey direction, and hence the width of the crack is simply twice the value given byEq. 2.33. The normal stress in the y direction is known inside the crack (it is the

(2.33)O::5x ~c.4 jC ~g(~)d~uy(x. 0) = - rrE' x (x2 _ ~2)1/2'

Note that ~ is a dummy variable having the same dimension as x. The functiong(~) has the same dimension as the pressure and it can be considered as a modifiedpressure summing up the effect of the pressure acting not only at the given locationbut at every other location. Once that function is known, the normal displacementof any point on the upper side of the crack line is given by

(2.32)o <~< c.t' p(x)dx

g(~) = Jo W - x2)1/2 •

at every location. For practical purposes, however, we are only interested in thedisplacement of the crack surface and the stress state at the tip and further along thecrack axes.

Mathematical solutions, based on the pioneering work of Muskhelishvili [9], havebeen accomplished by solving integral equations (England and Green [10], Green andZema [11]) or applying integral transformation (Sneddon [12]).

The solution procedure starts with the construction of a function g(~) accordingto

Figure 2.9 Pressurizedline crack

..' xcI ..

33Pressurized crack

The first condition states that the pressure acting on the line is compensated bythe normal stress of the same magnitude. (The second argument of the unknownfunctions is y. Clearly, y = 0 at the crack line.) The pressure is a function of thelocation x. It is supposed that the problem is symmetric with symmetry axes x = 0(the pressure acts on both faces) and y = 0 (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 equal to p(x).It is assumed that all 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. The complete solutionof the problem includes the construction of the stress state and the displacements

x::: o.X> C. (2.31)Uy(x, 0) = 0,

'1:"xy(x, 0) = 0,

l7yy(X,0) = - p(x), O::s x ::5 c,

An infinite plane with a hollow two-dimensional "crack" in it is an idealizationneeded for an analytically tractable problem. In addition, the crack is assumed to bewithout any appreciable opening. Pressure acts inside, trying to open the two sidesof the line. This is the famous Griffith [8} crack problem. We are interested in the(small but important) displacement of the points, i.e. in the shape of the fracture.Moreover we wish to describe the state of stress around the fracture.

For the sake of simplicity we assume that the infinite plane (in which the displace­ment occurs) is the x, y plane, and the line crack of length 2c is located along thex axis with its center in the origin (see Figure 2.9). The half-length of the line crack,c, is the characteristic size associated with the problem. The boundary conditions are:

2.4.1 Solution of the Line Crack Problem

Linear elasticity deals with static equilibrium of solids. Fracture propagation is adynamic phenomenon. Nevertheless, we can still use linear elasticity to describe astable fracture and its neighbours. Moreover, even a propagating fracture can beconsidered as transforming through equilibrium states. A "snapshot" of such anequilibrium state will be considered in the remaining part of this chapter.

2.4 PressurizedCrack

The governing equations of any problem in linear elasticity are the same and haveto be sati~fied at all points of the body. The specific problem is distinguished fromothers by its boundary conditions. In fracture mechanics often mixed boundary valueproblems are considered where at some part of the boundary the stress and at someother part the displacement are defined.

2.3.5 Boundary Conditions

Unear elasticity32

.~~---.--------.---.--- ..•--- ...- ...._.__ .

Interestingly, very few solutions have been published for any pressure distribu­tion different from constant. Here we give (some new) results for a polynomial

2.4.3 Polynomial Pressure Distribution

and hence the stress is infinite at x = c, i.e. at the crack tip. The phenomenon isoften referred to as stress singularity.

The solution of the constant-pressure line crack problem is illustrated byFigure 2.10 consisting of four plots. The first plot gives the pressure acting alongthe crack line. The second is the function g(~). It represents the integrated influenceof the pressure at a given location ~. The third plot shows the displacement (in otherwords the half-width,) u(x). Note the characteristic ellipsoid shape. Finally, from theplot of the tensile stress ahead of the crack tip, the dramatic effect of the crack canbe well seen. The stress induced by the inner pressure and amplified by the crackis infinite at the tip but decreases rapidly with the distance from the tip. The effectof the crack is hardly observable at distances larger than a few multiples of thecrack length.

(2.39)

While the shape of a pressurized crack is easily treatable, the stress distributionis more difficult to handle. The last term in Eq. 2.34 is zero since the function g(~)is constant. The remaining terms yield

O"yy(x,O) = Po [(X2 :c2)1/2 -lJ, x> c

Flgure 2.10 Constant pressure solution to the line crack problem

cc 4c2c 3cxx

b

xcc

35Pressurized crack

The reader may feel uncomfortable that the crack is treated as a line on the onehand and has an elliptical shape on the other. The given approach can be justifiedonly if the width is orders of magnitude smaller than the length. Fortunately this isthe case for hydraulically induced fractures.

Perhaps the most interesting feature of the solution is that the linear behaviorvalid locally is preserved for the whole crack as an entity. Indeed, the width dependslinearly on the opening pressure. The proportionality factor between pressure andwidth contains the elastic properties and a characteristic length.

(2.38)

The width is zero at the end of the crack (at x = c) and the maximum widthoccurs at the middle of the crack (at x == 0):

4cpoWO=P'

(2.37)4po ~w(x) = -y c2 -x2•E'

(2.36)o:::::x::::: c,4 Ie ~d~ 2po f"22.Uy(X, 0) = - E'rr x (x2 _ ~2)1/2 = Ii! V c: - .r-,

This is the well-known result stating that the line crack is of elliptical shape withthe width given by

Therefore, the corresponding g(~) function is also constant. The resulting displace­ment of the upper line of the crack is

(2.35)o < ~< c.

If the pressure opening the crack is constant, Po, the g(~) function is given by

($ podx porrg(~) = l« (~2 _x2)1/2 =T'

2.4.2 Constant Pressure

(2.34)2 [ xg(c) r g'(~) dS-- ]O"yy(x, 0) = -; (X2 _ C2)1/2 - g(O) - x Jo (X2 _ ~2)1/2 ' x » c.

The above equation is the one given by Sneddon (12, p. 318] with a modification. Itis derived for the case of differentiable g(~). If the pressure is a continuous functionof the location, then g(~) is differentiable. Fortunately, Eq. 2.34 gives correct resultseven for those cases where there are some distinct jumps in the pressure. Alongthe line y = 0 the normal stress in the x direction equals that in the y direction:O"=(x, 0) :;; O"yy(x, 0), X > c. Along the same line the shear stress, 't'xy(x, 0), disappears(see e.g. Green and Zema [11, p. 276]).

For several specific p(x) functions of practical importance, the above integralscan be solved in closed form.

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

Linear elasticity34

--- ---_ ..... _---_._. ---_. __ ..-.._-_ _._------------------

--~~-------

2.4.4 "Zipper" Cracks

The distributed forces, denoted by p(x), might be of cohesive nature as well. There­fore, we can allow negative p(x), at least in some part of the fracture. This "negative

In a fracture of half-length XI = 10m the pressure is Po = 0.5MPa in the center andlinearly drops toward the tips. At the tip the pressure is zero. Assume the state ofplane strain in (any) horizontal plane. What is the maximum width if E = 1 GPa andv =O.25?

Example 2.2 Line Crack with Linear Pressure Drop

The reader may easily verify that the above expressions reduce to the constant­pressure case, discussed previously, if the higher-order coefficients are zero.

Using the specified data

4{1- 0.252)Wo = 109]!" (5 X lOSJr - 5 X 104 X to) x 10= 12.8X 10-3 m = 12.8mm.

I! is interesting to compare the characteristics of the solution (Figure 2.11) withthose shown in Figure 2.10.The displacement, u(.x) has a very similar profile in bothcases. Also the shape of the stress distribution, u(x), is very similar near the tip in thetwo cases. 0

4Wo = -(poJr+ PIC)C.E'Jr

The maximum width is located at the center of the fracture, i.e. at x = O. If x tends tozero, the second term diminishes and the width is equal to

CTyy(X, 0) = po[x(x2 - cZ)-1/2 - 1]

+ 2Pl [cx(x2 _ C2)-1/2 _ x arctan(c(x2 _ c2)-1/2)]l(

+ P2 [ ( c~x _ ~ ) (x2 _ c2)-1/2 _ x2]

+ 2:3 [( ~x -cx3) (x2_c2rl/2_~ arctan(c(x2-C2)-1/Z)] + .... (2.43)

and the normal stress distribution is

Because of the horizontal plane strain we can use c = XI = 10 m. The pressure gradientis PI = -PO/xI = -50 kPaim while P2 = P3 = o. Figure 2.11 shows the importantcharacteristics of the fracture. The width is twice the normal displacement:

Solution

Figure 2.11 Linear pressure drop solution (Example 2.2)

uy(x, 0) =~,{ 2po(c2 _ x2)1/2

+ 2:1 [CCc2 _ x2)1/2+ x2 In C+ (c2x- X2)1/2]

2+ f_(cZ + 2x2)(? _x2)1/2

3

+ .~ [(~c3 + cx2) Ccl _ xZ)lf2 +x4ln c + CCZ-: X2)1/Z] + ... } (2.42)

40

8 1.2

6E <? 0.8c...§.4 e..~ '" 0.42 -,

0 00 2 4 6 8 10 10 20 30

x(m) x(m)

The normal displacement of the upper crack line is given by

Since any pressure distribution can be approximated by its Taylor series, the knowl­edge of the solution is of great importance.

Interestingly, the g(~) function is a polynomial in ;:

(2.40)0.5 0.80.4 ~0.6t ro0.3 a,

~ e. 0.4Q. 0.2 c:"

0.1 0.2

0 00 2 4 6 8 10 0 2 4 6 8 10

x(m) ~(m)(2.41)

distribution

Pressurized crack 37Linear elasticity36

..__ .._------------

Assume that 51 = 1 MPa pressure opens a line crack of half-length c = 10 m and thedistributed force (negative pressure), 52 = 2 MPa tries to close it near the tip. Whereis the jump of the pressure located if the ending is smooth? Calculate the maximumwidth if Young's modulus is E = 9.1 GPa and the Poisson ratio is v = 0.3.

Example 2.3 Zipper Crack with Piecewise Constant Pressure

(2.50)2 [ x(e2 - xt3)1/2]O'yy(x,O) =: -SI + - (Sl + S2) arctan 2 2 1/2 .

1( xO(X - xo)

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

. 1 [ql + q2 e2x5 - 2qlq2XQX + ?x2 - 2x5X2]uy(x,0)=-,-(Sl+S2) 4xoln---+xln 22 22 22'

E 1( q3 C Xo + 2qlq2xoX +C X - 2xOX

where(2.48)

ql = (e2 - X5)1/2,

q2 = (c2 - x2)1/2,

Final!y, the stress at the tip is(2.47)• 1(S2

Xo =esm .2(51 +S2)

(2.49)(

res2 )1+ cosx In 2(51 + S2)

. ( resz )sm2(SI + S2)

8(SI + s2)e . [ 1(S2 ]=: sm

E'tt 2(SI + 52)and the zipper crack equation, gee) = 0, is satisfied if

(2.46)x » Xo

8 (C + q1)Wo = £Ire (51 + sz)xo In ~

The maximum width is twice the normal displacement at x = 0, i.e.The third parameter is the location of the jump, Xo (see Figure 2.12). Note that allthe parameters are positive. The corresponding g(~) function is given by

if x ~ xo,

if x> Xo.(2.45)xo < x ~ c,p(x) = -S2,

and S2 is the cohesive distributed force (negative pressure) near the crack tip:Figure 2.12 Zipper crack solution (Example 2.3)

(2.44)p(x) =: S1, X ~ Xo,

to zippers.A glance at Eq. 2.34 may convince us that for avoiding stress singularity at the

tip, the function g(~) should be zero at ~ =: c. This is the zipper crack equation. Ifthe function g(~) is zero at the tip, the displacement computed from Eq. 2.33 has azero derivative at the tip with respect to x and hence the crack is closed smoothly.

A crack opened by constant positive pressure cannot have smooth closing. If,however, we allow for two levels of the pressure, one positive near the center of thecrack and one negative near the tip, the condition of smooth closing can be satisfied.This special pressure distribution plays a key role in several studies including those ofKhristianovitch and Zheltov [3,4], Barenblatt [13] and Geertsma and de Klerk [5].For brevity we refer to that pressure distribution as the jump function. The jumpfunction will be characterized by three constants: Sl is the repulsive distributed force(pressure) in the center part of the crack,

pressure" tries to close the fracture and may result in exotic fracture shapes. Ofparticular interest is a shape with smooth closing at the tip because then the stresssingularity disappears. Barenblatt [13J, who has built a whole theory on the condi­tion of smooth closing, notes that one of his colleagues compared such fractures

Pressurized crack 39

1.5 1.81

0.5 1.5

'" 0 -------------------- <0 1.2o, -0.5 D. 0.9~ -1 ~o_ -1.5 C» 0.6

-2 - 0.3-2.5 0

0 2 4 6 8 10 0 2 4 6 8 10x(m) ~(m)

20.6

1.5E <0 0.4.§. c,

::> ~0.5 b 0.2

4 8 10 20 ~o 40x (m) x (m)

38 Unear elasticity

--_ ... -_.__ ...--- ~--. "-_ ... _-_ ....~-.--.

Po1l'PI = -- = -78.54 kPa/m.

2c

Fracture mechanics has emerged from the observation that any existing disconti­nuity in a solid deteriorates its ability to carry loads (see Griffith [8J; Inglis [14]).A (possible small) hole may give rise to high local stresses compared to the onesbeing present without the hole. The high stresses, even if they are limited to a small

After substitution of P2 = P3 = 0 into Eq. 2.51 the pressure gradient satisfying thezipper equation is

2.5 Stress Concentration and Stress Intensity FactorSolution

Figure 2.13 illustrates the solution. Note the smooth closing at the tip. The normalstress at the tip is Po(-l +1l'/2) = 0.285 MPa. This is the same value as inside thecrack. The normal stress declines with the distance from the tip rather similarly toExample 2.3. 0

In a fracture of half-length XI = 10 m the pressure is po = 0.5MPa in the center anddrops linearly toward the tips. Assume the state of plane strain in (any) horizontal planeconsidering a line crack with C = xI. What is the "pressure" at the tip if the fracturecloses smoothly? What is the maximum width if E = 1 GPa and v = 0.25? (Note thesimilarity and difference with respect to Example 2.2.)

4 x (1 - 0.252)Wo = 09 (5 X 1<f1l'- 7.85 X 104) x 10 == 9.37 X 10-3 m = 9.37 mm,1 x rrExample 2.4 Linear Pressure Drop Causing Smooth Closing

Substituting the given parameters, we obtain

4Wo= -(PoJr + PIC)C.E'rr

Similarly to Example 2.2 the maximum width is given by

Any pressure distribution in the form of a cubic polynomial and satisfying Eq. 2.51results in smooth closing. Some care should be taken to ensure, however, that thedisplacement should be nonnegative. (Otherwise the crack is "overdosed" and theboundary conditions are no longer valid.)

The corresponding displacement and stress distribution can be read directly fromthe equations given in Section 2.4.3.

Po - PIC = - PoJr = 0.5 X 106 Pa - (78.5x 103 Pa/m)(10 m)2c

= -0.285 X 106 Pa = -0.285 MPa.

(2.51)3npo + 6CPl + 3nc2 p~+ 4c3p~= O.

Figure 2.13 Zipper crack solution with linear pressure drop (Example 2.4)

The "pressure" at the tip isApplying the zipper crack equation to the g(~) function given by Eq. 2.41, thefollowing simple condition is obtained:

2.4.5 "Zipper" Crack with Polynomial PressureDistribution

Figure 2.12 illustrates the solution. Note the smooth ending of the fracture. The normalstress is finite at the tip and equals to the normal stress acting on the fracture face insidethe tip (2MPa) but it still has negative infinite derivative. 0

8 x (3 X 106) x 8.66 ( 1+ 0.5)woo = 10 In --- = 3.63 X 10-3 m = 3.63 mm.10 it 0.866 .

0.25 r---------,0.2

lO.15:i-; 0.1

0.05o L-~""""':::::::==d

8 10 10 20 30 40x(m)

o ~~~~--~~~~o 2 4 6 8 10x(m)

0.6r0-a,e 0.40>

0.2

E' = 9.1 X 109 = 1010 Pa1- 0.32 '

and the maximum width according to Eq. 2.49 is given by

8 10-0.3

0 2 4 6x(m)

6

E'4§.::. 2

00 2 4 6

x (m)

The plane strain modulus is

. 1l'X 2 r:.:to= 10 x SID --- = 10",,3/2 = 8.66 m.2(1 +2)

~e. 0.1Q..

-0.1

The location of the jump is calculated from Eq. 2.47. 0.3

0.5 ~--------.Solution

41Stress concentration and stress intensity factorLinear elasticity40

·_- _-_._-,,---,,-_ .. __ _- ---_ _-_ -- _-_ .. -- _------------------- ----

Up to now we have considered a pressurized line crack in the absence of far-fieldstresses. How can we apply the results to real fractures where the far-field stressesare not zero? The general answer to this question is fairly complicated. Fortunately alarger fracture will align itself to the far-field stress field as discussed in Chapters 3and 4. Therefore, as a first approximation, we can represent the fracture by a linecrack located parallel to the-maximum principal stress of a given plane and openingup in the direction of the minimum principal stress of the same plane. The stressesare undisturbed only (relatively) far from the existing fracture and hence we callthem far-field stresses.

For consistency with the previous sections it is convenient to direct the x axis inthe direction of the maximum principal stress. Figure 2.14 shows the configurationof the line crack and the definition. We assume that at infinity the normal stressesare constant, satisfying ay < ax, and the shear stress is zero, Txy= O.

For this configuration the principle of superposition applies [2] as shown onFigure 2.14. In practice this means that an associated problem can be created. Theassociated problem has the same structure as the problem defined in Section 2.4.1.The only difference is that at the crack boundary the algebraic sum of two forceshas to be considered. The algebraic sum is the difference of the inner pressure andthe far-field minimum principal stress. It plays a central role in hydraulic fracturing.It is called net pressure.

2.6 Fracture Shape in the Presence of Far-field Stress.The Concept of Net Pressure

The above relations are of interest for determining the height of a fracture as will bediscussed in Chapter 11.

(2.55)

1 jC mx+xKl+ = r.;:;: p(x) --- dx,yl'CC -c c-x

1 t ~-xK/_ = r.;:;: p(x) -- dx.yl'CC -c c +x

Equation 2.53 was derived for a crack with symmetric loading, i.e. for p(x) = p( -x).If the loading is asymmetric, there are two different stress intensity factors at thetwo tips (see Rice [15]):

2.5.2 Stress Intensity Factor, Non-symmetrtc Loading

2g(e)cI/2 2 x 2.85 x UP x 101/2 _ 5 1/2K/ = = = 5.7':J X 10 Pa m . 0

7r l'(

Thus from Eq. 2.53 the stress intensity factor is

(2.54)PoT( .gee) = '2+ PIC = 2.85 X 10' Pa.

43Fracture shape in the presence of far-field stress

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

Substitutingc = 10 m, Po= 500 kPa, PI = -Po/X! = -50 kPa/m into Eq, 2.41, theend value of the g(~) function is obtainedas

Solution

Computethe stress intensity factor for Example2.2.

Example 2.5 Stress Intensity Factor

For a constant pressure line crack KJ= pocI/2. If the pressure distribution results ina zipper crack then the stress intensity factor is zero.

(2.53)

The stress intensity factor of a pressurized line crack is readily derived from Eq. 2.34:

2g(c)c1/2KJ=--­ T(

2.5.1 Stress Intensity Factor, Symmetric Loading

Not surprisingly, the stress intensity factor is strongly related to the function g(;),introduced in Eq. 2.32.

(2.52)

area, may lead to the rupture of the material. It is often convenient to look at mate­rial discontinuities as stress concentrators, especially if the nominal stress state (the"would be" stress state without a hole) can be scribed by a single scalar.

A circular hole in a plane induces stresses, the maximum magnitude of which isthree times the far-field stress. In other words a circular crack concentrates the stressand the stress concentration factor in this case equals 3.

The stress concentration factor is insensitive to the actual size of the crack. Thenature of the far-field stress is also of limited importance. What really matters isthe shape of the crack. If the radius of the crack in the x direction is considerablysmaller than the radius in the y direction, i.e. the crack has an elliptical shape, thestress concentration factor may be very large. Inglis [14] showed that in the limitingcase of a sharp crack the stress concentration factor tends to infinity since the stressat the tip becomes singular.

In fracture mechanics we have to deal with singularities. Two different loadings(pressure distributions) of a line crack result in two different stress distributions.Both cases may yield infinite stresses at the tip, but the "level of infinity", or simplythe "amplitude" (see Sneddon [12]) is different. We need a quantity to characterizethis difference.

Equation 2.34 shows that the singularity is always of the same type. In fact thesingular stress multiplied by rl/2, where r is the distance from the tip, will be alwaysfinite. Therefore the stress intensity factor K/ is defined as

Linear elasticity42

-~----------- ..-

where

2.7 Circular CrackThe mathematical treatment of the pressurized circular crack problem is si~nj~artothe line crack problem. For the sake of simplicity we assume that the crac~ IS l~ thex y plane, opening up in the z direction (Figure 2.15). The crack surface IS a ~I.rclelocated around the origin. The far-field stresses are zero. The boundary conditionsat z = 0 are

ql = (x} - X~)l/2,

q2 = (x} _~)1/2,

For this example the above expressiongives Wo == 3.3~ ~ (10~ e;,ror).The approx­imation improves rapidly if the ratio plO"y tends to unity, draggmg Xo toward xf' 0

8 _XO~2Wo = -p- Xf -xo·IrE' xf

By virtue of the superpositionprinciplewe can use the solutionof Example2.2 withSI= P - Oy = 1 MPa and Sz = (J"y== 2 MPa and half length c == xf == 10m. The loca­tion Xo is exactly the same as in Example2.3:

Xo == 10 x sin [ 1T x 52 ] = 10 x sin [ Ir x 2 ] = 8.66 m.2(sl + S2) 2 x (1+2)

The fracture width is obtained from Eq, 2.48, multiplying the normal displacementsby 2.

It is easily found numericallyor by doingsome algebraicmanipulationsthat the twoexpressionsgive identical result. ." . .

Geertsma and de Klerk [5) also give the following approxImationto wo,which ISvalid if Xo tends to xf:Solution

_2 2) 1/2Xf -xwhere Q4 = (-2--2

Xf - Xo[ ! XOq4! 14 _ 1 - "7 11 - q41w(x) == - p x In XOQ4 - Xo In 1+ QE'Ir 1+__ 4

X

Assume that the inner pressure, P = 3 MPa opens a line fracture of half length x f =10m against a far-field compressivestress Smi. = cry = 2 MPa. The exact magnitudeof the other principal stress is not known, but cry < a, and 'xy == O. Near the tips,Xo < Ixl ~ XL, the pressure is virtually zero (i.e. negligible). Assumingplane strain inthe X, y plane find the location Xo if the fracture ends smoothly. Give the width asa function of the location. Young's modulus is E = 9.1 GPa and the Poisson ratiois v = 0.3.

At x _ 0 the width is Wo == 3.63 mm as we know alrea~y from Example2.3..Thenormal stress distributionO"yy(x, 0) is essentiallythe same as m Example2.3, but shiftedwith the constant far-fieldvalue, (J"yo _. .

EquationA.2 of the paper by Geertsma and de Klerk [5] gIves the WIdthm thefollowingform:Example 2.6 Zipper Crack in the Presence of Far-Field Stress

The solution of the associated problem is then readily transformed to yield thesolution of the original problem.

The principle of superposition will be illustrated in conjunction with Example 2.3.The notations of the classical paper of Geertsma and de Klerk [5] will be used.

8.66 < x ~ 10m.Pressurizedcrackin thepresenceof far-fieldstress.The principleof superpositionFigure 2.14

o <x < 8.66 m

6- 1~-4 [ 5 + ,J100 - xi 150 - xl - xJ3(100 - X2)]x 20.J3 I +x In 'w(x)== n n .Jx2-75 150-x2+xJ3(100-x2)

Trivial problemAssociated problemOriginal problem

+ ------- ------_.=

6 10-4 [ 5+ ,JI00 - x2 150- x2 - xj3(100 - X2)]x 20.J31 +xln 'w(x) = n n .J75 _ x2 150_ x2 +x.j3(100 - x2)

p-ayp

Substituting E' == 10lDPa, 51 == 106Pa, S2= 2 X 106 Pa and xf == 10 m, the width isobtained as a function of the locationx:

a"y

xX---l--_'

yyy if x ~xoif x> Xo·

and

44 Unear elasticityCircular crack 45

-.,--_. -..-------.---~---.--------------------_.--",,-----_._._-- .--------_._-------_.-

Notice that here we do not have to consider a slice of thickness 8, because a circularcrack is a three-dimensional object in contrast to the two-dimensional line crack.

If the far-field stress is not zero, an additional term, Wi, appears in the elasticenergy. It is readily derived from the definition of work as the product of the minimum

(2.66)

(2.65)Wo= ~ t[G(~)fd~,rrE' )0

and for constant pressure this reduces to

(2.64)rrP5c2Wo=8--.2E'

The corresponding expression for one wing (i.e. one half) of a circular crack is

(2.63)Wo= 8~ r~[g(~)fd~.rrE'io

For constant pressure this reduces to

(2.62)Wo = !8foe Pn (.x)w(X) dx

is the work done while opening one wing of a two-wing "line crack" with half lengthc, if the far-field stress is zero. The factor 1/2 is necessary because the force to openthe crack at any location x varies linearly with the displacement. The pressure Pn (x)is only the final value. Sneddon [12] shows that an equivalent expression in termsof g(~) is

In this book every extensive quantity associated with the fracture is reported for onewing of a two-wing fracture. This convention is set to ease the further use of theformulas in hydraulic fracture models.

The elastic energy or, in other words, the strain energy is the excess energy storedin the medium due to the presence of the fracture. For a line crack in plane straincondition a slice of the elastic medium of thickness 8 is considered. The thicknessis necessary; otherwise, the line crack is only a two-dimensional abstraction. Thestrain energy is calculated from the definition of work: force multiplied by path.The integral

2.8 Volume and Strain Energy

Other cases may be treated similarly to the line crack problem.

(2.61)8RpowO=--·rrE'

resulting in the maximum width

47Volume and strain energy

------- .._----"" - .._-,,_ ..... _-

(2.59)

(2.60)

G(~) = po~

w(r) = ~ POv'R2 - ,2rrE'

For a "zipper" fracture G(R) should be equal to zero.The crack width is the normal displacement multiplied by 2. In particular, the

constant pressure case is characterized by (Green and Zerna [11]):

(2.58)o ~ r ~ R.

Once the G(~) function is known, the normal displacement of any point on the upperside of the crack disk is given by

(2.57)o < ~< R.G _ 2 r rp(~)d,. (~) - rr io (~2 - r2)1/2'

The first condition states that the pressure acting on the surface is compensated bythe normal stress of the same magnitude. The pressure is a function of the distancefrom the origin, r, It is supposed that the problem is symmetric with symmetryplane z = O.

The first step of the solution is again the construction of a function, G(~):

r;::: O.axz= 0,

(2.56)azz= -p(r), 0 ~ r ~ R,Uz = 0, r > R,

Figure 2.15 Pressurizedcircular crack

x

x

y

Linear elasticity46

_._---------_ ..,----------:------_. _.__ ._-. - - """'---

Finite difference, finite element and boundary element methods are available to solvethe linear elasticity problem with specified boundary conditions. Finite differences,the basic tool of solving differential equations on domains of simple geometry, haveonly limited application in elasticity calculations.

In finite element methods (see Hilton and Shih [16]) the region surrounding thefracture is divided into a network of elements. The "solution" consists of findingvalues of displacements and stresses at the mesh points of the network. The solutionbetween the nodes is expressed in a simplified way in terms of the node values. Asufficient number of algebraic equations is deduced relating the approximation to theoriginal partial differential equation. The resulting system of linear equations is largebut sparse.

In boundary element methods (see Crouch and Starfield [17]) only the boundaryis discretized. The numerical solution is sought as a linear combination of theknown analytical solution of simple boundary value problems. The method effec­tively reduces the dimension of the problem by one. The resulting system of linearequations is much smaller than in the finite element method but it is dense.

Whatever method is used, the singularity of the solution requires sophisticatedapproaches. The intuitive answer to the challenge is to use uneven mesh, high elementconcentration near the sharp edges and crack tips. Unfortunately, serious questionsarise concerning the reliability, convergence and convenience of this approach.

2.9 Computational Methods

During the fracturing process we have to provide the elastic energy to open thefracture and an additional energy that is dissipated (i.e. heats the formation). Asusual, the second. term is less known and depends not only on the final fractureshape but on the history of creating it.

2W = 2Wo+2W I = 2.4 x laJ +4.8 x 1(f = 5.04 X 104 J. 0

The total elastic energy of the circular crack is

2V16 X R3 PnO 16 x (1)3(3 x 106) 6 -3 3

= 3E' = 3 x 1010 = 1. x 10 m.

The superposed strain energy is

2WI = 2VUmin = 1.6 X 10-3 x 3 X 107= 4.8 X 104 J.

The volume of a circular crack can be determined as

The strain energy due to the net pressure for a circular crack is (from Eq. 2.66):

8p2 R3 8 x (3 X 106)2 x 132Wo = _n_O_ = = 2400 I.3£' 3 x 1010

2W == 2Wo + 2WI = 5.66 x laJ + 1.13 x 105 = 1.19.x 105 J.

The total elastic energy is

49Computational methods

The superposed elastic energy (work done against the minimum principal stress) is

2Wj = 2Vumin = 3.77 X 10-3 x 3 X 107 = 1.13 x 105 J.

From Eq. 2.64 the strain energy due to the net pressure is

rr:p2c2 rr: x (3 x 106i x 122Wo = 0_°_ = 2 = 5660 J.E' 1010

The volume of the two-wing fracture is

Solution

Calculate the elastic energy necessary to open two wings of a line crack of half­length 1 m contained in a 0 = 2 m thick slice of an infinite medium of plane strainmodulus, E' = 10 GPa. Compare it with the energy of a circular crack of radius 1 m.The minimum principal stress is Umin = 30 MPa. The opening pressure is Po = 33 MPa(i.e. the net pressure is 3 MPa).

Example 2.7 Energy of Constant Pressure Fractures

3rr:E'(2.71)8R3 PnO 16PnoRw---""':"".......,_=---

- 3E' (R~7r:)

(2.70)

(2.69)iii= rr:Pno8c2 _ rr:P"oCE'8c - E'

The volume of one wing (one half) of a circular crack is obtained from

V = -21 rR (2m) (8 P"o .; R2 _ r2) dr = 8R3 PnO ,Jo E' 3E'indicating an average width,

area, i.e

(2.68)v = 8 r 4Po J c2 _ x2 dx = rr:Pno8C2Jo E' E'

As a by-product we obtained the average width, which is the volume divided by the

Here we consider the volume of a fracture opened by constant net pressure. Fora line crack of half-length c the volume of one wing located in a slice of thickness8 is obtained from

(2.67)

principal stress and the volume of the fracture

Linear elasticity48

--- --- .._--- --_. ------

1. Billington,E.W. and Tate,A: The Physics of Deformation and Flow, McGraw Hill,New York, 1981.

2. Fenner, R.T.: Engineering Elasticity, Application of Numerical and Analytical Tech­niques, Ellis Horwood,Chichester, 1986.

3. Khristianovitch,S.A. and Zheltov,Y.P.: Formation of VerticalFractures by Means ofHighlyViscousFluids,Proc. World Pet. Cong., Rome, 2, 579, 1955.

4. Zheltov, Y.P. and Khristianovitch,S.A: On the Mechanismof HydraulicFractureof anOil-BearingStratum, Izv.AN SSSR, OTN, (No.5), 3-41, 1955.

5. Geertsma,l. and de Klerk, F.: A Rapid Method of Predicting Width and Extent ofHydraulicallyInducedFractures,JPT 1571-81, (Dec.), 1969.

6. Perkins, T.K. andKern, L.R.:Width ofHydraulicFractures,JPT 937-49, (Sept.), 1961;Trans. AlME, 222.

7. Nordgren, R.P.: Propagationof a VerticalHydraulic Fracture, SPEJ 306-14, (Aug.),1972; Trans., AlME, 253.

8. Griffith,AA: The Phenomenonof Rupture and Flow in Solids, Phil. Trans. Roy. Soc.A 221,163-198,1920.

9. Muskheiishvili,N.I.: Some Basic Problems of the Mathematical Theory of Elasticity,Nordhoof,Holland, 1953.

10. England,AH. and Green,AE.: SomeTwo-dimensionalPunch and Crack ProblemsionClassicalElasticity,Proc. Cambridge Phil. Soc., London, 59, 489-500, 1963.

11. Green, AE. and Zema,W.: Theoretical Elasticity, Oxford University Press, London,1968.

12. Sneddon, LN.:IntegralTransformMethods,inMechanics of fracture I,Methods of anal­ysis and solutions of crack problems, Ed. Sih, G.C. Nordhoff International, Leyden,1973.

13. Barenblatt,G.I.: MathematicalTheory of EquilibriumCracks, in Advances in AppliedMechanics, 7,55-129, 1962.

14. Inglis, C.E., Stresses in a Plate due to Presence of Cracks and Sharp Corners, Trans.Inst. Nav. Arch. 55, 219-230, 1913.

15. Rice, J.R.: Mathematical Analysis in the Mechanics of Fracture, in Fracture,Leibovitz,H. (ed.), Vol.2, Academic Press, NewYork, 1968.

References

16. Hilton, P.O. and Shih, G.c.: Applicationsof the Finite Element Method to the calcu­lations of Stress Intensity Factors, in Mechanics of fracture I, Methods of analysis andsolutions of crack problems, Sih, G.C (ed.), Nordhoff International,Leyden, 1973.

17. Crouch, S.L. and Starfield,AM.: Boundary Element Methods in Solid Mechanics,Unwin Hyman, London, 1990.

18. Clifton, R.J. and Abou-Sayed,AS.: On the Computation of the Three-DimensionalGeometry of HydraulicFractures, paper SPE 7943, 1979.

For a given finite element, calculation improvement on the accuracy can beachieved by retaining higher-order terms in the asymptotic expansion of thedisplacement components about the crack tip [16]. In boundary value methods theaccuracy is improved by applying special crack tip boundary elements [17]. Inboth cases previous knowledge on the shape of the analytical solution to simplifiedproblems is built into the solution algorithm and the "numerical solution" is in facta combination of numerical and analytical techniques.

Solving the elasticity problem at specified loading conditions is still only a part ofthe solution of the fracturing problem itself. The main difficulty is still ahead: Onehas to deal with the propagation phenomenon (see Clifton and Abou-Sayed [18]).This will be the subject of subsequent chapters.

51ReferencesUnear elasticity50

--__ -- - ------ - ------,--,,-------- --- -----------

(3.3)and

(3.2)

Since an infinite amount of planes can be drawn through a given point and althoughthe resultant force acting on these planes is the same, the stresses acting on them aredifferent because of the varying inclinations of the individual planes. For a completedescription of the stress field it is necessary to specify not only the acting forcemagnitude and direction, but also the surface upon which the force acts.

Stress, applied on a surface at any angIe, can be decomposed into three vectors, anormal stress, ax (along the x axis and normal to the x plane) and two shear stresses,Tyx (along the y axis) and TIS (along the z axis). In a three-dimensional body thestress tensor was given in Chapter 2 by Eq. 2.3.

Suppose that in a two-dimensional system as shown in Figure 3.2, ax and ay arethe normal stresses on the y and x planes respectively, and rxy = Tyx are the shearstresses. Any other direction, at angle e from the y axis, would result in a normaland a shear stress (See Roegiers (1]). These were shown in Chapter 2 to be

(3.1).a = lim (b..F) ,M ......O M

Formations that are candidates for hydraulic fracturing are often at great depth,overlain by earth formations of considerable thickness. This "overburden" causesthe formation of interest to be subjected to stresses. These stresses are causednot only from geological depositional mechanisms but also from historical tectonicphenomena.

As shown in Chapter 2, stresses are vectors. Away from a well they are oftenreferred to as far-field stresses, The drilling of a well creates a different near-wellstress concentration which may be much more complicated than the far-field stresses.

In considering a randomly oriented plane of area D.A at a point P within a body,and across which a resultant force b..F acts (Figure 3.1), the stress at the point isdefined as

3.1 Basic Concepts

STRESSES IN FORMATIONS3

· _. __ --,----------------

(3.11)

and in a three-dimensional porous medium the following can be written

(3.10)0" = Ee,

From the general elastic stress/strain relationship, using Young's modulus, E,

(3.9)

where p is the pore pressure.It is important here to realize that the overburden absolute stress remains constant

throughout the time of interest whereas the effective stress may change profoundlywith fluid production or injection and the associated reservoir pressure changes. Also,overpressured formations may have much smaller effective stresses than underpres­sured ones.

The Poisson ratio, V, is the ratio of the lateral strain to the longitudinal strain,

(3.8)

A well-known value in petroleum field units is 2.49 x 104 Palm (1.1 psi/ft).The calculated stress and stress gradient from Eqs. 3.6 and 3.7 are absolute, and

in a porous medium they are carried by both grains and the pore-inhabiting fluid.Biot [3] has introduced the poroelastic constant, a, such that an effective stress, a',on the grains is

(3.7)dO'. 4- = (9.8)(2650) = 2.6 x 10 (palm).dH

where p is the density of the overlaying strata. Density logs can readily provide thedensity values from the surface to the formation of interest and integration of thedensity log provides a.:

Typical values of rock density range from 2500 to 2750 kglm3 and, assuming thatall overburden consists of sandstone with p = 2650 kg/nr', a common first approxi­mation of the gradient of the overburden stress is simply

(3.6)a; =g1"pdH,

A formation at depth H can be considered as a system subjected to three principalstresses, one vertical and two horizontal. These are also the far-field stresses. Ananalysis of the effect of these stresses in hydraulic fracturing was introduced byHubbert and Willis [2]-

The easiest to understand is the absolute vertical stress, O"y, which is simply theweight of the overburden

3.2 Stresses at Depth

55Stresses at depth

------_ ....--- .

The stresses calculated from Eq. 3.5 are referred to as principal stresses andcoincide with the directions where shear stresses vanish. All other directions havenonvanishing shear stress components.

(3.5)

and by substitution in Eq. 3.2

(3.4)o _ 1t -1 ( 2-cyx )- 2 an ---,O"x - O"y

From Eqs. 3.2 and 3.3 it can be concluded that there exists an angle e where theshear stress vanishes and, thus, r =: O. From Eq, 3.3, setting it equal to zero,

Figure 3.2 Normal and shear stresses acting at an angle q from a direction with known stressvalues

y

tyx

~i Yor

0 x--<xy

uyt

-ryx

Figure 3.1 Force and resulting normal and shear stresses on a surface

Tzx

Stresses in formations54

--~----- ----_ .._ ...--. __ ...._-----_.

vah= --(ov - ctp) + etp.

1-vi.e. to the left of the y axis in Figure 3.2.

During the reservoir pressure depletion, Eq. 3.13 can be used to examine the effects onthe absolute horizontal stress:

e = ! tan"! ( (2)(7 x 106) ) = -31.7.2 3.5 x 107 - 4.2 X 107

The effective minimum stress is then

a' . = 3 8 X 10' - (0.72)(2.5 X 107) = 2.0 X 107 Pa.h.mm ~

To calculate the principal stress magnitude, first the direction of the principal stressesvs the direction of a)" can be calculated.From Eq. 3.4

= 3.8 X 10' Pa,

From Eq. 3.3

and, thus, from Eq. 3.8

a~= 7.8 x 10' - (0.72)(2.5 X 107) ~ 6.0 X 10' Pa.

2. Equation 3.13 provides the absolute minimum horizontal stress as a function of thevertical stress. The vertical stress is equal to the weight of the overburden. Furthermore,the pore pressure has been assumed to be hydrostatic. Thus, at 3000 m,

0.25 [7.8 x 107 _ (0.72)(2.5 X 107)] + (0.72)(2.5 X 10')ah.min = (1- 0.25)

The stresses a and T on a fault at an angle B = 30· can be calculated as follows: FromEq.3.2

a = (3.5 x 107) cos2 30· + (2)(7 x lW)sin 30· cos 30· + (4.2 X 107) sin2 30·= 4.28 X 107 Pa.

Since no information on the reservoir pressure is given. a minimum reservoir pressurecan be assumed to be simply hydrostatic:

p "'"PogH = (850){9.8)(3000) = 2.5 x 107 (Pa),Solution

1.The absolute vertical stress is equal to the overburden pressure, and from Eq. 3.6

a; = PtgH = (2650)(9.8)(3000) = 7.8 x 107 Pa.Assume that the two-dimensional stresses ax. O"yand 'xy on Figure 3.2 are 3.5 x 107 Pa,4.2 x 107 Pa and 7 x 106 Pa, respectively. Calculate the stresses 0" and T on a fault at anangle ()= 30·. Calculate the principal stress magnitudes and directions in this system.

SolutionExample 3.1 Stresses on a Fault and Principal Stress Magnitudes

and Directions

where O"leCI is the net tectonic stress component.

(3.14)

This stress is the minimum horizontal stress deriving from the Poisson ratio trans­lation from the overburden to lateral. However, tectonic phenomena may resultin additional horizontal stress components and, thus, two horizontal stresses, oneminimum and one maximum, can be considered, such that

Example 3.2 Stresses vs. Depth

1. If a reservoir is 3000 m deep, the poroelastic constant a = 0.72, the formationdensity Pr = 2650 kg/nr', the oil density Po= 850 kg/nr', and all saturating fluid isoil calculate the effective vertical stress.2. Calculate the minimum horizontal stress at 3000m depth for the formation describedin Part 1. Assume that the Poisson ratio is 0.25. Will the absolute minimum stressincrease or decrease during reservoir depletion?3. Assuming that the Poisson relationship is in effect and assuming that the. horizont~1stresses are "locked" in place, what would the critical depth be above WhICha hori­zontal fracture would be generated if 500 m of overburden were removed by somegeologic means?

(3.13)V

O"h= --(O"v - aP) +aP.1- v

Substitution of the effective by the absolute stresses leads to

From Eq. 3.5

O"max.min = ~(3.5 X 107+ 4.1 X 107) ± )(7 x 106)2 + ~(3.5 X 107 - 4.2 X 107)2

= 4.58 X 107 and 3.01 x 10' Pa. 0(3.12)! f f V f

O"h= 0" =O"y= --O"v'1- V

If the lateral movement is constrained, i.e. Cx = 0, if a; - O"y (perfect isotropy inthe horizontal plane) and O"r = O"y, then

57Stresses at depthStresses in formations56

Figure 3.3 Criticaldepthfor horizontalhydraulicfracturesfor Example3.2 (3.20)and

(1e = (O"x + (1y) - 2{O"x - O"y)cos(2e),

(3.18)

(3.19)a, =0,

At the borehole wall, where r -+ rw, Eqs. 3.15 to 3.17 simplify to

(3.17)] (2r; 3r!) . (28)rr() = -2(O'X - O"y) 1+ 72 - -;=4 SIll .

and

0

E -500 0

8~;) -1000 -500VI...c:'"e"" -1500OJ -1000 Ec:'0 S'I:0 0.

."E -2000 -1500 0g~~ -2500 -2000

-30000 20x10" 4Ox10" 60x10" 8Ox10"

Stress, Pa

The magnitude and orientation of the in situ stress field can be altered locally,as a result of the drilling of a well. These induced stresses often result in stressconcentrations that are significantly different from the original values.

The drilling of a borehole distorts the preexisting stress field. Figure 3.4 offers adescription and conventional nomenclature for the far-field stresses (Figure 3.4a) andthe stresses in cylindrical coordinates created by the drilling of the well (Figure 3.4b).In all cases the first subscript denotes the direction of the force and the second denotesthe plane upon which the force acts. The index of the plane corresponds to the normaldirection. Thus, stress 0"16 means a stress along the r direction and normal to the 9plane. In all cases normal stresses with two identical subscripts can be replaced bya single subscript.

If the rock is assumed to remain linearly elastic and the borehole is drilled parallelto one of the principal stress directions (e.g. a vertical well) the following expressioncan be obtained for the near-well stresses. (Note: In this section stresses are effective.)

a, = 4(0',. + O"y) (1 - ;) + 4(0',. - cry) (1 - 4;; 3~!) cos(29), (3.15)

O"e = !(O"x +O"y) (1+ ~) - !(O"x - O"y) (1 + 3;) cos(28) (3.16)

3, Figure 3.3 provides a graphical solution to the problem. While the "original"minimum horizontal stresses remain largely the same, the weight of the overburdenhas been reduced and the vertical stress profile vs depth has been shifted to the left.The slope remains the same since the density is the same. The critical depth from thisconstructionhappens at the intersectionof the original horizontal stress and the newvertical stress.This is at approximately1000m from the originalsurfaceor 500 m fromthe current.At depthsshallowerthan this a hydraulicfracture is likely to be horizontal;at larger depths the fracturewill be vertical.

Interestingphenomenacan happen around this critical depth, where a fracturemayinitiate horizontallyor vertically and then may tum by 90°. This would be caused byan additional fracturepropagation-inducedpressure.

3.3 Near-wellbore StressestiO"h,min:; (0.48)(5.0 X 106) = 2.4 X 106 Pa.

Setting O"v = O"h.mie, critical depth H = 977 m from the "original" ground surface or477 m from the current. 0

From the expressionabove it is obvious that as the pore pressure decreases, theO"h.min decreases. For a 5 x 106 Pa reduction in the reservoir pressure, the minimumhorizontal stress for this formationwill decreaseby

a; = (2650)(9.8)(H - 500)= 2.6 X 104(H - 500) Pa.

and the new vertical stress equation

((0.72)(0.25»)

CJ.O"h.min = 0.72 - 0 tip =O.48tip Pa.1- .25

and for this problem,

O"h.m,n=G:~~)[(2650)(9.8) - (0.72)(850)(9.8)]H + (0.72)(850)(9.8)H

= 1.27 x 104H Pa.

CJ.O"h.min = (a - ~) tip,1-v

The problemcan be solved also by the simultaneoussolutionofIf the reservoir pressurechanges CJ.P, then the horizontal stress will changeby

59Near-wei/bore stressesStresses in formations58

----.--.~--- --_--~-. - ----- .._----- ----- .------~------------------------~

Suppose that in Example 3.3 the tensile strength of the rock was approximatelyequal to 4 x 106 Pa. The calculation shows the possibility for tensile failure to occurin a direction perpendicular to the minimum principal stress solely as the result ofdrilling the borehole. Such a value of tensile stress is in the range of typical tensilestrengths 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 providesthe stresses in one direction, Figure 3.6 provides their profiles around the well. Ascan be seen 0"1) (Figure 3.6a) and 0", (Figure 3.6b) have several minima which wouldprovide the "venue" for the second branch of a fracture. Once initiated from a pointor a plane of the well the fracture starts propagating in a first branch. Friction pressureresistance and other retardation effects could allow the fracture to find another pointof minimum stress concentration for a second fracture branch. Depending on therequired path length (and thus the perforation phasing) it is conceivable that incertain cases a second branch may not develop. The near-well stress concentration

and

From Eqs. 3.21 and 3.22 the effective stresses at e = 0 and e = rr/2 are

ay = 3.6 x 107 - 2.0 X 107 = 1.6 X 107 Pa.and

ax = 2.4 x 107 - 2.Q X 107 = 0.4 X 107 Pa,

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

Solution

A well has been drilled in a reservoir where the pore pressure is 2.0 x 107 Pa and theabsolute minimum and maximum horizontal stresses are 2.4 x 107 Pa and 3.6 x 107 Pa,respectively. Calculate the effective stresses ae at e = 0 and e = :rc/2. Assume thatct=1.

Example3.3 Calculationof the Stressesat the Borehole

(3.22)and

(3.21)

Considering only the directions parallel and perpendicular to the minimum horizontalstress directions, i.e. e = 0 and e = 1</2, Eq. 3.19 simplifies further:

61Near-weI/bore stresses

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

(b)

r,o

Z O"zz

r,,~rzr

(a)

xy

z

Stresses in formations60

Figure 3.7 Problem decomposition: (a) Kirsch's problem, (b) antiplane loading, (c) internallypressured hole

Figure 3.6 Three-dimensional view of stress concentration around the wellbore

P.... (c)(b)(a)

y

)-,z

a;a: sin2 f3 cos2 f3 cos2 a cos2 f3 sin2 aayy 0 sin2a cos2aOzz cos- f3 sin2 f3 cos? a sin 2 f3 sin2 a {::} (3.23)=T~ 0 - sin a cos a sin f3 sin a cos a sin f3TXl - sin f3 cos f3 sin f3 cos f3 cos2 a sin f3 cos f3 sin2 a ay

Txy 0 - sin rxcos rxcos f3 sin rxcos rxcos f3where ax, 0Y' and az are far field stresses.

(a)

Radial distance from welJ, m

Stress concentration around and away from a wellboreFigure 3.5

1000100

The solution for the stresses and displacements because of an infinitely long circularhole in a homogeneous, isotropic, linearly elastic medium is given by the superpo­sition of Kirsch's solution, the antiplane solution, and the solution for an internallypressured hole (Figure 3.7, Deily and Owens [4], Bradley [5], Richardson [6]).

To represent this solution mathematically, it is necessary to define the orientationof the borehole with respect to the in situ stresses. Define a as the angle betweenthe ax(ah,min) direction and the projection of the borehole axis onto the ax - Oyplane, and f3 as the angle between the borehole axis and the 0::. direction as shownin Figure 3.8. The rotation of the stresses from the in situ system of coordinates tothe borehole local system of coordinates is obtained from the following:

10

3.4 Stress Concentrations for an ArbitrarilyOriented Well

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

63Stress concentrations for an arbitrarily oriented well

._._--_._ ...----

0.1

".= 3.3x107Pa"y=3.3X107PaUz= 3.3X107 Pa'w=0.1 m

K o

(fe

~( ",

120x10·

100x10·

'" ' aOx10'c..rti., 60x10'~ij)

40x10·

20x10·

0

Stresses in formations62

where P is the formation pore pressure and T is the formation tensile strength.This equation is valid only in the case of no fluid penetration (Detournay and

Cheng [9]) and it gives an "upper bound" for the breakdown pressure. Also, itassumes that the initiation and propagation directions are identicaL However, if

(3.36)Pb.upper ::: 3a~ - a~+ P + T,

Equation 3.35 is well known and often referred to as Terzaghi's criterion [8]. Theabsolute stress is used in Eq. 3.35. In terms of effective stresses, a~and a;,

(3.35)Ph.upper = 3ax - ay - P + T with ay ::::fJ",.

The upper limit of the stress concentration at the borehole can be calculated byEq, 3.22 and, therefore,

Pbd = stress concentration at the borehole + tensile strength of rock.or

breakdown pressure - stress concentration at the borehole= tensile strength of rock.

For a vertical well, one of the principal stresses, the vertical stress, is parallel tothe borehole axis. If this stress is the minimum stress, the fracture will initiate andpropagate horizontally. However, if the vertical stress is not the minimum, then thefracture will propagate vertically and normal to the minimum horizontal stress. Thebreakdown pressure for an uncased, smooth borehole can be written as

3.5 Vertical Well Breakdown Pressure

The solution above can be further simplified, with little loss of accuracy, byassuming the Poisson ratio, v::: 0 (Yew and Li [7]). The major difference with aborehole (horizontal or otherwise) drilled parallel to a principal axis is that one ofthe shear components, r(Jz, remains finite at the borehole wall. The magnitude of thisshear component will affect the overall stability of the borehole. More importantly,the principal stress tensor will be rotated in the neighborhood of the circular opening.In other words, the stress condition at the borehole wall will differ in magnitude andin orientation from the far-field conditions.

r~ = 2(-rxz sine + TyZ cos 8).

0'(18 = (axx + ayy - p",) - 2(axx - ayy) cos(28) - 4rxy sin(2e),

au ~ au - 2v(axx + ayy) cos(28) - 4VTxy sin(28),

rrfJ = Tn = 0,

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)

At the borehole wall (r = rw), these expressions simplify to

65Vertical well breakdown pressure

--------_ ..._._--_._-..__ _---_.- _--

(3.29)

(3.28)

(3.27)

(3.26)

(3.25)

(3.24)

---_ .._-_ ....

_ [ 1(. ( 2"; 3r!)rrf}- -Zaxx-ayy)sm(28)+rxvcos(28)] 1+---. r2 r4 '

(3r4) 2

-rx), 1+ r4w Sin(28)-:;Pw,

(4r2 3r4) r2+ Tx)' 1- r2w+ r4w sin(28) + r; P»,

al)f)= !(axx + ayy) (1+ :; ) - 4(axx - ayy) (1+ 3~!) cos(28)

1( ) (1 ~) 1 (42 3r4)a.; = 1 0'= + ayy - r2 + l(aD - ayy) 1- rZw+ rt cos(28)

The stress field resulting from the in situ stress tensor and the internal boreholepressure Pw is given by the following:

Figure 3.8 Pertinent parameters for inclined wellbore geometry

.: :.: "

Stresses in formations64

where p is the pore pressure at the considered point, and by minimizing Pw withrespect to e.

(3.43)0'3 = -T+ p,

Therefore, the trace of the fracture at the borehole wall forms an angle y withthe borehole axis. This has been observed experimentally by Kuriyagawa et at. [11].It must be emphasized that the tests mentioned by these authors were conducted ata slow pumping rate, which favors multiple fracture development. At higher rates,fracture coalescence will probably occur almost instantaneously, aligning itself withthe borehole axis, near the wellbore.

The location of the failure on the borehole wall, e and the breakdown pressureare obtained by solving for Pw using

and0'3 is also tangent to the borehole wall and deviates from the borehole axis by

(900 - y).

(3.42)1 -1 ( 2rez )y = :2 tan ,O'(j() - azz

0'1 is in the radial direction;0'2 is tangent to the borehole wall (Figure 3.9) and deviates from the borehole axis

by an angle

They are defined with respect to the local (borehole) system of coordinates in thefollowing manner:

Figure 3.9 Stresses at the borehole wall of an arbitrarily inclined well

67Breakdown pressure for an arbitrarily oriented well

(3.41)

(3.39)

(3.40)

Tensile failure will occur when the minor principal stress (in tension) reaches thetensile strength of the medium. The principal stresses at the wall of the borehole aregiven by Daneshy [10].

3.6 Breakdown Pressure for an Arbitrarily Oriented Well

The fracture will initiate and propagate perpendicular to the least resistance, i.e, theminimum horizontal stress. 0

Thus,

The breakdown pressure for open hole can be calculated with Eq. 3.35, where

O"x = O"h.min = 3.8 X 107 Pa and C7y = O"h.max= 5.0 X 107 Pa.

Solution

A reservoir is 3000 m deep, the absolute vertical stress, C7v, is 7.8 X 107 Pa, the absoluteminimum horizontal stress, C7h,min, is 3.8 X 107 Pa, and the absolute maximum horizontalstress, O"h.ma>, is 5.0 X 107 Pa. The reservoir pressure at this depth is 2.5 x 107 Pa andthe tensile strength is 4.0 x 106 Pa. Calculate the open hole breakdown pressure.

Example 3.4 Breakdown Pressure for a Vertical Well

(3.38)a(l - 2v)

Tj = ~-:--~2(1 - v) .

It should be noted that an increase in the pore pressure in the vicinity of thewell corresponds to a decrease in the effective stresses of the rock, and hence adecrease in the breakdown pressure. Therefore the use of low-viscosity fluids and/orlow pumping rate may decrease the pressure for breakdown.

where Tj is a parameter, defined by

(3.37)

leakage occurs prior to breakdown, Eq. 3.36 becomes more complex and it is neces­sary to define a "lower bound" for the breakdown pressure:

p _3ax-ay-2rJP+Tb,lower _ 2( ,

1- v)

Stresses in formations66

Figure 3.10 The impact of the non-vanishing shear stress on the breakdown pressure of awell inclined along the fracture plane

The stress expressions in the coordinate system corresponding to a horizontal well(fJ = 90°) are simplified from Eq. 3.23

Solution

Use the data in Example 3.5 and calculate the breakdown pressure for two limitingcases of a horizontal well. In Case 1 the well is drilled along the minimum horizontalstress which would result in transverse fractures, and in Case 2 the well is drilled alongthe maximum horizontal stress resulting in a longitudinal fracture.

Example 3.6 Horizontal Well Breakdown Pressure

60x106

50x106

<Ua.. 40x106"ii~

30x106

20x1060 20 40 60 80 100

13. degrees

The breakdown pressures are calculated with rnimmum horizontal stress equal toEq. 3.43, by minimizing p", with respect to e. The well stays on the plane normalto the minimum horizontal stress implying that ex= 90·. For this particular case,au, ayy,au, flY' !v and !%in Eq. 3.23 are simplified:

For the case of a horizontal well, production engineering considerations have identi­fied two limiting cases [12]. The first possibility is for the well to be drilled along theminimum horizontal stress, resulting in transverse hydraulic fractures. The secondoption is for the well to be drilled along the maximum horizontal stress resultingin a longitudinal hydraulic fracture. Depending on the reservoir permeability one orthe other configuration may be desirable. Low reservoir permeabilities would requirethe transverse configuration whereas higher to moderate reservoir permeability wouldindicate the longitudinal configuration [12].

Equations 3.23 and 3.30 to 3.34 can be adjusted for these two limiting cases. Fora horizontal well the angle f3 = 90°, and therefore the tensor in Eq. 3.23 will begreatly simplified. The three stresses (lz, <Tx, and (ly and can be substituted by (Iv,

(lh.min, and (lh.max respectively. For a well along the minimum horizontal stress theangle a =0° and along the maximum horizontal stress a = 90°.

With these adjustments to Eq. 3.23 and Eqs. 3.30 to 3.34 the breakdown pressurecan be calculated using Eqs, 3.40 and 3.41. The breakdown pressure will be thesmallest value calculated from the two expressions.

Solution

3.7 Limiting Case: Horizontal Well

The results are plotted in Figure 3.10. 0Use the data in Example 3.4 (but with <Th.min = 4.2 X 107 Pa) and calculate the break­.down pressure of a deviated well. Assume that the angle fJ changes from O· (verticalwell) to 90· (horizontal well) but the well stays on the plane normal to the minimumhorizontal stress.

a3 = -T+ p

Example 3.5 The Impact of Non-vanishing Shear Stress onBreakdown Pressure

The stresses in the coordinate system corresponding to the deviated well, <Till}'<T=.,

and !~ are calculated from Eqs. 3.31, 3.32, and 3.34. a3 is calculated from Eq. 3.41.The calculation of breakdown pressure is tedious and a computer program is used here.For a given well deviation angle /3, Pbd is calculated by minimizing Pw with respect toe using

!z:.r = -<Tz sin fJ cos fJ +ay cos fJ sin /3.!}~=O.

The above calculations allow a specific prediction of the breakdown pressure andthe angle of initial fracture growth according to the general principle of energy mini­mization. Such a fracture will ultimately reorient itself to propagate most efficiently,perpendicular to the minimum principal stress.

·Umiting case: horizontal well 69Stresses in formations68

-_ ....--- ------_."-----"--"."._---_._,, "'---" .__ ... -----

(3.46)ky = [Byln a; ]3ax-ap

and

(3.45)k, = [B In a; ]3x O"y - a p

where a" has been labeled as the closure quality and B tbe conductivity/porosityfactor.

The closure quality, a" has been postulated to be affected by the type of fissureporosity, degree of mineralization, tortuosity and shape of the flow path. For a singlefissure, the factor B is a strong function of the asperity height and, thus, width ofthe flow path. For a system of fissures the factor B can incorporate their cumula­tive effect.

If the reservoir permeability can be considered as largely the result of fissures,Eq, 3.44 yields

(3.44)

As discussed in Chapter 1, the reservoir permeability plays a crucial role in theproduction performance of a hydraulically fractured welL It is also well understoodthat the permeability tensor exhibits considerable anisotropy (Ramey [14]; Warpinskiand Branagan [15]) which may go unnoticed in radial flow into a vertical well,Muskat [16], in a classic study, has shown that kx and ky permeability anisotropyresults in an average radial permeability, k = -jkJZ;. Obviously, the same value ofcan be obtained from an infinite combination of k; and ky values,

A hydraulic fracture which follows a preferential direction and, especially, theemergence of horizontal wells and the requirement to optimize their drilling directionhave forced a new awareness on the issue (Ben-Naceur [17]; Deimbacher et al. [18J).Warpinski [19] has presented field evidence where horizontal permeability anisotropyis as much as 100:1. Anisotropy from 2:1 to 3:1 is considered common, especiallyin naturally fissured media.

Walsh [201 has suggested that fissure permeability is related to the effective stressnormal to the fissure direction. This idea is based on asperity contacts and an assump­tion of exponential distribution of the asperity heights. Walsh's model was simplifiedby Buchsteiner et al. [21] who arrived at the following expression for the fissurepermeability, kf:

3.8 Permeability and Stress

For a horizontal well, f3 = 90·,Using the equations listed in the solution of Example 3.5the far-field stresses are transformed to the coordinate system corresponding to thehorizontal well. With a varying from 0' to 900. the breakdown pressures are calculatedby the same procedures as in Example 3,6, They are plotted in Figure 3,11 as functionsof the angle a. 0

Solution

71Permeability and stress

----------_. _ .•._._ ..__-.-------------

Figure 3.11 The impact of a on the breakdown pressure of a horizontal well

60x10"

50x10"

'"c,-0 4Ox10·.0Q_

30x10"

20x10"0 20 40 60 80 100

a, degrees

Use the data in Example 3.5 and calculate the breakdown. pressure for an arbitrarilyoriented horizontal well, with a changing from 0' to 90·.

Example 3.7 The Impact of a on the Breakdown Pressure of aHorizontal Well

In drilling horizontal wells off of platforms or "drilling pads" it is often not possibleto select either one of the two limiting cases. This problem may create interestingcomplications from the production of arbitrarily oriented fractured horizontal wells(Owens et al. [13]).

The angle a formed between the minimum horizontal stress directions and thehorizontal well trajectories varies between O· and 90·. The following example shows

. the impact of a on the well breakdown,

3.7.1 Arbitrarily OrientedHorizontal Well

The results are Phd = 5.1 X 107Pa for a = 0' and Phd = 2.7 X 107Pa for a = 90·,0

0'3 = -T+ p.

The stresses O"fJ(j,O'::z, and r:;jI at the borehole are calculated from Eqs. 3,31, 3.32,and 3,33. 0'3 is calculated from Eq. 3.41. The breakdown pressures, Phd, for the twolimiting cases (a = 0' and a = 90') are calculated by minimizing p.; with respect toe using the formula

rzx = r}:X = O.

rOY= -O"h.min sin a cosa +O'h,ma. sin o cos «,

Stresses in formations70

..---.-- ...~.~~-----

(3.49)I V Ea Edci vEdcj

dah = -- d(av - a) + -- dT + -1 2 + -1 2'I-v I-v -v -v

where E is Young's modulus, a is the poroelastic constant, T is the temperature andCi and 6j are tectonic strains in the i and j directions. Obviously, the knowledge ofall these variables is difficult and very expensive to acquire.

For hydraulic fracturing what is often more interesting is the closure pressuredefined by Nolte [24] as the minimum fluid pressure to open an already existingfracture. This closure pressure may be equal to the minimum horizontal stress if ahomogeneous and single layer is fractured. In field applications this is often not thecase because the hydraulic fracture may traverse both lateral and vertical hetero­geneities and other strata. The concept of the closure pressure is intended to accountfor all these effects.

Below, an outline of common stress measurement techniques is presented, asapplied to hydraulic fracturing.

For hydraulic fracturing applications two stress variables are of the greatest interest:the minimum stress at the target and the direction of the principal stresses. The formeris related directly to the breakdown pressure and the fracture propagation pressure.The latter controls the ultimate fracture azimuth and has major implications on thenear-wellbore fracture path.

In Section 3.2 the three far-field stresses were introduced and related. For thevast majority of petroleum formations the' minimum in situ stress is the minimumhorizontal stress, and hence fractures are vertical (see Example 3.4).

The values of the minimum stresses vs depth, especially in overlain and underlainlayers to the target interval, are essential in predicting both the lateral and verticalfracture geometry. As will be addressed in Chapter 11 the fracture height greatlydepends on the stress contrasts between the target and adjoining layers.

Prats [23] has shown that the differential effective horizontal stress is influencedby depth, temperature, tectonic strains and pore pressure.

3.9 Measurement of Stresses

(3.48)

half-length, XI' and a real fracture half-length, XI, that obeys

XI = 4fk:,XI V t;

where k; is normal to the hydraulic fracture and k.. along the fracture.From'the results of Example 3.8 the apparent half-length, xI' would be approxi­

mately 0.75 xf' In certain reservoirs this could cause a substantial reduction in theexpected fracture performance. Thus, stress-induced permeability anisotropy must beconsidered in fracture design.

To determine the parameters Bx, By, a; and a;,Economides et al. [22] have intro­duced a well-testing technique that uses conventional pressure transient interpretationmethodologies.

73Measurement of stresses

.....- ....... _--_._. ----

This model was matched with field data from two large and well-known fissuredreservoirs, and the parameters Bx, By, a;, and a; were obtained. The model ofEq. 3.47 suggests the possibility of permeability reduction in the life of a reservoir asthe reservoir pressure declines and the effective stresses increase. More importantly,the nature of permeability anisotropy may change, and, depending on the relationshipbetween B; and By, a; and a;, it may even "flip-flop" its direction at some time inthe life of a reservoir [21J.

These ideas have considerable implications in the planning of horizontal welltrajectories, hydraulic fracturing and the design of fracture lengths. Ben-Naceurand Economides [17J have shown that for the production from infinite conduc­tivity hydraulic fractures there exists a relationship between an apparent fracture

(3.47)k=[BxByln( a; )In( a; )]3/2ay + ap ax -ap

3.8.1 Stress-sensitive Permeability

Following Muskat's permeability definition [16], Buchsteiner et al. [21] havepresented an expression for reservoir permeability in a "stress-sensitive" formation:

implying a 3:1 permeability anisotropy. 0

k.= [1.51n 1x1OS ]3_) 6.2 x 107- (0.72)(3.5 x 107) - 3.4 md,

Similarly, from Eq. 3.46

[Ix 108 ]3kx == 1.51n _

4.8 x 107 - (0.72)(3.5 X 107) - 11md.

From Eq. 3.44

Solution

Assume that a; == 0'; == 1 X 108 Pa and B, == By == 1.5 (for permeability in md). Calcu­late k, and ky if ax == 6.2 x 107 Pa, uy == 4.8 X 107 Pa, a == 0.72 and p == 3.5 X 107 Pa.

Example 3.8 Permeability and Stress Anisotropy

Assuming that a; ~ a; and Bx ~ By (this is not always true, as shown by Buchsteineret al. [21]) then the maximum permeability is normal to the minimum stress directionwhich would result in the least favorable situation for flow into the hydraulic fracture,which is also normal to the minimum stress. The permeability normal to the fractureface is likely to be the minimum value.

Stresses in formations72

where p is the bulk density of the rock.Newberry et al. [32] presented the logging technique using Eq. 3.52 in conjunc­

tion with measurements of rock densities to calculate horizontal stresses vs depth.Their calculation is exactly the one suggested by Eq. 3.13 requiring the knowledgeof pore pressure and, especially, an assumption of the poroelastic constant, Ct.

This approach has been criticized by several investigators both because of thefrequent lack of data on poroelastic constants [29] and the use of dynamic acousticmeasurements in inferring the response of a quasi-static operation such as hydraulicfracturing. Yet, while the technique may be flawed in detecting exact stress contrastsbetween overlain, underlain and target layers, it can clearly delineate deflections inthe stress values. "Calibration" of the acoustic measurements with independentlyobtained values from, e.g., an injection stress test, can provide relatively reliabledata in specific formations. Logging data are considerably less expensive than injec­tion tests.

(3.53)2 [3U~ - 4U;]E = PUs 2 2 '

Uc - Us

and

(3.52)

As an acoustic wave propagates through a rock, it causes a rock deformation. Thisdeformation affects the "slowness" of the component compressional and shear waves(Howard and Fast, [31]). The compressional wave slowness, u-, is an indicator ofthe rock response to a longitudinal stress. Shear wave slowness, Us. measures therock response to a transverse stress.

Modem sonic logging tools with sufficiently spaced transmitting and receivingpoints allow the determination of the sonic wave velocity slowness through thereservoir rock (Newberry et al. [32]).

Both the Poisson ratio and Young's modulus can be calculated from the sonicwave slowness by (Jaeger and Cook, [33])

3.9.2 Acoustic Measurements

possible and, therefore, subtle changes in the pressure record, aided by the moresensitive pressure derivative, are employed for the detection of Pis·

There has been considerable debate as to whether the maximum horizontal stresscan be determined from these tests. In general, while the concept is plausible,it requires 'independent analysis of the poroelastic behavior of the rock [26,29].Thus, the determination of O"h.max from these tests is considered unreliable and israrely done.

75Measurement of stresses

Time (min)

Figure 3.12 Determination of Pis from stress tests (After Warpinski [30])

324 02 3

Injection #3Injection #26

Injection #1

5.9

S~ 5.8!b.~ 5.7

'"'"~Q.. 5.6

Pis=5.73x107 Pa

5.5Deplh: 2456.7

102457.3 m5.42 3 4 5 6

Figure 3.12 after Warpinski [30J shows a succession of injection tests with clearlymarked instantaneous shut-in pressures. Often, such a clear demarcation is not

(3.51)Pis::::::Urnin =Uh.min·

For a vertical fracture in a single layer it can be surmised that

(3.50)Pis == Urnin·

A widely practiced technique is the successive mechanical isolation of relativelysmall intervals (1 to 3 m) and the injection of small volumes (5 x 10-3 to 10-1 m3)

with each injection lasting 1 to 2 minutes. The important element is the ensuingshut-in pressure decline. The rationale of the technique is based on the conceptsintroduced by Hubbert and Willis [2J and expounded upon in a series of publicationsby Haimson and Fairhurst [25-28].

While these tests are particularly useful in open hole completions, they have beenapplied to cemented and perforated wells. Care must be taken to assure enoughopen perforations (undamaged) with appropriate phasing. Warpinski and Smith [29Jsuggest 13 perforations per meter (actually four per foot) and at least a 90° perforationphasing, although ideal perforations, of course, should be aligned with the expectedfracture azimuth. The injected fluid is of low viscosity, usually a KCl solution.

The main variable extracted from these tests is the instantaneous shut-in pressure,Pis, which is obtained from the pressure decline following the termination of injec­tion. The Pis is the pressure in the hydraulic fracture at the instant of shut-in. In alarge treatment this value could be considerably larger than the closure pressure andit will decline to the latter value following fluid leakoff into the formation.

However, because of the very small fracture propagation in the injection testsdescribed here, and because no masking phenomena such as induced stresses orfracture retardation effects are anticipated, it is postulated that

3.9.1 Small Interval Fracture Injection Tests

Stresses in formations74

..__ _-----

Measurements of stresses using native cores require the directional extraction ofthe cores and extensive laboratory equipment and instrumentation. Techniques fallunder two general categories, acoustic, such as the Differential Wave Velocity Anal­ysis (DWVA) and, static, such as the Anelastic Strain Recovery (ASR) and theDifferential Strain Curve Analysis (DSCA). The latter two offer the possibility tomeasure the relative (and, at times, the absolute) magnitudes of the principal in situstresses. This section presents a brief synopsis of the rationale and the methodologyof interpretation of ASR and DSCA. .

Friedman [36] has suggested a mechanism for the origin of the in situ state ofstresses which is the essence of the core analysis methods. As grains are buried andundergo lithification, they are compressed and distorted. They are, thus, stressed.Cementing material percolates through the porous medium, sets and holds the stressedgrains in their distorted shapes. This situation leads to large amounts of storedenergy, which is likely to vary depending on the amount of stress exercised ineach direction.

3.9.4 Core Stress Measurements

permeability and the filtercake resistance. They will be described extensively inChapter 8.

Frequently, in the Mayerhofer et al. [35] log-log diagnostic plot, the closure pres­sure can be detected as a sharp change in the pressure derivative curve as shown inFigure 3.14. The explanation is simple, Before closure, an open fracture provides aconsiderably different pressure response than that from the reservoir which wouldbe controlling once the fracture is closed. Pressure derivatives are far more sensitivethan the corresponding pressure in detecting such phenomena.

Time, min

Figure 3.14 Log-log diagnostic plot of pressure and pressure derivative of a fracture injectionpressure decline (Mayerhofer et al. [35])

10 lQ20.1

• pressure• derivative • ~r- .

,.......--.....~ V~· I)

Closure

..

'iilQ.

cO 10~:>1ij:>·cQ)"'0"'0cell 10~

103

rrMeasurementof stresses

Figure 3.13 Closure pressure determination from step-rate and flow-back tests (After Nolte [24Jand Smith [34]) .

TimeInjection rate

Injection pressure

~""" P""'"

Fractureextensionpressure

~

~

The determination of the closure pressure is important not only because of its intrinsicv~lue b~t also because of design considerations (injection pressure demand) and thediagnosis of the net fracturing pressure during the treatment execution. The netfracturing pressure is defined as simply the fracturing pressure minus the closurepressure. Nolte [24] and Smith [34] have suggested relatively qualitative techniques,tempered by field experiences, to determine the closure pressure. The rationale of themeasurement is based on the observation of pressure profiles both during injectionand shut-in. The idea is that the pressure patterns are likely to be different while thefracture is open compared to when the fracture is closed.

A c~mmon injection test is the "step-rate" test involving the injection of discretelyescalating rates mto an already initiated fracture. Recommended injection rates startfrom 1 barrel/min (2.6 x 10-3 m3/s), in increments of 1 barrel/min for about tensteps. Injection time is equal for each step (1 to 2 minutes) except for the laststep which may last 5 to 10 minutes. The observed bottomhole injection pressure isplotted against the injection rate as shown in Figure 3.13. The break point in the plotcorresponds to the fracture extension pressure which is the maximum possible valuefor the closure pressure. This extension pressure is usually 7 x lOS to 1.5 X 106 Palarger than the closure pressure because of near-well friction and other phenomena.

The step-rate test is frequently followed by a "flow-back" test, the two tests almost:u;,ay_s conducte~ in tandem. The well is flowed back immediately after the step-ratelllJe~tlOn.To aVOIdrate transient effects masking the pressure response, the flow-backtest IS conducted at a rate about 1/6 to 1/4 of the last injection rate. The bottomholepressure falloff is observed and plotted against time as shown in Figure 3.13. Thepressure decline will show a characteristic reversal of the slope, with the inflectionpoint coinciding with the closure pressure. The test is repeated several times to ensurereproducibility of the results.

Recently, Mayerhofer et al. [35] have introduced a robust model for the pressuretransient analysis of fracture injection tests. The main purpose of these tests is toestimate the controlling variables for fracturing fluid leakoff, namely the reservoir

3.9.3 Determination of the Closure Pressure

Stresses in formations76

-_.._._. __ ..._----...._- ..._-----_ .._-._-_., ..._------._----

and

sv(t) = (2av - 0'1 - (2)J 1(1 - e-t/tt) + (0'1 + 0'2+ o; - 3p )]z(1 - e-t/tI), (3.60)

where J1 is the distortional creep compliance, J2 is the dilatational creep complianceand tl and t2 are the relaxational time constants for the distortional and dilata­tional creeps, respectively. These are rock properties that can be determined in thelaboratory.

From Eqs. 3.59 and 3.60, measurement of strains at different angles e, and fordifferent times, would lead to the calculation of 0'1. 0'2 and a.,

Section 3.9 presented the most established techniques for the measurement of stressesand stress directions that have an impact on hydraulic fracturing. Other techniquesexist and there is reasoning in outlining the present methods.

The determination of closure pressure with an injection test is by far the mostreliable technique, and the measurement is important for the analysis of the fracturenet pressure, i.e. the proper determination of the propagating fracture ge?metry.Injection stress measurements along a vertical column are important to diagnoseintralayer stress contrast and, thus, predict fracture height migration. ~onic logs,properly calibrated, can be a reasonable and relatively inexpens~ve substItute.. .

The determination of stress directions in the target interval IS for the predictionof the fracture azimuth. This is essential information in the hydraulic fracturing ofhighly deviated and horizontal wells. ASR and DSCA can readily fulfill this t~k.

The various tests done together for critical wells can provide all stress magnitudesand stress directions.

Clearly, not all tests are necessary in every well that is to be hydraulically =:tured. In geologies that are not chaotic, a set of cores from one well may be sufficientfor the entire field. Also, in. reservoirs where continuity of layers is established fromlogs in interwell correlation, stress tests can be done in only a few select~d wells..Fracture closure tests should be done in almost all wells to be hydrauhcally frac­

tured. The closure pressure could vary considerably. Conveniently, these tests areoften part of an overall injection strategy for leakoff determination.

(3.58)

(3.57)3.9.5 Critique and Applicability of Techniques

( )(1 - V)LlSl + vCl\S2+ Llsv)

0'1= O'\)-ap +ap(1 - v)l\sv +V(LlS2+ LlSl)

( )(1 - V)LlS2+ V(l\Sl + l\sv)

0'2= av-ap -s a p,. (1- v)l\sv + v(l\sz + l\sJ)

In Eqs. 3.57 and 3.58, Sv is the overburden strain.A viscoelastic model, using time-dependent analysis, was constructed by Warpin­

ski and Teufel [39] for the radial and vertical strains, respectively, er(r) and Sv(/),using strain values from any angle, e. The forms of their solution are

srCt) = (20'1 cos2 e+ 20'2 sin2 e - 0'1 sin2 e - 0'2 cos2 e - av)h (1 - e-t/t1)

+ (0'1+ C12 + av - 3p)]ZC1- e-t/tl) (3.59)

(3.67)

(3.66)Cn = ,(1- v12 - 2V31V13)

E3(1 - V12)(3.56)

(3.65)C12 = •(1+ V12)(1 - v12 - 2V31V13)

E3V13

(3.55)

and

en = .(1+ v1z)(1 - V12 - 2V31V13)

E1(V!2 - V31V13)So + S90 .../2/ez = - - (co - S45)2 + (S45 - E90)22 2 .

The direction of the principal strains is given by

e 1 -1 {2£45- EO - E90 }= 2: tan .So - S90

Blanton [38] has solved the problem of the stress magnitudes from the principalstrains and the value of the vertical, overburden, stress O'v:

(3.64)(3.54)

where

(3.61)

(3.62)

(3.63)

0'1 - ap = Clle1 + C12E2+ C13e3,

0'2 - a p =C12e1+ C1lS2 + C13e3.

0'3 - ap =Cl3el + C13c2 + C33e3.

DSCA is essentially a reverse procedure from ASR by measuring the inducedrelative strains under a confining pressure (Ren and Roegiers, [40]). Measurementof strains and calculation of the principal strains similarly to Eqs. 3.54 and 3.55 leadto the following solution for the three principal stresses, 0'1, 0'2 and 0'3·

Grains may undergo changes in the state of stresses in geologic time. While thecurrent state of stresses is controlling, historical evidence, such as the presence ofcertain fissures, may reflect past states of stress.

Thus, strain relaxation, related to the release of stored energy, is presumed to beproportional to the present state of stress. Oriented relative strain relaxation allowsthe determination of the oriented stresses' relative magnitudes. The maximum straindirection coincides with the maximum stress direction.

Supposing that three strain gauges on a horizontal plane, at 0·, 45° and 90·,measure strains eo, S45 and c9O, then the principal strain directions are (Obert andDuvall [37])

79Measurementof stressesStresses in formations78

--- ----- ._--_. --------._--_.._-_ ..---------_ ......_._------ ._----------

son, TX, 1970.32. Newberry, B.M., Nelson, R.F. and Ahmed, U.: Prediction of Vertical Hydraulic Fracture

Migration Using Compressional and Shear Wave Slowness, SPEIDOE 13895, 1985..33. Jaeger, J.C. and Cook, N.G.W.: Fundamentalsof RockMechanics, Chapman and Hall,

New York, 1979.34. Smith, M.B.: Stimulation Design for Short, Precise Hydraulic Fractures, SPEJ., (June)

371-379,1985.35. Mayerhofer, M..J.., Ehlig-Economides, C.A.. and Economides, MJ.: Pressure Transient

Analysis of Fracture Calibration Tests, Paper SPE 26527, 1993.36. Friedman, M.: Residual Elastic Strains in Rocks, Tectonophysics,15,297-330,1972.37. Obert, L. and Duvall, W.E.: RockMechanics and theDesignof Structures inRock, John

Wiley New York, 1967. .38. Blanton, T.L.: The Relation Between Recovery Deformation and In-Situ Stress Magru-

tudes, Paper SPE/DOE 11624, 1983. . . .39. Warpinski, N..R..and Teufel, L.W.: A Viscoelastic Model for Determining In-SHu Stress

Magnitudes from Anelastic Strain Recovery of Core, SPEPE, (Au..g.), 273-280, 1989 ..40. Ren, N.-K. and Roegiers, J.-c.: Differential Strain Curve AnalYSIS- A New Method

for Determining the Pre-Existing In-Situ Stress from Rock Core Measurements, Proc.Fifth CongressInt. Soc. Rock Mech., Melbourne, 1983.

31.

30.

29.

28 ..

27.

26.

25.

24.

Prats, M..: Effect of Burial History on the Subsurface Horizontal Stresses of FormationsHaving Different Material Properties, SPEJ, (Dec.), 658-662, 1981. ..Nolte, KG ..: Fracture Design Considerations Based on Pressure AnalYSIS,SPEPE, (Feb.),23-30, 1988.Haimson, B.e. and Fairhurst, C.: Initiation and Extension of .Hydraulic Fractures inRocks, SPEJ, (Sept.), 310-318, 1967. .Haimson, B.C. and Fairhurst, c.: Hydraulic Fracturing in Porous Permeable Matenals,IPT, (July), 811-817, 1969. .Haimson, B.C.: The Hydrofracturing Stress Measuring Method and Recent Field Results,Intl.J, RockMech. Min. Sci., 25,167-178,1978.Haimson, B..C.: Confirmation of Hydrofracturing Results Through Comparisons withOther Stress Measurements, Proc. 22nd US. Rock Mechanics Symposium Massachu-setts Inst.. of Technology, Boston, (June), 379-385, 1981. .Warpinski, N.R. and Smith, M.B.: Rock Mechanics and Fracture Geometry, IIIRecentAdvances in HydraulicFracturing,Gidley, J.L. et at. (eds.), SPE Monograph 12, SPE,Richardson, TX, 1989..Warpinski, N.R.: In-Situ Stress Measurements at U.S. DOE's Multiwell Experiment Site,Mesaverde Group, Rifle, Colorado, (March),IPT., 527-537, 1985. .Howard, G.C. and Fast, C.R.: HydraulicFracturing, SPE Monograph 2, SPE, Richard-

23.

81References

1.. Roegiers, J..e..:Elements of Rock Mechanics, in ReservoirStimulation,M.J. Economidesand K.G. Nolte (eds ..), Prentice Hall, Englewood Clift, N.J., 1989.

2. Hubbert, M.K. and Willis, D.G.: Mechanics of Hydraulic Fracturing, Trans.AlME, 210,153-166, 1957.

3.. Biot, M.A: General Solution of the Equations of Elasticity and Consideration for aPorous Material, 1.Appl. Mech.., 23, 91-96, 1956.

4. Deily, F.H. and Owens, T.e.: Stress Around a Wellbore, Paper SPE 2557, 1969..5. Bradley, W.B.: Failure of Inclined Borehole, 1. EnergyRes..Tech.,233-239, 1972.6. Richardson, R.M.: Hydraulic Fracture in Arbitrarily Oriented Boreholes: an Analyt­

ical Solution, Proc. Workshopon HydraulicFracturingStressMeasurements,Monterey,California, (Dec.), 1981.

7. Yew, c.a, Li,Y.: Fracturing of A Deviated Well, SPEPE, (NOV),429-437, 1988.8.. Terzaghi, K: Die Berechnung der Durchlassigkeitsziffer des Tones aus dem Verlauf der

hydrodynamischen Spannungserscheinungen, Sber, Akad. Wiss., Wien, 132, 105, 1923.9. Detournay, E. and Cheng, AH-D.: Poroelastic Response of a Borehole in a Non­

hydrostatic Stress Field, Int. 1.Rock Mech., Min. Sci. and Geomech. Abstr., 25 (3),171-182, 1988.

10. Daneshy, AA.: Experimental Investigation of Hydraulic Fracturing Through Perfora­tions,IPT, (Oct.), 1201-1206, 1973.

11.. Kuriyagawa, M., Kobayashi, H., Matsunaga, I., Yamaguchi, T. and Hibiya, K:Application of Hydraulic Fracturing to three-dimensional In-Situ Stress Measurements,Proc ..Second Int. Workshopon HydraulicFracturingStressMeasurements,HFSM 88,Minnesota University, (June), 307-340, 1988.

12. Brown, J.E. and Economides, M. J.: An Analysis of Hydraulically Fractured HorizontalWells, Paper SPE 24322, 1992.

13.. Owens, K.A., Andersen, SA. and Economides, M.J.: Fracturing Pressure for HorizontalWells, SPE 24822, 1992.

14. Ramey, H.J. Jr..: Interference Analysis for Anisotropic Formations - A Case History,IPT, (Oct), 1290-1298, 1975.

15. Warpinski, N.R. and Branagan, P.T..: Altered Stress Fracturing, JPT, (Sept.), 990-997,1989.

16. Muskat, M.: The Flow of HomogeneousFluids ThroughPorousMedia, McGraw-Hill,New York, 1937.

17. Ben-Naceur, K.. and Economides, MJ.: Production from Naturally Fissured ReservoirsIntercepted by a Vertical Hydraulic Fracture, SPEFE, (Dec.), 550-558, 1989.

18. Deimbacher, F.X., Economides, M.J.., Heinemann, Z.E. and Brown, J.E.: Comparisonof Methane Production from Coalbeds Using Vertical and Horizontal Wells, Paper SPE21280, 1990.

19. Warpinski, N.R.: Hydraulic Fracturing in TIght, Fissured Media, IPT, (Feb ..), 146-152,1991.

.20. Walsh, J.B.: Effect of Pore Pressure and Confining Pressure on Fracture Permeability,Int.1. of Rock Mech.,Min. Sci. & Geoph..Abstr., 18, 429-435, 1981.

21. Buchsteiner, H.., Warpinski, N.R. and Economides, MJ.: Stress-Induced PermeabilityReduction in Fissured Reservoirs, Paper SPE 26513, 1993.

22. Economides, M..J., Buchsteiner, H. and Warpinski, N.R.: Step-Pressure Test for Stress­Sensitive Permeability Determination, Paper SPE 27380, 1994.

References

Stresses in formations80

--- -----------_.. -,,_,,---- _---_ _ _--- ----- --- .

Two models, both assuming constant height and two-dimensional (2D) propaga­tion, have dominated the routine prediction of hydraulic fractures. These are thePerkins and Kern (PK) [4J and the Khristianovich and Zheltov (KZ) [5] models. InSection 2.3.4, plane strain was introduced as a 2D (x, y) condition with no displace­ment in the z direction.

Implicit to the two 2D models is the presumption of plane strain, on the verticalplane for the Perkins and Kern [4] case and the horizontal plane for the Khris­tianovich and Zheltov [5J case. Figure 4.1 is a depiction of the two plane stressconditions. Vertical plane strain, along a fracture, with considerably larger length thanheight, allows vertical parallel planes to "slide" against each other. This condition [4J,further developed by Nordgren [6], is often referred to as the PKN geometry.

4.1 The Perkins and Kern and Khristianovich andZheltov Geometries

The physics described in Chapter 2 and the analysis of stresses in Chapter 3 dictatecertain conceptualizations of hydraulic fracture geometry. Such conceptualizationsare necessary not only in the modeling of the fracturing process itself but, equallyimportant, in the prediction of the flow performance of the created fracture.

Simplified geometry is often tractable mathematically. Some of the best knowndepictions have been alluded to already in Chapter 2 and will be expounded uponin this chapter. Deviations from the idealized geometry have given rise to numericalsimulation schemes which purport to account for out-of-plane fracturing, complicatedgeometries, multi-layered formations and formation heterogeneities.

There is considerable value in using clearly defined assumptions, consistent lawsand constitutional equations to delineate the basic behavior of hydraulic fractures.Traditionally, the Griffith-Sneddon (see Griffith [1]; Sneddon [2]) crack has formedthe basis of most fracture geometry models while the work of Hubbert and Willis [3]has cleared the notion of fracture orientation. The former has been presented inChapter 2 while the latter has been dealt with extensively in Chapter 3.

FRACTURE GEOMETRY4

The PKN and KGD geometries have been used in a great number of engineeringdesign schemes as reasonable (macro-) approximations of induced fractures. Thehydraulic shape, maintained with the addition of propping materials (propped frac­tures) or etched by acids (acid fractures) corresponds well with the production-typemodels of Gringarten and Ramey [8} and Cinco-Ley and Samaniego [9], describedin Chapter 1.

The same fracture models, along with their limiting approximation (radial) havebeen used for the interpretation of observed fracturing pressures and the extractionof propagation and leakoff variables [10-12].It should be emphasized here that the two models cannot be used interchangeably.

They are mutually exclusive, albeit elegant, approximations. Fractures in formations

(4.2)

assuming an elliptical shape in both directions. An alternative way of writing Eq. 4.1is to replace (rr/4)2 with a shape factor, y, characterizing the assumed relation­ship between average and maximum width. This has been used often as 3rr/16.In Chapter 9 this shape factor will be found to be equal to n:/5 using the originalassumptions of Perkins and Kern [4].

The horizontal plane strain condition of the KGD geometry would result in afracture with a rectangular profile at the well as shown in Figure 4.3. Again, thefracture height, hf is constant. The rectangular shape of a cross section further fromthe well has a smaller width, decreasing to zero at the fracture length, xf. Theaverage width is then related to the maximum width, Wo, simply by approximatingthe horizontal cross section with an ellipse:

(4.1)

The assumption of horizontal plane strain, i.e., an infinite number of "sliding"parallel planes traversing the height of a fracture, is a plausible simplification fora short but considerably taller fracture. The horizontal plane strain condition [5]further developed by Geertsma and deKlerk [7] is often referred to as the KGDgeometry.

The PKN geometry depicted in Figure 4.2 is of an approximate elliptical shapein both the vertical and horizontal axes. In both directions the width is much smallerthan the other dimensions (of the order of a few millimeters compared to tens orhundreds of meters.) The elliptical geometry is not entirely true and a more rigoroustreatment will be presented in Chapter 9. The height, hf' is constant and the length,x f' is considerably larger.

One of the most important characteristics of the fracture is the average width,W, defined as the total fracture volume divided by the area of one face of the twofracture wings (or, equivalently, the volume of one wing divided by the area of oneface of the wing.) The maximum width at the wellbore (Section 2.4.2) Wo wouldthen be approximately related to the average width by

85The Perkins and Kern and Khristianovich and Zheltov geometries

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

Figure 4.2 The PKN [4,6] geometry

Figure 4.1 Vertical and horizontal 2D plane strain condition

Vertical planestrain condition

Fracture geometry84

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

Figure 4.4 Plane strain approximation to a circular crack

x

z

(4.8)c(x) = VR2 - x2;

We take the above exact solution as a basis and compare it to an approximatesolution of the problem. The approximate solution is obtained a~p.lying th~ planestrain assumption in every vertical plane normal to the plane contammg the CIrcle ~shown in Figure 4.4. In other words we compute a line crack ("Sned~on crack"). Inevery vertical plane. The characteristic length c of the line crack vanes dependingon the location, x. From elementary geometry

(4.7)

(4.6)

(4.5)16R3 pnOV:;; 3E'

_ 16RpnOW:::':~'

(4.4)8RpnOWo = nE' '

(4.3)WeT) = _3__pnOVR2 - ,2,nE'

87The Perkins and Kern and Khristianov/ch and Zheltov geometries

------ -.-------~--------------------' ...-'.--.-- .... _. -_ .._--

Both the PKN and KGD geometries are based on the plane strain assumption. It isnatural to ask what are the consequences of such a simplifying assumption. Will thewidth volume energy computed from a plane strain assumption be larger or smallerthan the "true" values? Does the vertical or the horizontal plane strain assumptiongive "more realistic" fracture shape? To answer these questions we refer to the resultsdescribed in Chapter 2, where we considered mathematical solutions for both thecircular crack and the line crack problems. Our approach here may seem somewhattheoretical but the conclusions will be of practical importance.

We consider a circular crack (in other words a radial fracture) of radius R, locatedin a vertical plane (the x,z plane) and opened by a constant net pressure PnO. Theshape, width, volume, average width and elastic energy of the crack have been givenin Sections 2.7 and 2.8:

4.1.1 The Consequences of the Plane Strain Assumption

that are clearly bounded at the top and bottom by lithologies likely to contain thefracture height could be approximated with the PKN model. Relatively uncontrolledfracture height or small fracture treatments could be approximated with the KGDmodel. In general, KGD-type fractures are not interesting from a production pointof view.

Figure 4.3 The KGD [5,7] geometry

..-- .......--",-

Fracture geometry86

As explained in Chapter 3, induced hydraulic fractures away from the well willpropagate normal to the minimum stress. In the vast majority of petroleum appli­cations, this would result in vertical hydraulic fractures, normal to the minimumhorizontal stress. While some petroleum formations would be shallow enough thata hydraulic fracture could be horizontal (Section 3.2 and Example 3.2) it wouldbe a rare undertaking. Such reservoirs would not be candidates for hydraulic frac­turing by virtue of their higher permeabilities. In some cases high reservoir pressuresmay cause complications (see Example 3.2) but these should be considered as rareexceptions.

Of course, the near-well stresses in the radial well geometry are different fromthe far-field, Cartesian stress components (Section 3.3.) Fracture initiation at thewell. is likely to follow a plane that is different from that of the final fracture prop­agation. The plane of fracture initiation is also affected greatly by the perforationpattern.

Figure 4.5 depicts a plausible fracture initiation from a vertical well (a) and aturning fracture (b) en route to its final direction, normal to the minimum hori­zontal stress. The illustrations in Figure 4.5 (a and b) are idealizations. Severalnear-well cracks are likely to develop coalescing into one dominant fracture. Thenear-well effects result in additional friction pressure during the fracturing process(often lumped as "tortuosity effects") and would almost certainly lead to "choke"effects of varying severity during production.

The distribution of the induced fracturing pressure would greatly depend on thedissipation of this energy against the various resistances. In the presence of one planeof initiation (e.g. zero perforation phasing) a single wing of a fracture will be createdfirst. A second wing will initiate when the resistance in the first wing, because oftortuosity friction losses and other retarding effects, exceeds the fracture initiationpressure of the second wing. The latter will depend on the adhesion between cementand casing or cement and formation and the distance the fluid has to travel betweenthe perforation and the point of the second wing initiation. The stress profile arounda well has been shown in Section 3.3 and the point of the likely second wing canbe predicted. Excessive resistance ahead of a second wing may result in only onewing of a fracture since the fracture propagation pressure demand may be lower than

4.2 Fracture Initiation vs. Propagation Direction

Figure 4.5 Fracture initiation from a vertical well (a), turning normal to the least resistance,in most cases, the minimum horizontal stress (b) and, finally, once the resistance isovercome, evolving into a two-winged fracture (c)

(c)(b)(a)

Compare Eq. 4.9 to Eq. 4.3 and keep in mind that r2 = x2 + Z2 reveals' that theplane strain approximation results in the same shape as the exact solution but thewidth is 11:/2times larger everywhere. The error of the plane strain approximation isther.efore 57~. Cl~arly the maximum width, volume and average width from the planestram .approxIma~IOnare also It /2 times greater than their exact values, respectively.Knowing the straightforward relation between volume and elastic energy, it is obviousthat the elastic energy estimated from the plane strain approximation is also It/2 timesgreater than the exact value given by Eq. 4.7.

The plane strain approximation overestimates the width because it "feels" therestrictive effect of the tips only in the z direction while it disregards the similarrestric.tion from the x direction. The exact solution of the circular crack problemtakes into account the effect of the tips correctly in every direction.

Intuitively it is obvious that in general a plane strain approximation will overes­timate the width. The error becomes less if the characteristic dimension normal tothe plane in which the plane strain assumption is applied exceeds considerably the?the: characteristic dimension. In particular, the KGD geometry is a good approx­imanon for a short fracture with large height and the PKN geometry is a goodapproximation for an elongated fracture with height significantly less than its length(Barree, [13]). In the light of this result the recently appearing practice of creatingmodels which additively sum widths calculated from different plane strain approxi­mations is not correct.

(4.9)

therefore from Eq. 2.37

89Fracture initiation vs. propagation directionFracture geometry88

.--.--- ..-.~ ---------------------------- ----

To obtain a geometry resembling the one in Figure 4.6, t~e en~ry.from the wellto the fracture must be minimized. Otherwise a configuratIOn ~lmll~r to the onede icted in Figure 4.7 is likely to occur where, even if the well IS drilled properly,th: fracture will initiate longitudinally and then turn through a very tortuous path tobecome transverse and normal to the minimum horizontal stress.

Figure 4.7 Turning fracture in a horizontal well from longitudinal initiation to the transversedirection [15]

Figure 4.6 Transverse vertical fractures from a horizontal well

91Fracture initiation vs. propagation direction

-----, ....._--_ ...._-----------------

Horizontal wells have emerged since the early 1980s as important additions topetroleum activities, capturing an ever increasing share of all wells drilled.

Production engineering considerations suggest (Brown and Economides [14]) thatmany horizontal wells should be hydraulically fractured. In some cases they shouldbe drilled to accept a longitudinal fracture while, in some others, they should bedrilled to accept multiple transverse fractures. The delineation of this issue and theoptimum well direction based on reservoir characteristics have been dealt with byBrown and Economides [14].

The difference between an ideal transverse fracture initiated from a horizontalwell (Figure 4.6) and a longitudinal one initiated from a vertical well relates to theproduction characteristics. In a vertical well/vertical fracture configuration the flow offluids is linear from (or into) the reservoir normal to the fracture face and then linearwithin the fracture. For a horizontal well/transverse vertical fracture configuration,while the linear part between fracture and reservoir is maintained, inside the fracturenear the well, the flow reverts from linear to radial, This effect on fractured wellperformance has been quantified [14], and, in general, it results in a considerablereduction in the flowrate from each individual transverse fracture compared to afracture in a vertical well. However, the composite fiowrate from multiple treatmentscould be much larger than the one from the single fracture in a vertical well.

4.2.1 Fractures in Horizontal Wells

the friction pressure demand around the wellbore. To avoid this problem perforationphasing becomes important and a 120· phasing is deemed as minimum. Currentperforating guns are configured routinely with 30· phasing.

Figure 4.5(c) shows a fully propagating two-winged fracture with the final direc­tion normal to the minimum horizontal stress.

More tortuous paths are to be expected from deviated and horizontal wells.Although unfractured deviated wells generally have production advantages oververtical wells, fractured deviated wells, at best, have no advantages over fracturedvertical wells and, at worst, they pose considerable disadvantages. The reason is thenear-well tortuosity.

Thus, in general, it is not recommended to fracture deviated wells, both because ofthe non-vanishing share stress components and the invariably larger fracturing pres­sures and, more critically, because of the aforementioned near-well tortuosity. Thiswould lead to a highly undesirable reduction in the fractured well production. There­fore, wells that must be drilled at an angle from a platform or a drilling pad shouldbe turned vertical in approaching the target formation if they are to be fractured. Thetechnology for this type of drilling is readily available.

Horizontal wells can, and at times should, be hydraulically fractured. Thefollowing subsection will describe fracture geometries induced from horizontal wells.In general, wells to be fractured, from a production point of view, should followone of the three principal stresses: vertical or maximum horizontal stresses forlongitudinal fractures or minimum horizontal stress for transverse fractures.

Fracture geometry90

'---'~-"~" __ "---'-- _-- .._-- ..--_."

Figure 4.9 Vertical fracture profile through a three-layer formationFigure 4.8 Vertical fracture initiated from an arbitrarily oriented horizontal well at an angle [¥

from the minimum horizontal stress

where WI and WI are the widths of the longitudinal-to-transverse and the ideal trans­verse fracture, respectively, D is the well diameter and L is the length of the contact.Equation 4.10 suggests that if L > 1.5D a detrimental reduction to the fracture widthis likely to occur. Thus, transverse fractures should be executed through very shortopen well sections. This can be accomplished either by abrasive jet cutting or multiplepasses with very short perforating guns. Multiple transverse treatments must be doneindividually with proper zonal isolation.

A longitudinal fracture from a horizontal well is relatively easier to execute. Inmost cases this configuration would not offer production advantages over a fracturedvertical well. This would be particularly true in lower-permeability reservoirs, whichare the normal candidates for hydraulic fracturing. However, in relatively higherpermeability formations a longitudinally fractured horizontal well could be moreattractive than a fractured vertical well (Economides et al. [16]).

At times, in offshore operations, it may not be possible to execute one or theother of the limiting cases in fracturing horizontal wells. Configurations such asthe one shown in Figure 4.8 are then likely. The production behavior of arbi­trarily fractured horizontal wells has been presented by Deimbacher et al. (15].The breakdown pressure for these geometries was explained in Section 3.7.1 andExample 3.7.

Fracture height migration and the variables affecting it will be addressed inChapter 11. A vertical fracture profile can be visualized as in Figure 4.9 wherethe middle is the target interval with fracture growth in the overlain and underlainstrata. Different elastic properties are likely to result in different fracture widths atthe various layers.

Fracture height propagation generally is an undesirable occurrence. Beyond thewasted fracture growth in potentially impermeable layers, there are real dangers fora proppant "screenout" during execution. Figure 4.10 depicts a plausible occurrencewhen the target interval (e.g., the bottom layer in Figure 4.10) is separated fromanother similar layer (top) by a relatively thin layer of larger modulus.

While the fracture height may migrate through the connecting layer, the widthmay be such that fracturing fluid can leak off but propp ant may not. Thus, thefracturing fluid slurry in the bottom target interval may become dehydrated and

(4.10)• WI nDWI =ZL'

4.3 Fracture Profiles in Multi-layered FormationsThis is an undesirable event and Deimbacher et al. (15] have shown that a fracturewidth reduction is likely to occur when the contact length of the turning fracture andthe well is excessive. A simple approximation of this effect is

93Fracture profiles in multi-layered formationsFracture geometry92

·.-~~-.--- ..-- -----.---

1. Griffith, A.A: The Phenomenon of Rupture and Flow in Solids, Phil. Trans. Roy. Soc.,A 221, 163-198, 1920.

2. Sneddon, I.N.: The Distribution of Stress in the Neighborhood of a Crack in an ElasticSolid, Proc. Roy. Soc., A 187, 229-238, 1946.

3. Hubbert, M.K. and Willis, D.G.: Mechanics of Hydraulic Fracturing, Trans., AIME,210, 153-166,1957.

4. Perkins, T.K. and Kern, L.R.: Width of Hydraulic Fractures, 1PT, (Sept.), 937-49, 1961;Trans., AIME, 222, 1961.

5. Khristianovitch, S.A. and Zheltov, YP., Formation of Vertical Fractures by Means ofHighly Viscous Fluids, Proc. World Petroleum Congress, Rome, 2, 579-586, 1955.

6. Nordgren, R.P.: A Propagation of a Vertical Hydraulic Fracture, SPEJ, (Aug.), 306-314,1972; Trans., AIME, 253, 1972.

7. Geertsma,J. and de KIerk, F.: A Rapid Method of Predicting Width and Extent ofHydraulically Induced Fractures, JPT, (Dec.), 1571-81, 1969.

8. Gringarten, A.C. and Ramey, A.J., Jr.: Unsteady State Pressure Distributions Createdby a Well with a Single-Infinite Conductivity Vertical Fracture, SPEJ, (Aug.), 347-360,1974.

9. Cinco-Ley, H. and Samaniego, F.: Transient Pressure Analysis for Fractured WelIs,1PT,(Sept.), 1749-1766, 1981.

References

Figure 4.11 A T-shape fracture

References 95

unable to transport the proppant. This would lead to a near-well screenout, rapidpressurization and termination of the treatment.

Another, relatively common, undesirable occurrence is the T-shape fracture.Consider a target layer (Figure 4.11) at a depth where the minimum horizontalstress is relatively near the vertical stress. Vertical fracture initiation may resultin fracturing pressure that may exceed the overburden. If, in addition, the overlainlayer is relatively difficult to fracture, a T'-shape fracture may be created in thetarget layer.

Again, the width of the horizontal branch may be large enough to accept fluid butnot large enough to accept proppant, leading to a treatment-terminating screenout.

Phenomena such as the above have contributed to the need for pseudo-3D and 3Dfracture simulation schemes. The elegant but simplified PKN and KGD geometrieswould no longer be adequate.

Figure 4.10 Fracture height migration and associated width reduction through an adjoining layerof large modulus

94 Fracturegeometry

is called shear rate. One can consider the shear stress as the response of matter to theshear rate. For Newtonian fluids the shear stress varies linearly with the shear rate.The coefficient of proportionality is called viscosity. The larger the viscosity the moreresistant the fluid is to flow. If the linear relation does not hold but the shear stressis still a unique function of the shear rate, we speak about a general (or generalizedNewtonian) fluid. The shear stress versus shear rate relationship expressed in an

(5.2)

which is force divided by the area.Since the external force and stress-induced force are balanced, it is sufficient to

use one of them to characterize the state of the fluid. The change in velocity Su isproportional to the distance of the layers tiy. The limit

. tiuy=­

tiy

(5.1)F

r= -A

Matter responds by a finite deformation to external force applied normal to its outersurface. From this point of view solids and liquids act rather similarly while gasessustain less resistance, i.e. they are more compressible. This difference of behavior ismore quantitative than qualitative. Though less perceptible, a force acting on a planemay have a direction parallel to it. The response of a solid to such a force is, again,a finite deformation. Liquids and gases react to such a force differently, manifestinga continuous deformation called flow [1-3]. The common name of matter able toflow is fluid.

It is convenient to consider flow as the sliding of parallel layers relative to oneanother. Figure 5.1 illustrates this concept. The external forces originate from thedifference of pressures and/or from gravity (Poiseuille flow) or from torque (Couetteflow). The shear stress keeping the system in equilibrium acts in the opposite direc­tion and has the magnitude:

5.1 Basic Concepts

RHEOLOGY ANDLAMINAR FLOW

510. Nolte, K.G. and Smith, M.B.: Interpretation of Fracturing Pressures, IPT, (Sept.),

1767-1775,1981.11. Nolte, K.G.: Fracture Design Considerations Based on Pressure Analysis, SPEPE, (Feb.),

23-30, 1988.12. Mayerhofer, M.J., Ehlig-Economides, c.A. and Economides, M.J.: Pressure Transient

Analysis of Fracture Calibration Tests, Paper SPE 26527, 1993.13. Bartee, R.D.: A Practical numerical Simulator for Three-dimensional Hydraulic Fracture

Propagation in Heterogeneous Media, Paper SPE 12273, 1983.14. Brown, J.E. and Economides, M.J.: An Analysis of Hydraulically Fractured Horizontal

Wells, Paper SPE 24322, 1992.15. Deimbacher, F.X., Economides, M.J. and Jensen, O.K.: Generalized Performance of

Hydraulic Fractures with Complex Geometry Intersecting Horizontal Wells, Paper SPE25505, 1993.

16. Economides, M.J. and Deimbacher, F.X., Brand, C.W. and Heinemann, Z.E.: Compre­hensive Simulation of Horizontal Well Performance, SPEFE, (Dec.), 418-426, 1991.

96 Fracturegeometry

Finally, it reduces to the Bingham plastic model, if n is equal to unity. To illustratethe versatility of the yield power law, some rheological curves generated from it,are shown in Figure 5.3 (Cartesian coordinate system) and Figure 5.4 (logarithmiccoordinates). Several interesting conclusions can be drawn comparing the two figures.The linear plot is not a very efficient tool to diagnose the presence of a yield stress.Conversely, a logarithmic plot deemphasizes the differences at high shear rates. Notethat in the Cartesian system the base case (1) and the power law case (2) seem similarwhile on the logarithmic plot the base case (1) and the Bingham plastic case (3) arealmost indistinguishable.

Several other constitutive equations have been suggested in the rheological litera­ture. It is useful to keep in mind that these idealized models are convenient devicesand that reality may be far more complex. The selection of model should be dictatedby the field of the application, by the accuracy of the measurements and by otherconsiderations such as computational ease.

The shear stress/shear rate relationship is not the only way to characterize therheological behavior. In some cases it is more convenient to present the apparent

(5.5)

where the three constants are ty = yield stress, K = consistency index, and n = flowbehavior index.

Equation 5.4 reduces to the Newtonian model if the yield stress is zero and theflow behavior index is unity. The limiting case where Ly = 0 leads to an expression inwide use for the description of polymer solutions which constitute the vast majorityof fracturing fluids. This is the well known power law

(5.4)

Fluids whose rheological behavior obeys a constitutive equation other than theexpression in Eq. 5.3 are referred to as non-Newtonian. Almost all fluids used inhydraulic fracturing behave in this manner, except for water.

In this chapter we make extensive use of the yield power law model (often referredto as the Herschel-Bulkley model)

(5.3)L = fLY.

line passing through the origin. If there is a positive shear stress necessary to initiateflow we call this stress yield stress and the behavior is called plastic. Pseudoplasticbehavior means that the fluid has no yield stress but the slope of the rheologicalcurve decreases with increasing shear rate. Dilatant behavior means that the slopeincreases with shear rate. A real fluid may show a combination of different behaviorsdepending on the considered shear rate interval. In addition, the exact behavior atthe two ends of a shear rate spectrum is always subject to transitional uncertainties.

In some cases it may be convenient to use only the rheological curve (or anothergraphical presentation of the same information) without a parametrization. In the eraof computer applications, however, it is more straightforward to deal with parametricmodels, i.e. with constitutive equations.

The straight line, passing through the origin in Figure 5.2, corresponds to a Newto­nian fluid with the "constitutive equation"

99Basic concepts

Shear rate, yFigure 5.2 fluid types and their rheological curves

Fluids can be classified by the shape of their rheological curves. Figure 5.2 illustratessome rheological features. A fluid is Newtonian if the rheological curve is a straight

5.1.1 Material Behavior and Constitutive Equations

algebraic form is the rheological constitutive equation and its graphical representationis the rheological curve. The rheological behavior is a material property, independentof the geometry of the flow conduit.

At increased flow rates, the concept of sliding parallel layers is no longer appli­cable because of the appearance of more complex movements leading to turbulence.While Newtonian fluids exhibit a marked transition to turbulence, most polymersolutions show a gradual change in the flow regimes.

Figure 5.1 Sliding layers concept of fluid flow

Rheology and laminar flow98

Shear rate. yFigure 5.5 Concept of apparent viscosity

(5.7)lAO -·lAs +lAo= ( )a lAs,

1 + lAo ~ lAsY

(5_6)

101Basic concepts

Figure 5.4 Yield power law model - logarithmic scale

10310210'

Shear rate, Y (1/S)

10°10-'

'r (Pa) n K(Pas")1 5 0.6 162 0 0.6 163 5 1 024 0 1.8 0005

800 1000400 600

Shear rate, )"(1/S)

Figure 5.3 Yield power law model - Cartesian coordinates

200o

1',. (Pal n K(Pa·s")1 5 0.8 1.62 0 0.6 1.63 5 1 0.24 0 1.8 0.005

130

'"CLp 60

oi'":!?w~ 40Q)s:CJ)

20

viscosity

Rheology and laminar flow100

---- - - .._--_._- ---- .__ .._-_.- ----- .. -- ._-_ .. --_.

""'10'

~(-> I-'oi .,;'" ..~ d>W .;;~ lii'":ll d>U5 10' s::.

(/)4 f.J - Ta-y

rlAa= 7

Y

as a function of the shear rate. Graphically, the apparent viscosity is the slope of aline drawn through the origin and the point on the rheological curve correspondingto the given shear rate, as shown on Figure 5.5. Note that the apparent viscosity isindependent of the geometry of the flow conduit. The same name is used for anothervariable defined in conjunction with the Hagen-Poiseuille law (see below). To avoidmisinterpretation we will use another name (equivalent Newtonian viscosity) for thatpurpose. An apparent viscosity curve can be obtained from the rheological curve byapplying the definition point by point.

Figures 5.6 and 5.7 show the apparent viscosity curves for the four differentcases of the yield power law model considered previously. The typical behavior ofa polymer liquid is more complex. Figure 5.8 is a logarithmic plot of the apparentviscosity of two different solutions of hydroxypropylguar (HPG) polymer in water.Note the different low and high shear rate behavior, emphasized by the logarithmicscale. The straight line in the middle indicates the domain where the power law is"valid" while at low and high shear rates the horizontal portions show Newtonian-likebehavior. A possible algebraic representation of the above behavior is the modifiedCross model (Chakrabarti et al. [4]);

(5.8)LPr =Ac!!"p.

Certain flow conduits obey a specific symmetry leading to a simple velocity profile.Pressure driven flow between parallel plates and in circular tubes has the impor­tant property that for a given cross section the wall stress is constant along theperimeter. The force balance around a' body with cross sectional area, Ac, wettedperimeter, P, and length, L, provides the basic relation between wall stress andpressure drop:

5.1.2 Force Balance

where J-to= low shear rate viscosity (at room temperature 0.0291 Pa- s for the 0.25mass % HPG solution and 0.410 Pa- s for the 0.5 mass % solution); J-ts = high shearrate viscosity (0.89 x 10-3 Pa- s), r" = stress parameter (3.72 Pa) and a = exponent(0.641) (Chakrabarti et al. [4]).

The determination of a rheological curve in graphical form and/or its representa­tion by an appropriate constitutive equation is the main task of rheometry. While theshear stress is readily available for measurement, at least at the walls of a device,the determination of the shear rate is more complicated. A curve similar to therheological curve but constructed from observable quantities is called a flow curve.Understanding of the relation between a flow curve and the rheological curve iscrucial for any application of rheology including the computations related to pressuredriven flow.

10'lCP 10" 103 10'

Shear rate.y (1/$)

Figure 5.8 Apparent viscosity of different HPG solutions

10-'

103Basic concepts

-----------------------_ _.-._-_ _-_ ------------~

,ao 10'

Shear rate, y(lIS)

Figure 5.7 Apparent viscosity of yield power law fluids - logarithmic scale

10"let'

10'<i)mIJ..

'y (Pal n K(Pa·s")1 5 0.6 1.62 0 0.6 1.63 5 1 0.24 0 1.8 0.005

102 ~-------------------------------------------,

Figure 5.6 Apparent viscosity of yield power law fluids - Cartesian coordinates

Shear rate, y (liS)1000BOO600400

3

D.B4

<i)m!!:.., 0.6~~'<i;00

'":; 0.4ECD(iiC-o.«

., (Pal II K (Pa·S,,)1 5 0.6 1.82 0 0.6 1.83 5 1 0.24 0 1.8 0.005

Rheology and laminar flow102

Figure 5.9 Schematic of slot flow geometry(5.13)

(5.12)NRe = pUavgl ,

f-J-

where the characteristic length, I, is defined by

1 = 4AcP

In some of the European literature, the Weisbach friction factor is preferred. Denotedusually by A, the Weisbach friction factor is equal to four times the Fanning frictionfactor. In this book we use only the Fanning friction factor and omit the nameFanning in front of it. The friction factor is dimensionless. The average velocitymeans flow rate divided by the cross sectional area, i.e. the averaging is carried outwith respect to the area.

The Reynolds number, NRe, is a dimensionless quantity characterizing the ratioof inertial and viscous forces:

In this section we derive flow characteristics for the yield power law fluid and thenapply them for some special cases. The solution strategy is as follows: (1) derivethe velocity distribution, (2) calculate the maximum velocity, (3) calculate the flow(5.11)

(5.15)

and the characteristic length to be used in Eq, 5.12 is

I =2w.

will be useful in energy calculations.The flow rate vs pressure drop relation can be presented as friction factor vs

Reynolds number as well. The Fanning friction factor is defined as the proportionalitycoefficient between the wall stress and a suitable approximation of the kinetic energyper unit volume expressed in terms of the average velocity [2]:

(5.14)

One way to consider a hydraulically induced fracture is to envision a channel ofrectangular cross section with width w and height h, where wlh. _,. O. Figure 5.9shows the schematic of flow between parallel plates of "infinite" height. The ratioof the cross section to the wetted perimeter is

Ac hw w-= _,.-P 2(h+ w) 2'

(5.10)

5.2.1 Derivation of the Basic Relations

5.2 Slot Flow

The right-hand side of the equation represents the force driving the flow and the left­hand side is the force arising at the outer surface of the body keeping the balance ofthe system. If applied to the whole cross section, the equation gives the relation ofthe wall stress to the pressure gradient

For a circular pipe, I is the diameter, and for other geometries it is often called thehydraulic diameter. The cross section divided by the wetted perimeter is sometimesreferred to as the hydraulic radius. (The name is somewhat misleading because fora circular pipe it is half of the actual radius.)

While all the other variables are general in the sense that they do not refer to aspecific constitutive equation, the Reynolds number is an exception. It is inherentlyconnected with Newtonian behavior. Since for non-Newtonian fluids the "viscosity"is not unique, several choices are possible to extend the definition of the Reynoldsnumber. Whatever choice is preferred, for laminar flow a friction factor vs Reynoldsnumber relation is simply a restatement of the flow rate vs. pressure drop relationderived from the rheological model. For turbulent flow characterization, however,the concept of Reynolds number plays a central role.

(5.9).w = (;) ilZ·If applied to any smaller cross section symmetrically located around the center plane(parallel plates) or around the center line (circular tube) Eq. 5.8 states that the stressvaries linearly with the distance from the symmetry axis, since Ac/P is a linearfunction of that distance and the pressure gradient is a constant.

Once we know that the stress varies linearly along the coordinate representingthe distance of a contour line from the symmetry axis, we can obtain the shear rateand, by integration, the velocity profile. Knowing the velocity profile, a subsequentintegration enables us to determine the flow rate, q.The resulting flow rate vs pressuredrop relationship is the basic equation for the given flow geometry. Several additionalflow characteristics are of interest. These include the maximum velocity, Umax andthe average velocity uavg, where the averaging is carried out along the cross sectionavailable for flow; the ratio of the maximum to the average velocity and the kineticenergy correction factor, CtKE. The latter two quantities are different measures forthe flatness of the velocity profile. The kinetic energy correction factor, defined by

105Slot flowRheology and laminar flow104

------ ---------------_- __ -----. __ ..__.__ - __ ._--- ----- --------

Figure 5.10 Velocity profiles(S.23)Centerlinee[(W/2-y' )/ ]1/nu(y) - u(O) = Jo w/2 Tw - Ty K dj/,

(5.22)du _ [(W/2- y _ ) / ]1/ndy - w/2 Tw Ty K

The velocity profile is obtained by integrating Eq. S.22 from the wall to a givenlocation y:

Substituting the shear stress from Eq. S.17, we arrive at the differential equationdescribing the velocity field:

(S.21)

The shear rate is simply the derivative of the velocity with respect to the distancefrom the wall. Rearranging the specific constitutive equation (Eq. 5.4), we obtain

:; = c~TYr/"

(S.28)

Substituting Eqs. 5.24 and 5.25 into the latter two equations and carrying out theintegration, we arrive at the following expression for the total flow rate:

= hw2n(l- ¢)(1+ n + n¢) [Tw(1 - ¢)] 11"q 2(I+n)(1+2n) K

(5.20)(1- ¢)w2Yo =

which has to be less than unity; otherwise no flow occurs. In terms of the newlyintroduced variable, Eq. 5.18 takes the form

(5.27)

(5.26)qi = hew - 2yo)umax,

while qz is obtained integrating the velocity profile:

q2 = 2h 1)'0 u(y)dy.(5.19)Ty

¢=-,Tw

i.e. with increasing ratio of yield stress to wall stress the plug is wider and wider. Itis convenient to introduce a new variable for this ratio:

At a certain distance from the wall, Yo, the stress is equal to the yield stress. Inthe domain inside Yo the stress is not enough to cause velocity change and thereforethe fluid moves as a plug with the uniform velocity Urn•x. Clearly,

It is instructive to plot the different velocity profiles corresponding to the differentcases of the yield power law model studied in Figures 5.3,5.4,5.6 and 5.7. Assumingw = 20 mm and Tw = 25 Pa, the resulting profiles are shown 00 Figure 5.10. Theconstant velocity plug starting at a distance from the wall, Yo = 8 mm, can be wellseen for cases (a) and (c).

The total flow rate is the sum of the flow rate in the plug, qi and in the outerpart, q2. The plug flow rate is simply

(S.18)(Ty) WYo= 1- - -,Tw 2

2

(5.17)T=

With a linear relationship, the shear stress at a distance y from the wall (and actingin the direction normal to the flow path) is

(5.25)

where u(O) is zero because of the no-slip condition. The solution to Eq. 5.23 is notwell known in the literature:

u(y) = 2(n: 1) {W(l- ¢) [Tw(lK- ¢)] lin

_ [w(1- ¢) - 2y] [T",(1- ~- 2Y/W)fln}. (5.24)

The maximum velocity is obtained by substituting y = Yo into Eq. 5.24:

n [Tw(1 - ¢)] l/nUmax= 2(n + 1)w(1 - ¢) K

(5.16)

rate, (4) obtain the average velocity, and (5) determine the quantities characterizingthe smoothness of the velocity profile.

For slot flow the force balance takes the form

w6.pTw=U'

107Slot flowRheology and laminar flow106

_._ _--_._---------------------------------::,------

(5.41)_ [ 1+3n + 2n2 ] 6uavgYw= --,3(1-¢)n(1+n+¢n) w

The Reynolds number has been defined with respect to Newtonian fluids. The productj x NRe equals 24 for the Newtonian slot flow (as we will see soon). One possibleway to generalize the Reynolds number for non-Newtonian fluids is to preserve this

can be deduced from Eqs. 5.36 and 5.3. The right-hand side of Eq. 5.40 is an observ­able quantity, even if the fluid is non-Newtonian. It is called nominal Newtonianshear rate. WaH stresses, found experimentally, are often presented as a function ofthe nominal Newtonian shear rate. Such a curve is called a flow curve.

The general expression of the wall shear rate for the yield power law fluid, interms of the nominal Newtonian shear rate, is

(5.34)f = 4(1 + n)(1 + 2n)t~-1/n

(1-¢)u«:

uavgpw(1 - ¢)n(1 + n + ¢n) ~

(5.40). 6uavgYwN=-­

w

(5.33)¢= 2"CyL.f;.pw

Unfortunately, we cannot express the pressure gradient as a function of the flowrate. Si~c~ in en~inee~ng applications we most often wish to calculate the pressuredrop, this is a senous disadvantage, Nevertheless, any suitable root-finding numericalmethod can be applied.

Another form of the same relationship is the combination of Eqs. 5.11 and 5.29:

where

It is obvious that the corresponding relations can be obtained easily for the specialcases of Bingham plastic, power law and Newtonian fluids by substituting the specificvalues for the yield power law parameters. Table 5.1 gives a summary of the results.

When the wall stress (pressure drop) is not known, the corresponding relationsshown in Table 5.2 are more convenient to use.

In Table 5.2 we also show the relationship between the average velocity and wallshear rate. The shear rate at the wall for a Newtonian fluid

(5.32)q = hw2n(l- ¢)(1 +n + n¢) [W(I- ¢) f;.P] lin

2(1+ n)(l + 2n) 2K L

(5.39)

(5.38)wn(l- ¢)(1+ n +n¢).

uavg = 2(1+ n)(1 + 2n) Yw

I 6uavgpw(l- ¢)n(l + n + ¢n)NRe = (1+ n)(1 +2n)J1-w .

where the averaging operation is carried out on the cross sectional area available forflow.

The flow rate vs wall stress relationship can be written in several forms. Oftenthe pressure gradient is preferred over the shear stress. Then, a convenient formula­tion is

The latter two equations enable us to reformulate the expressions for the averagevelocity and for the Reynolds number:

(5.37)(K ) l/n

/I = <1-1/n __.....w w 1-¢

CXKE = _CU_av_g)_3= (2+ 3n )(3+ 4n )(1+ n + ¢n)3(u3)avg (1+ 2n )2(6+ 18n + II¢n + 18n2 +28¢n2 + 6n3 + 18¢n3)

(5.31)

Consequently, the apparent viscosity at the wall (wall shear stress divided by wallshear rate) is

(5.36)_ _ [(1- ¢)<w] l/nYw- K

(5.30)Umax 1+ 2n-uavg 1+ n + dm '

where, interestingly, this ratio is a function of nand ¢ only. The same is true forthe other measure of flatness, the kinetic energy correction factor

From Eqs. 5.4 and 5.19 we can express the shear rate at the wall:

Equation 5.29 does not contain the height, indicating that the flow has the samecharacteristics in any horizontal plane. This is a consequence of the assumptionwfh. -lo- O. The usual measure of the flatness of the velocity profile u II' can, max "avg,be obtained from Eqs. 5.25 and 5.29,

(5.35)( 1_¢)l/n6uavgpw(1 - ¢)n(l + n + ¢n) ~

N~e=(1+ n)(1 + 2n)<~-1/n

(5.29)Uav =!!._ = wn(l- ¢)(1+n + n¢) [<w(1 _ ¢)] l/n

g hw 2(1+ n)(1 + 2n) K

property. Therefore, NRe = 241j, andfrom which the average velocity is

109Slot flowRheology and laminar flow108

---~-,,-- ..,,--------------~-------------------------------, """-',,,.,,-,,------_._----------------------

, 2wu.vgP 12u;vgP<PNRc = --- = ---

Il-e Ty

2"-1 (1+2n)"" _ _ -- Kwl-"u"-1f'Ve - 3 1l 3'\18

( ) "-1 ( )"-11 + 2n 6uavg/).w=K -- --3n W

..:.( 1_----'-t/!):_:.(2_+_..:..<I>:__)wf,L", = Il-p + Ty 12uavg

T=KV

Power lawBingham plastic

Table 5.3 Different viscosities of non-Newtonian fluids for slot flow

(6uavg)T",=1l- --

W

A. 2 'f"'T'b 4rr + arccos[(1 +bo)-3]'I'=- V 1+ uo cos 3 xw-(n+1Ju~vg

4f,Lpuavgwhere bo= ---

WTy

6p = 2Ty

L w¢

. 6u.vgyw=-­

W

A fracture is h = 30 m high and w = 10 mm wide. Flow of a yield power law fluid(ry = 5 Pa, n = 0.6, K = 0.2Pa _5") results in a pressure gradient t:1p/L = 1.8 kPa/m.

. _ (1 + 2n) 6u,vgy",- ~ -;- Example 5.1 Flow in Fracture Assuming Slot Geometry

T=Jl.YT=KY'

Newtonian

Table 5.2 Wall shear rate, wall stress and pressure drop for slot flow

PowerlawBingham plastic

24j=­«;/=~

- N'p,<24

/=­NR~

Often it is convenient to characterize the fluid by the viscosity of a Newtonian fluidcausing the same pressure drop. This viscosity is called here equivalent Newtonianviscosity (but the terms effective viscosity and apparent viscosity have also beenused in the literature.) The equivalent Newtonian viscosity (for slot flow) is definedby the Hagen - Poiseuille law:N' _ 6nwuavgP

R.. - (1+ 2n)KI/"-r;-I/n w2 /:!,.pMe = ----, (5.43)

12uavg Land it is a property associated both with the fluid and the geometry of the flowchannel. From Table 5.2 it can be expressed both for a Bingham plastic and for apower law fluid. Table 5.3 shows both the equivalent Newtonian viscosity, J.Le =rw/YwN and the apparent wall viscosity, J.l.w= rw/yw' Also included is the Reynoldsnumber. Substituting the explicit expression for ¢ from Table 5.2, one can obtainexplicit expressions for a Bingham plastic fluid. This exercise is left for the reader.

N _ 2wu.vgPR<- ---

f,L

hw2n (w i!>p) lInq = 2(1 +2n) 2K T

(2+ 3n)(3 + 4n)«xe = _;_-;6-::(1:-+~2-n::::)2~

35(2 + ¢)3aKE = 27(16 + 19t/!)

Uavg 2U",a. 3-=-

Uavg 2 + t/! 5.2.2 Equivalent Newtonian ViscosityUrn"" 1+ 2nU.vg= l+n

wn (Tw)I/nUavg = 2(1 + 2n) K

WT..Uavg=-

61l-

In the section describing tube flow, we give a detailed discussion of the relationbetween the flow curve and the true rheological curve.

Equation 5.42 is an implicit relationship because ¢ is defined in terms of the wallstress. For a Bingham plastic fluid an explicit expression is available for ¢ (in termsof the average velocity) as seen from Table 5.2. For a power law fluid, the expressionis explicit because ¢= O.

= hwzn (TIV)]/.q 2(1 +2n) K

wn (T•.) I/.Um"" = 2(1 +n) K

(5.42)T =Kyn

and therefore the dependence of the wall stress on the nominal Newtonian shear ratecan be given as

[2] n n1+ 3n + 2n 6uavg

r",= r +K ---y 3(1-¢)n(1+n+</m) (W)Newtonian

Table 5.1 Bingham plastic, power law and Newtonian formulas for slot flow

PowerlawBingham plastic

111Slot "owRheology and laminar flow110

"~---------------------------------------_--

All the equations above are easy to use if the wall stress (pressure gradient)is known. In engineering practice usually the flow rate (average velocity, nominalNewtonian shear rate) is specified and the wall stress (pressure gradient) has tobe determined. This requires an iterative solution procedure. Efficient root-findingmethods such as Newton's method can be applied.Figure 5.11 Schematic of tube flow geometry

ao = 36n5 + 159n4 +279n3 + 243n2 + 105n + 18,

al = 108n5 + 390n4 + 522n3 + 306n2 + 66n,

oz = 216n5 +477n4 + 350n3 + 85n2.

where(5.50)

= 4'pand

(5.44)D

(5.49)Uavg

Umax (1+ 2n)(1 +3n)~ = -::--::-----,.-.:.....,..--'-:::-----::------::---;;--;;-(2n2 + 3n + 1)+ (2n2 + 2n)¢ + 2n2</J2'If D is the diameter of the pipe (see Figure 5.11), then the ratio of the cross section

to the wetted perimeter is

The flatness measures take the (somewhat complex) form:5.3.1 BasicRelations(5.48)5.3 Flow in CircularTube

nD [TwClK- ¢)r/n (1 - ¢)[(2n2 +3n + 1)+ (2n2 + 2n)¢ + (2n2)¢2]

uavg = 8(1 + n)(1 + 2n)(1 +3n)

Neglecting the yield stress, but retaining the same K and n, less than one-third of theoriginally given pressure gradient is calculated. 0

The derivation of the relationships follows the pattern introduced for slot flow. Theaverage velocity is given by

At the same flow rate (q = 0.0324 m3/s) but neglecting the yield stress, the pressuregradient from Table 5.2 is

(5.47)2

(1- ¢)DYo =

Similarly to the flow between parallel plates, at a certain distance from the wall,Yo, the stress is equal to the yield stress. Within Yothe stress is not enough to causea velocity change and the fluid moves as a "plug" with the uniform velocity, umax.The distance Yo depends on the ratio of the yield stress to the wall stress, 4>, and canbe expressed as

= hw1n (1 - .p)(l + n + n4» [weI - 4» c,.p] lin = 0.0324 m3/s (12.2 b m).q 2(1+n )(1+ 2n) 2K L P

2therefore, flow rate is determined (from Eq. 5.32) as

(5.46)(~-y).

D Tw·.=2ryL4> = -- = 0.556;tipw

From Eg. 5.33

The other consequence of the force balance is the linear variation of the shear stress.The shear stress at a distance y from the wall is

Solution

(5.45)Df1p·w= 4[.

and, following Eq. 5.13, the characteristic length is D. The force balance applied tothe total cross section gives

What is the flow rate assuming slot flow? What is the calculated pressure gradient atthe same flow rate if the yield stress is neglected? (Assume laminar flow. We will seein Chapter 6 how to check the validity of that assumption.)

113Flow in circular tubeRheology and laminar flow112

tJ.pL and r/! as above, where

bo = 6IJ.puavgDry

and, thus, the wall stress

(6n3 + lln2 + 6n +1 ) It (8uavg)"

t - T +K --w - Y 8n3 + 12n2 + 4n - (4n2 + 4n)cp - 8n2<J>2 - 8n3cp3 D

(5.52)

(5.51)

The 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. The wall shear ratefor a yield power law fluid depends on both nand <J>

. 6n3 + lln2 + 6n+ 1 (8Uavg)y - ---w - 8n3 + 12n2 +4n - (4n2 + 4n)<J> - 8n2<J>2 - 8n3<J>3 D

5.3.2 Flow Curve

6.: = 23n+2)f-nK ( 1: 3nrxD-(3n+l)qn

wherebo = 24IJ.pq

1rIJ3r:)'

$._ _8(;_1-=+=-b.:.:_o)_ bl$.4> = --!...._.._~2:...:....:...-~

bl =2(b;-I/3+ b~/3)bz=l+2bo+b~

+ J-4b-o-+-6'-,b5,....+-4-b~03-+-b-ri

6.p 4ry

L Dr/!

For the specified flow behavior index, the dissipation rate, D., decreases with the 4.5thpower of the diameter, if the flow rate, q is fixed (see first row). Similarly, if the pressuregradient is kept constant, D. increases with the cubed diameter (see the second row).Finally, from the last row the interesting observation is made that D; is constant, if thenominal Newtonian shear rate is fixed. 0

r -!LYr = KY"Binghamplastic Powerlaw Newtonian

Table 5.5 Wall shear rate, wall stress and pressure drop for different fluids

We can eliminate either the pressure gradient, the flow rate, or both. From Tables 5.4and 5.5 the following expressions are derived:

o, = [23n+4)f-"-1c:3n ) " KqJ.;·n] D-(3n+3) = [(1:3n) (4%) (Hn)/.] D(l+·)/'

NRe = DuavgpIJ.

16f=­

N~,

Solution

1rD3n (D 6.p) lin

q= 8(1+3n) 4KL

(1+2n)(3 + 5n)CiKE = 3(1+ 3n)2

Um"" 1+ 3nuavg == l+n

35(3 + 2cp+qi)3CtKE = 54(35 + 5&/1+47r/!2)

q x 6.pDv = :rrD2L .

4Derive an expression for the dissipation rate of a power law fluid flowing in a pipe.How does the dissipation rate vary with the diameter if n = 0.57

u""'" = __ 6__uavg 3 + 2cp+ (P

Dn «,,)1/"U,vg = 2+ (1 + 3n) K

The power required to pump an incompressible fluid in a straight horizontal conduitis given by q x tsp, The power per unit volume that goes into viscous losses is thedissipation rate (Denn, [5]):

o-;uavg =s;

Example 5.2 Dissipation Rate

Dr",Umax = -

4/L

:rrrYn (-r,,) 11"q = 8(1 + 3n) K

Dn ('w)l/"Urn"" = 2(1 + n) K

D(1-rpf,,,Umax =

'=IJ.Y, = KY"Powerlaw

The special cases of interest are summarized in Table 5.4. Several equations shownin the last columns are often called the Hagen-Poiseuille law.

Table 5.5 shows appropriate equations when the flow rate (or average velocity) isgiven. Note the explicit solution available for a Bingham plastic fluid.

Newtonian

Table 5.4 Bingham plastic, power law and Newtonian formulas for tube flow

Binghamplastic

115Flow in circular tube

-------_ .._--.. _..-..__ ...-_.-.... _ .._--_ ..._-----_ ..-._--_ ......__ ..._ ..._.- .._ .. --------------

Rheology and laminar flow114

Figure 5.12

which states that the actual wall shear rate can be obtained from measurable quan­tities: the flow rate and its semi-logarithmic derivative with respect to the pressuregradient.

10310" 1~ 1~

Shear rate, Y (1/s)

Rheological curve and tube flow curve

10-' 10"

(5.59). 1 [ (~P) dq jYw = rrD3 24q + 8 L d ( ~:) •

10"

'Cij'!e:_p

.s<I) 10'~u;~(I).c:en

and K' is defined to give the measured wall stress if substituted into Eq. 5.57.PNaturally, both parameters are varying along the flow curve except for a powerlaw fluid.

From Eq. 5.51 we see that the wall shear stress is a unique function of the nominalNewtonian shear rate for a rather complex fluid such as the yield power law fluid.One of the main results of rheology is that a similar correspondence exists for anyconstitutive equation. Knowing the flow curve obtained in a tube of any diameter,the information is sufficient to determine laminar pressure drops in (other) pipesprovided we do not leave the range of nominal Newtonian shear rates correspondingto the experiments. Moreover, knowing the flow curve (or a part of it) we canconstruct the true rheological curve (or a part of it). This exercise is based on theRabinowitsch-Mooney equation [6,7]:

10'

(5.5S)(5.55)

In this case, however, n' is defined as the log-log slope of the flow curveor, in a simpler form

(5.57)_ K' (8Uavg)n''w - PD'(5.54)[ ( 1+3n) (8U)]log(,w) = log(K) +n log ~ + log ;Vg .

Unfortunately for other fluids, the flow and rheological curves differ more signif­icantly. Formally, it is possible to write the wall stress vs nominal Newtonian shearrate relationship for the general fluid in the same form as for the power law fluid:

(5.56)(4n )n

K = 1+3n Kp.The advantage of the nominal Newtonian shear rate is that it can be observed.

The plot of the wall stress as a function of the nominal Newtonian shear rate is theflow curve. The difference between the true rheological curve and the flow curve isshown in Figure 5.12.

The flow curve coincides with the true rheological curve only if the fluid isNewtonian. The flow curve and the true rheological curve have still similar shapesin the case of a power law fluid. Indeed, on log-log paper both curves will be straightlines. From Table 5.5 we have

Equation 5.55 tells us that the slope of the log-log straight line is the flow behaviorindex, n, the same as in the case of the true rheological curve. (No base is given forthe logarithm since both the ten-based or natural logarithm can be used.) The interceptat unit nominal Newtonian shear rate is (1 + 3n /4n)n K, and it is often denoted byKp- As seen from Eq. 5.55, the true consistency index, K, can be obtained from theintercept by applying the relation

rJ1-w = y (5.53)

¢ ( 6n3 + lln2 + 6n + 1 ) (SUDavg)Sn3 + 12n2 + 4n - (4n2 + 4n)¢ - 8n2¢2 _ 8n3¢3

is not a simple function of the nominal Newtonian shear rate. The apparent viscosityat the wall for a yield power law fluid can be given as

117Flow in circular tubeRheology and laminar flow116

---- - ._--- -_. - -- -----------_._. ---- ----

N' _ uavgpDR ---, J.1.e

N' _ 4nuavgpDRe - (1+3n)JLw

N' _ UavgpD(l - t,b)(3 + 2¢ +¢2)R. - JLw

(1+ 3n)"JL =2"-3 -- KD1-nun-1

t n avg

( ) "-1 ( )"-11+3n 8uavgJLw=K ~ D

r= Kv"

PowerlawBinghamplastic

Table 5.6 Viscosities and Reynolds number for non-Newtonian Fluids (lubeflow)

In other words f1.e = <wlY",N. The name apparent viscosity is also common for thesame quantity but we prefer the adjective "equivalent" to avoid confusion with thegeometry independent apparent viscosity defined by Eq. 5.6. The apparent viscositycalculated at the actual wall shear rate is called wall viscosity, Jot", = <",IY",.Table 5.6shows the different viscosities and their relation to the generalized Reynolds number,N~ for tube flow. In fact all these equations are mere restatements of the wall shearstress vs. average velocity relationship. The apparent viscosity curve on Figure 5.13is a property of the fluid while the equivalent Newtonian viscosity curve is validonly for the given geometry. The definition of the equivalent Newtonian viscositymay also be applied to non-laminar flow conditions. This practice, however, is notrecommended.

(5.65)D2 D.p

Jote= ---.32uavg L

By virtue of the Hagen-Poiseuille law for tube flow, the equivalent Newtonianviscosity for a non-Newtonian fluid is defined as

5.3.3 Equivalent Newtonian Viscosity for Tube Flow

is applied to "adjust" the pipe value. Note that in this book, subscript p is used if thecoefficient is defined in terms of nominal Newtonian shear rate for pipe flow. Themeaning of the prime is somewhat different. It indicates an experimentally derivedvalue valid only for a portion of the flow curve. Other conventions are also usedin the literature. Some authors do not use the subscript p, and other authors mightprefer the superscript prime for the true power law parameters. The reader shouldbe cautious if data denoted by K' and n' are given without explanation.

119Flow in circular tube

(5.64), (4n' )n' ,K= -- K1+ 3n' P

holds, since for a general fluid the two slopes are different. If, however, there isan interval of nominal Newtonian shear rates, in which n' is constant, then the truerheological curve has a part in which N is also constant and agrees with n',

In principle, Eq. 5.59 can be used to convert a flow curve into a true rheo­logical curve. The graphical or approximate numerical differentiation to obtain n'should be carried out at every point. Then the true rheological curve (wall stressvs. true wall shear rate) can be constructed from point to point. Because of theuncertainty in graphical and numerical derivative calculation, this procedure is notrecommended.

The application of computers necessitates models with numerical parameters ratherthan the traditional, graphically presented, curves. This leads to the problem of patternrecognition and model identification. If the fluid is Newtonian, the flow curve is astraight line drawn through the origin. If the fluid obeys the power law, we obtain astraight line when plotting on log-log coordinates. For the Bingham plastic and yieldpower law fluid, neither type of plot can give a well recognizable pattern. Thus,the selection of a constitutive equation is not trivial. Once an equation is selected,a nonlinear parameter estimation procedure can be applied to fit the solution to agiven constitutive equation (e.g. Eq. 5.48) to the measurements.

A further observation on K~ can be made. As defined by Eq. 5.57 it refers to pipeflow. Often the transformation

(5.63)• _J. (1 + 3N) 8Uavgy", -r- 4N D

In general the inequality

(5.62)n'

N = 1 dn'1-----

3n' + 1 d(ln <",)

and compare the two derivatives at the same stress value, we obtain the relationship(Babok and Navratil [8])

(5.61)d(ln r)

N=--d(ln y)'

may suggest the misleading impression that every fluid behaves like a power lawfluid. This is not true. If we define N as the similar derivative of the true rheologicalcurve,

(5.60). = (1 + 3n') 8uavgYw 4n' D'

It is tempting to use Eq. 5.57 to characterize a general fluid. The well known formof Eq. 5.59,

Rheology and laminar flow118

-----.- _-------------

(3 - if} _.p2 - ¢)D . (3 - 0.673 - 0.672 - 0.67) x 0.03f-Lw= f-Lp+ Ty = 0.21 +5.4..;__--------.;.._--

24uavg 24 x 0.025

=0.64Pa,s.D

and

Dry 0.03 x 5.4f-Le= 8uavg¢ = 8 x 0.025 x 0.67 = 1.21 Pa- s,

The required viscosities are calculated from the equations in Table 5.6:

bo= 6iJ..puavg = 6 x 0.21 x 0.025

-c:-:-,----,-- = 0.194,Dry 0.03 x 5.4

b2 = 1+ 2bo +b5 +V4bo + 6b6 +4b6 + b~= 2.44,

b! = 2(b2"IJ3 + b~/3) = 4.18.

8(1 +ho)----;o=---b1

,,;r;; = 0.67.2

The key quantity to determine is the ratio of the yield stress to the wall stress, .p. FromTable 5.5

Solution

Calculate the equivalent Newtonian viscosity, /le, and the wall viscosity, iJ..w, of aBingham plastic fluid (ry = 5.4 Pa, f-Lp= 0.21 Pa-s") flowing in a tubing of innerdiameter, D = 3 em. The average velocity is uavg = 0.025 m/s.

Example 5.4 Viscositiesof Bingham Plastic Fluid

A plot of the function (1 + 3n/4n)" (see Figure 5.14) reveals that the maximum occursat 71= 0.241, where (1+371/471)" = 1.15. Thus f-Le::: 1.15 K[sl-n]. 0

(1+3n)" (1+371)"u, = 4;;- Ky~:;:/ = ~ x K X [51-"].

From the definition of the equivalent Newtonian viscosity and using the expression forthe wall stress in Table 5.5

Solution

Show that if the nominal Newtonian shear rate is l/s, the equivalent Newtonianviscosity of a power law fluid cannot be greater than 1.15 [sl-"]K.

Example 5.3 A Limiting Property of Power Law Fluids

121Flow in circular tube

Figure 5.14 Plot of the [(1 +3n)/(4n)]" function

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Flow behavior index, n

0.1o

Figure S.13 Apparent and equivalent Newtonian viscosity

Shear rate,Y (l/s)lao 10210'10"

Rheology and laminar flow

1.16

1.14

1.12

1.1s,~::!:- 1.08<,

'2C')

+ 1.06~~~1.04

1.02

~ 10''"tee:..:::t~'"0o'"s lao

120

where £(m) is the complete elliptic integral of the second kind and m = 1 - DVD~.(The above equation gives the perimeter ofa circle for m = 0 since £(0) = rr/2.)Applying the definition, the characteristic length is

(5.69)with perimeter

12u.vg [ 4K L ] lIn

Dz(l - w) Dz(! - w) 6.p

3n= 1+2n

48J.LpUavg LD~(l- w2) 6.p

= 1- 6ry LDz(l- w) 6.p

+0.5 [ 4ry ~] 3D2(1- w) /:,p

NewtonianPower law(approximate)

Bingham plastic(approximate)

If we denote the smaller diameter by DJ and the larger diameter by D2 (seeFigure 5.16) the cross sectional area is given by

Table 5.7 Annulus flow

5.4.2 Flow in Elliptic Cross SectionFigure 5.15 Schematic of flow geometry in annulus

Using trial and error (or some modern computer software) it is readily determined thatthe solution is (I) = 0.78. Since, co = DdO.Ol m, the inner diameter is DJ = 7.8 mm. 0

0012 (~ 1- w2)1= . x 9.81 x lif 1+ or: + -1-- .32 x 0.001 nw

i.e.

~ (6.p) ( 2 1-W2)Uavg=- - 1+w +--32JL L friction In w

From Table 5.7

( 6.p) = pg = 9.81 kPa/m.L friction(5.67)

where D2 is the larger and DJ is the smaller diameter. The velocity profile dependson the ratio

Solution(5.66)

Water (p = 1000 kg/rn", JL= 1 m Pa- s) flow is driven by gravity (g = 9.81 mls2) in avertical annulus of larger diameter, D2= 1 em. Calculate the smaller diameter to assureuavg = 1m/s,For the flow geometry shown in Figure 5.15 the characteristic length is

5.4.1 Flow in Annulus

Example 5.5 Annulus Design

In hydraulic fracturing we have to deal with flow through channels of more complexshapes. The annulus between two concentric cylinders is of some interest because'fracturing fluid is often pumped down through the annulus between the tube andcasing of a hydrocarbon well. For the description of the flow in the fracture itself achannel of elliptical cross section and infinite aspect ratio can be postulated.

For Newtonian fluids a closed form solution can be obtained, while for the Binghamplastic and power law fluids, good approximations are available (Whittaker [3];Savins [9]). The approximate formulas in Table 5.7 give less than 1% error forw > 0.5 and n > 0.2.

The product f x NRE varies from 16 (if w -+ 0) to 24 (if co -+ 1).

5.4 Flow in OtherCrossSections

123Flow in other cross sectionsRheology and laminar flow122

-.-~..---..- ..--- .. _..-----------..--~.-..-----~-----------~--

(5.80)Llp 2n+5 (1+ 2n) n n -n -(2n+l)- = -- --- Kq h w ,L 3rr n

Solutions for non-Newtonian fluids are not available for the limiting ellipsoid crosssection. The importance of the power law equation for fracturing fluids necessitatesa formula similar to the one available for slot flow. Attempts have been made tosuggest a reasonable solution to this problem. In the petroleum engineering literaturethe reasoning of Perkins and Kern [11] is accepted. They compared the pressuregradient for the flow in a slot with width wand in a limiting ellipsoid cross sectionwith maximum width Wo = w, provided that the flow rate of the Newtonian fluid isthe same. The ellipsoid pressure gradient 16/(31l') times larger than the slot one asseen from Eq. 5.74 and from Table 5.2. Perkins and Kern assumed that the samerelationship holds for power law fluids and therefore

(5.79)-r- (2TrUavg)Yw= --- .Wo

showing that the average wall shear stress is

(5.78)_ (2TrUavg)Tw=J.L --- ,Wo

An important rearrangement of Eqs. 5.73 and 5.76 is

(5.77)

The wall stress is not constant along the perimeter and the definition of the frictionfactor refers to the average wall stress, denoted by rw' Thus,

(5.76)rrwQuavgPNRe = -----"'-211-

where

Figure 5.17 Schematic of limiting ellipsoid flow geometry

L

125Flow in other cross sections

(5.75)f X NRe =2rr2,

and

(5.74)Llp 64J.Lq-=--L rrhw~'

i.e.

(5.73)w2 LlpU __ 0 _

avg - 16J.LL '

For hydraulic fracturing applications, the case with infinite aspect ratio, i.e. m = 1,is of great importance. To see the analogy with slot flow, we use the notation Wo forthe smaller diameter (i.e. the maximum fracture width) and h for the larger diameter(fracture height) as indicated in Figure 5.17. Substituting E(l) = 1 into the aboveequations we obtain the expressions for Newtonian flow in limiting ellipsoid crosssection

5.4.3 Limiting Ellipsoid Cross Section

(5.72)x N = 2rr2(2 - m)

f Re [E(mW'

and the same relationship rewritten in friction factor vs Reynolds number form is

(5.71)

For Newtonian fluids the average velocity is a simple expression of the pressuregradient (Happel and Brenner [10]):

(5.70)1= TrDl .2E(m)

Figure 5.16 Schematic of elliptic cross section flow geometry

Rheology and laminar flow124

K = 0.004 Ibf-s" /ft2 = 0.192 Pa s",

We have to decide how to interpret the given rheological parameters. In the absence ofadditional information the best we can do is to assume that K' is an adjusted (geom­etry independent) power law consistency index. Therefore, we use power law withparameters K' and n = n', Using SI units the data are

Solution

A fracturing fluid is reported to have properties K' = 0.004 lbf-s" /ft2, n' = 0.5 andp = 65.6 lbm/fr'. For a limiting ellipsoid cross section of maximum width, Wo = 0.3in, calculate the Reynolds number, the friction factor and the pressure gradient. Theaverage linear velocity is uavg = 2 ftJmin. Compare the pressure gradient with flow ina slot of width w = 0.3 in.

Example 5.6 Flow in Fracture Assuming Limiting Ellipsoid and SlotCross Section

can be readily obtained. While for Newtonian fluids (n = 1) all the listed expressions(Eqs. 5.80, 5.83, 5.87) give the known pressure gradient, the limiting behavior at lowflow behavior indexes is different as shown in Table 5.8.

From Table 5.8 it is obvious that only Eq. 5.87 results in the desired expression fornear zero flow behavior index. Some other consequences of the suggested equationare given in Table 5.9.

(5.87)~p =~ [1+ (rr - l)n]n (2lTUavg)nL lrWo rrn Wo

from which

[ In-I ( )"-1f.l.w =K 1 + (1T - l)n 2Jtuavgitn. Wo

/).P (8)"+1 [l+(rr-l)n]n K "h-" -(z,,+I)T = -; n q Wo

6.P -23+" -1 [l+(rr-l)n]n K n -1"+1)T - it n UavgWo .

Table 5.9 Limiting ellipsoid flowof power law fluids accordingto Eq. 5.87

127Flow in other cross sections

------_. __ .._._ - ..

Tube Slol Limiting Limiting Limitingellipsoid ellipsoid ellipsoid

(Eq. 5.80) (Eq. 5.83) (Eq. 5.87)

2uavg 2u,vg ( ~) lIn JrUavg 1tUavg 2uavg

Dn wn 3 2wOn 2won wOn

Table 5.8 Wall shear rates at near zero flow behavior index forpower law fluid

(5.86)

In other words, the same dependence of the wall shear rate on the flow behaviorindex is assumed for limiting ellipsoid flow that has been found for tube flow. Theanalogy is, however, not satisfactory if n is nearly zero as seen from Table 5.8.

A closer look at the "small n behavior" of the wall shear rate both for pipe andslot flow (see the first two columns of Table 5.8) reveals that (if the problem has asolution at all) the limiting ellipsoid wall shearshould behave as 2uavglwon. Basedon this analogy in this work we suggest the following wall shear rate expression

(5_85)

i.e. the wall stress is assumed to be

(5.84)~p = 8K (1 + 3n ) n (2rruavg) n ,L lrWo 4n lTWO

from which one can reveal its origin:

(5.83)/::;.p = 8K (0.3048 + o.9253n)n (16Uavg)nL l!WO n lrWo

Such a wall shear stress dependence on the flow behavior index seems to be unusualbecause of the exponent lin. (At small n the wall shear rate tends to infinity muchfaster than for a slot flow.)

Kozicki and Tin [12J suggest another formula for limiting ellipsoid flow

(5.82)

The explicit expression of the (average) wall shear rate can be readily recon­structed from Eq. 5.81:

(5.81)~p = 25-n3-1 n-l (1+ 2n) k n -(2n+l)L it n uavgw .

i.e.

Rheology and laminar flow126

-----------_._" .._--------_---------_.

1. Metzner, A.B., Flow of Non-Newtonian Fluids, in Handbook of Fluid Mechanics,V.L. Streeter (ed.), McGraw-Hili, New York, 1961.

2. Ullmann'sEncyclopedia of IndustrialChemistry (ed. Barbara Elvers) Vol. B,1. Funda­mentals of Chemical Engineering, Chs 4 and 5, WCH Weinheim, FRG, 1990.

3. Whittaker, A. (ed.): Theory and Application of Drilling Fluid Hydraulics, IHRDC,Boston, MA, 1985.

4. Chakrabarti, S., Seidl, B., Vorwerk, J. and Brunn, P.O.: The Rheology of Hydroxypropy­Iguar (HPG) Solutions and its Influence on the Flow Through a Porous Medium andTurbulent Tube Flow, Respectively (Part 1), RheologicaActa, 30, 114-123, 1991.

5. Denn M. M.: ProcessFluidMechanics,Prentice Hall, Englewood Cliffs, N.J., 1980.6. Rabinowitsch, B. Z. Phys. Chem.,1, 145A, 1929.7. Mooney, M., 1.Rheo!., 2,210, 1930.8. Bobok, E. and Navratil, L.:Folyasi gorbek turbulens tartomanyanak meghatarozasa cso­

viszkozirneterrel, (in Hungarian) Koolaj esFoldgtiz, 24, 135-144, 1991.9. Savins, J.G.: Generalized Newtonian (Pseudoplastic Flow in Stationary Pipes and

Annuli, Pet. Trans.AIME, 213, 1958.

References

Thus, slot flow at the same average velocity results in less pressure drop than limitingellipsoid flow if the width of the slot is equal to the maximum width of the ellipsoid. 0

sr (1+2n)"L = 2"+1Kw-(n+l) -n- U~vg= 521 Pa/m (0.0231 psi/ft).

The pressure gradient assuming slot flow is obtained from the expression contained inTable 5.2:

The pressure gradient can be determined either directly from Eq. 5.86 or from

t:>.P _ 8 I 2 8 1 2-L - -f 'j_UavgP = X 0.371 x 2" x 0.102 x 1050

1lWo :rr x 0.00762

= 673 Palm (0.0297 psi/ft).

and

From Table 5.9

10. Happel, J., Brenner H.: Low Reynolds Number Hydrodynamics,Prentice Hall, Engle­wood Cliffs, N.J., 1965.

11. Perkins T.K. Jr. and Kern, L. R.: Width of Hydraulic Fracture, JPT, (Sept.), 935-49,1961; Trans.AIME, 222, 1961.

12. Kozicki, W. and Tin, c.: Parametric modeling of Flow Geometries in Non-NewtonianFlows, in Encyclopediaof FluidMechanics, (Cheremisinoff, units ed.), Gulf, Houston,Vol 7, pp. 199-252, 1986.

n =0.5,

Wo = 0.00762 m,

uavg = 0.102 m/s,

P ;: 1050 kg/nr'.

129ReferencesRheology and laminar flow128

........._-_.- .._----_._._--------------------------:::------------

The relative roughness can be considered as a value on an empirical scale, origi­nally defined by Nikuradze [4] who created artificial roughness by gluing sand to theinner wall of smooth pipes. In practice, the relative roughness is either assumed or

_1__ -410 {..!._ _ 5.045210 [81.1098 (7.149)0.8981]}.Jl- glO 3.7 NRe glO 2.8257+ NRe . (6.2)

provides a basis to calculate the friction factor numerically. The inherently iterativemethod can be substituted by an explicit approximation offered by Chen [3]

(6.1)1 ( e 1.255).Jl = -410g10 3.7 + NRe.Jl

Transitional and turbulent flow has been studied extensively for simple geometriesand fluids. For a wide class of Newtonian fluids the friction factor, f, in a tubeis determined by the Reynolds number, NRe, and the relative roughness of thewall, e. The log-log plot of f versus NRe, shown in Figure 6.1, is the well knownMoody diagram [1] generated with an explicit relationship presented below. TheColebrook- White [2] equation

6.1.1 Newtonian Fluid

6.1 Non-laminar Flow

The transitional and turbulent behavior of fracturing fluids has not been studiedadequately and a similar statement is even more true for the transport of solids bythese fluids. The general approach is to find the essential variables and/or their combi­nations and to present the relationship between them in graphical and/or algebraicform. This chapter is devoted to relations relevant to the flow and solids transport inthe fracture and in the well tubulars.

NON-LAMINAR FLOWAND SOLIDS TRANSPORT

6

Before the actual calculations we have to interpret the given data carefully. Since thesubscriptp is given explicitly,we use this value as a geometrydependent(pipe) consis­tency index: K~ = 0.012 Ibf-s"'/ft2 = 0.575 Pa s"'. Note that the adjusted (geometryindependent)valuewould be K' = 0.216 Pa- sn', but we do not need it, because Eq. 6.4canbe used directly.The Reynolds numberis obtainedfrom thediameter,density,linearvelocity and the pipe consistency index:

(6.4)

Solution

A fracturing fluid (K~ = 0.012 lbf-s" /ft2, n' = 004, p = 65.5 Ibm/fr') is pumpedthroughD= 2.259 in. tubing. The flow rate is q = 20 bpm. Neglecting possible dragreduction and assuming a relative roughnessof 0.001, calculate the frictional pressuregradient.

(6.3)

Example 6.1 Turbulent Flow of a "Generalized" Fluid

Naturally, the BNS equation reduces to the Colebrook-White equation for n' = 1.

(6.7)y =0.8295 + 1.405 _ 1.5111/n' (0.3535 + 1.061) .n' n'

where

(6.6)1 [lOY e ]- --410 --J7 - g10 (Re,)1/n' f(2-n')/(2n') + 3.715 '

Both f.l-~ and NRe reduce to the corresponding power law values (see Tables 5.4and 5.6) if the fluid obeys the power law. The basic assumption is that from the pointof view of the turbulent behavior any fluid can be replaced hypothetically by a powerlaw fluid. The replacing power law fluid is selected to give locally similar pipe flowbehavior in the corresponding shear rate range but under laminar conditions. In thisformalism the use of the symbol K~ underlines the practical character of the approach.Indeed, only observable variables are involved in the transfer of information fromthe laminar to turbulent regime. The specific use of the power law is not an absolutenecessity and other authors prefer to use a generalized Reynolds number derivedfrom other models (but still obeying fNRe = 16, if the flow is laminar.) Such aconstruction is given in Table 5.6 for the Bingham plastic fluid.

For a Bingham plastic fluid the above "generalized power law" treatment can becompared numerically to more accurate turbulent flow description. Hanks [6] showsthat the deviation could be significant and, therefore, the generalized power lawconcept should be used with caution.

The Dodge-Metzner approach has been extended for rough pipes by Szilaset al. [7]. Their BNS equation is rewritten here in terms of the Fanning frictionfactor:

(6.5)(8 ) n'-l, = K' uavgf.l-e P D

where

133Non-laminar flow

, uavgpD gl-n'u~;tpDn'N - -- - -____;~--

Re - /I' - K'r:« p

Here the meanings of NRe and n' require more clarification. First, the nominal Newto­nian shear rate is calculated. Note that it is nominal not only because the fluid isnot Newtonian but also because, in spite of the different velocity profile due toturbulence, the calculation involves a hypothetical laminar velocity profile. Second,it is also assumed that the laminar flow curve in the vicinity of the nominal Newto­nian shear rate can be described by Eq. 5.57 introduced in Chapter 5. The Reynoldsnumber is calculated from

_1 4_ ' (2-n')/2 _ ~VI - (n,)O.75 loglO(NRef ) (n')1.2 .

For a wide range of non-Newtonian fluids flowing in a smooth pipe the approach ofDodge and Metzner [5] can be applied:

6.1.2 GeneralFluid

determined experimentally from the Colebrook-White equation using a fluid (e.g.,water) known to obey it.

Reynolds nurnosr, NRe

Figure 6.1 Frictionfactorvs. Reynoldsnumber

10" 10610~1(t3

1()2

O.OS0.040.03 ;;00.02 !!!.0.015

~0.Q10.006 (1)0.004 ;;00.002 0c::;0.001 <C0.0006 :T

::J0.0002

(I)<J)

0.0001 !"0.00005 M0.000010.000001

~

~

~ I- ~IIiIII

I'IIItiiII[:::I I'

10"'

Non-laminar flow and solids transport

.......:~Sc::;

~1~

CD.sc::;c::;asLL

132

(6.10)DuavgpNRew= ---.

, tJ-w

Explicit expressions for the wall viscosity of the Bingham plastic and power lawfluids were given in Table 5.6. Note that not even for a power law fluid the wall

Most fracturing fluids have one or two orders of magnitude higher polymerconcentrations than the solutions studied by early investigators of the drag reductionphenomenon. Therefore, it is natural .ro assume that maximum drag reduction isin effect for these shear thinning fluids. Nevertheless, the question is still openwhether the MDRA can be applied numerically to shear thinning fluids. The principalreason of the uncertainty is that one has to decide which Reynolds number to usein conjunction with the above equation. Virk [8] criticized the early practice to usethe solvent Reynolds number (i.e., water Reynolds number for water based polymersolutions) and stressed the importance to correct for the shear thinning. Moreover,he stated explicitly that "a Reynolds number formed with the apparent wall viscosityseems preferable to the generalized power-law Reynolds number". The wall Reynoldsnumber is calculated with the wall viscosity

(6.9)

to Eq. 6.8 is available (Denn [9]):

In(!) = 28.135 + (-29.379 + (8.2405 - 0.86227x)x)x,

where x = In[ln(NRe)].

Wall Reynolds.number, NRe,w

Figure 6.2 MaximumDrag Reduction Asymptote

10· 107105101

MDRA11111111

II IIIIv

135Non-laminar flow

----._,,_ ..__ ..

providing the lowest possible friction factor at a given Reynolds number. The plotof the MDRA is shown in Figure 6.2. A sufficiently accurate explicit approximation

(6.8)

Most fracturing fluids which are polymer solutions exhibit drag reduction in turbulentflow. The term drag reduction can be applied if a certain fluid shows Newtonianbehavior in laminar flow, but for higher Reynolds numbers the friction factor is belowthe value calculated from the Colebrook-White equation. As a first approximation,a similar definition might be applied to a general fluid: drag reduction means thatthe turbulent friction factor is significantly less than the one calculated from theDodge-Metzner relation. A somewhat simplified view of drag reduction envisions thelong polymer molecules dampening the instabilities and hence reducing the turbulentenergy dissipation. As a consequence, the transition from laminar to turbulent regimeis gradual.

The concept of drag reduction was originally used to indicate the reduction offriction pressure when small quantities of high molecular weight polymers wereadded to turbulent water flow (Virk, [8]). When the addition of the polymer is solimited that the water viscosity remains constant, the comparison can be made atthe same flow conditions characterized by the solvent Reynolds number. It has beenobserved that the addition of polymer decreases the friction factor only to a limitingvalue. For dilute polymer solutions Virk [8] established the Maximum drag reductionasymptote (MDRA)

6.1.3 DragReduction

which is 7.6 times larger than the pressure gradient due to gravity. 0

Do 4 .: = D [0.00490 X Opu2) J = 7.53 x lit Pa/m (3.33 psi/ft),

The solution of the BNS equation (Eqs, 6.6) for n = 0.4, e = 0.001, Nn. = 2.54 x 105is f = 0.00490.

If we neglect that the Reynolds number has been obtained from a power law, wecould use the Colebrook-White equation (Eq, 6.1). Its solution for e == 0.001, Nne =2.54 x 105, would yield f = 0.00519. The same value could be obtained directly fromthe Chen approximation (Eq. 6.2).

The pressure drop is calculated from the definition of the friction factor:

81-nJ 2-n' tr'N' - u p = 2.54 X 105.R. - K'

p

D = 0.0574 m,

p = 1050 kg/m",q

II= --W = 20.5 m/s,d2_

4

Non-teminsr flow and solids transport134

q = 20 bpm = 0.0534 m3/s,

h = 60 ft = 18.3 m,

Wo = 0.4 in = 0.0102 m,

IL = 1 cP = 0.001 Pa s,

p = 62.4 lbm/tt? = 1000 kg/m".

The data in SI units are:

Solution

Water (IL = 1 cp, p = 62.4 lbm/fr") is pumped into a two wing fracture as "prepad".Assume a limiting elliptic cross section with height 60 ft and maximum width Wo =0.4 in. Calculate the pressure gradient if the total flow rate is 20 bpm. Use reasonableassumptions concerning the roughness of the fracture surface.

Example 6.3 Turbulent Flow in Ellipsoid Cross Section

Turbulent flow in other geometries such as a flow channel with an elliptic crosssection has not been investigated extensively. A general rule of thumb is suggestedas follows. The relevant Reynolds number for the given geometry is determined andthen a friction factor for tube is obtained according to one of the above methods.The friction factor is modified according to the ratio (f x NRe) /16 valid for thegiven geometry in the case of Newtonian flow. For slot geometry the ratio is ~and for limiting ellipsoid cross section it is 2rr2/16. Finally, the pressure gradient iscalculated from the friction factor.It is usually accepted that the kinetic energy correction factor, aKE, can be taken

as unity for turbulent flow in any geometry since the velocity profile is very fiat incomparison to the laminar profile.

6.1.4 Turbulent Flow in Other Geometries

The calculated frictional pressure drop is less than the pressure increase due togravity and much smaller than the one calculated in Example 6.1. The above resultshould be treated with some caution because of the still open questions whether theMDRA can be applied and whether the wall Reynolds number should be preferred.Nevertheless, from the results of Examples 6.1 and 6.2 it should be clear that the dragreduction phenomenon cannot be neglected. 0

~p 4 I 0 4L = '0[0.00565 x (zpu-)] = 0.868 x 10 Palm (0.384 psi/ft).

The solution of Eq. 6.8 is f = 0.565 x IfY. (The explicit Eq, 6.9 yields virtually thesame value: f = 0.563 x 10-3.) The corresponding pressure gradient is

137Non·laminar flow

N . = (1 +3n') r _ ~.5Re.w 4n' N R. - 3.50 x liT.

From Eq. 6.10 the wall Reynolds number is

81-n' 2-n' TVI'

N' - u p,-, = 2.54 x lOsRe - K'

p

In Example 6.1 we found that the generalized Reynolds number is

Solution

Consider the same conditions as in Example 6.1. The viscoelastic fracturing fluid (K' =0.012 lbf-s" /ft2, n' = 0.4, p = 65.5 Jbm/fr') is pumped through D = 2.259 in. tubing.The flow rate is q = 20 bpm. Neglecting pipe roughness but taking into account thedrag reduction phenomenon calculate the frictional pressure gradient.

Example 6.2 Pressure Drop from Drag-reduction Correlation

The definition of the wall Reynolds number can be applied to any other constitutiveequation, as well. For example, for the yield power law first a fictitious laminar wallstress (or pressure drop) has to be determined from Eq.5.48 and then the wallviscosity can be calculated dividing the wall stress by the wall shear rate (Eq. 5.51).Knowing the wall viscosity we can proceed to obtain the wall Reynolds number.From the wall Reynolds number the friction factor and the turbulent pressure gradientcan be calculated according to Eqs. 6.10 and 6.9, respectively.

There have been several interesting attempts to characterize the turbulent behaviorof viscoelastic fluids involving another dimensionless group, the Deborah number(Dem [9J). The somewhat general definition of the Deborah number is the ratioof the characteristic relaxation time of the fluid to the characteristic process time.When the Deborah number is small, the fluid behaves like a purely viscous liquid.When the Deborah number is large, the elastic property of the fluid dominatesits behavior.It is not straightforward to translate the concept into a readily observable quantity

and therefore the Deborah number is calculated extremely differently by differentauthors. Studying the flow behavior of a fracturing fluid (HPG) Chakrabarti et al, [lOJsuggested that the MDRA might be the NDe -+ 00 limit of an f vs (N Re, NDe)correlation. Analyzing the friction factors for the same fluid Keck [11] arrived ata conclusion that the Deborah number should be in the range 22 to 30 for a0.48% HPG solution and in this range the friction factors lie considerably belowthe MDRA.

(6.11)(1+3n) ,

NRe.w == 4;- NRe·

Reynolds number and the generalized Reynolds have the same values:

Non·laminar flow and solids transport136

(6.15)CDNRe.p = 24.

Combining the above three equations we arrive at another form of Stokes' law:

(6.14)(4F ) 1 2d~Jr = CD X CzPfU ).

less than 0.01 the drag specified by Stokes' law is only 2 percent less than a valuebased on rigorous small Reynolds number expansion of the Navier -Stokes equations[9]. In engineering practice Stokes' law is considered valid if NRe.p < 1. The dragon a sphere can be characterized through the drag coefficient, CD, defined to relatethe induced stress (force divided by the projected area of the sphere) with kineticenergy per unit volume

(6.13)

At particle Reynolds number

(6.12)

According to Stokes' law the drag on a sphere of diameter dp moving steadily withvelocity u through an unbounded fluid is given by

6.2.1 Settling of an Individual Sphere

a part of the proppant bed. The equilibrium height concept is a suitable means todescribe proppant transport under such conditions (Babcock et al. [15]).

Figure 6.3 Components of proppant velocity

Proppant bed

_ ...._ ..._._-_ .._------_

One of the main functions of the fracturing fluid is to transport proppant. Theproppant carrying capacity of the fluid is limited by gravitational settling. Figure 6.3shows a possible trajectory of the proppant particle in the fracture. A simple vectorialview of the horizontal and vertical velocity component suggests the requirement thatthe settling velocity should be less than the horizontal transport velocity multipliedby the ratio of height to half-length. This requirement has led to the application ofhigh-viscosity fracturing fluids (Daneshy [13]; Novotny [14]).If proppant settling is significant a bank of solids is formed leaving less free space

for flow of the slurry. The resulting larger horizontal velocities lead to remobilizing

6.2 Solids Transport

For an order of magnitude analysis it is reasonable to assume that the relative roughnessis minimum 0.001 and maximum 0.1. The results are given in Table 6.1.

The interested reader may compare these results with the measurements ofWarpinski [12], who found that observed in situ pressure losses in a fracture werefrom 1.4 to 3.1 times higher than those calculated neglecting the roughness of thewalls. 0

tJ.p _ ( 8 ) 1 2L - tt x 0.0102 (f)(2:0.182 x 1000).

The corresponding pressure gradient is obtained by applying the definition of the frictionfactor (Eq. 5.1l) and the relation between pressure drop and wall stress (Eq. 5.77):

_2n2 [ {e 5.0452 [el.1098 (7.149)O.S981]}]-2f - - -41oglQ -- - --loa -- + --16 3.706 NRe 010 2.8257 3040

q

U = -h 2 = 0.182 m/s,won4

The resulting Reynolds number is (see Eq. 5.76)

N = n x 0.0102 x 0.182 x 1000 _Re 2 x 0.001 - 3040.

Thus, turbulent flow regime can be assumed. The friction factor is determined (assumingan appropriate relative roughness) from Eq. 6.2 taking into account the factor 2n2/16:

Since the flow is equally divided between the two fracture wings, the linear velocity is

.----~-------.--.- ...----------=

138 Non-laminar flow and solids transport Solids transport 139

Table 6.1

e f i [Palm) !J.p [psilft]L

0.1 0.0329 142 0.006300.01 0.0159 691 0.003060.001 0.0136 59.0 0.00261

therefore the power law form of Eq, 6.18 is

(6.26)

21 } (1-n)/2

"0 ~ ~::g{G:)' + [~% Y1 "[2(l+"~)"'" rWe can solve (numerically) the above equation at any y and hence a profil~ of settli~gvelocities can be determined. If only one representative value of the setthng velocityis needed, the above equation may be solved at y = w/4.

(6.21)

where the function f (n) is given by

fen) = 3(3n-3)/2 [33n5 - 63n4 -lln3 + 97n2 + 16n] ;4n2(n + l)(n + 2)(2n + 1)

Non-Newtonian Fluid

(6.25)y = [(;:r + (Ydf/2where Yt is the shear rate which would be generated by the fluid flow only. Using aparallel plate approximation the shear stress induced by the fluid flow can.be obtainedfrom the constitutive equation (see Table 5.2). Since Yl depends on the distance fromthe wall, the effective shear rate is higher in the vicinity of the wall than in the center.Assuming power law behavior the equation becomes

(6.20)

Results of the drag force acting on a moving sphere are available for power lawfluids. Acharya [17] obtained

F = 37l'd~K (:J11 fen),

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

For hydraulic fracturing applications we are interested in settling under the additionalshear induced by the flow. The flow induced shear results in further changes ofthe draa force if the fluid is non-Newtonian. To account for the thinning effectNovotny [14] suggested the use of Stokes' law with an apparent viscosity calculatedat an "effective shear rate"

(6.19)

6.2.2 Effect of Shear Rate Induced by Flow

(6.18)d~fl.pg

Uo=---·181£

The more general relation equivalent to Eq. 6.18 in the Stokes regime but valid forany Reynolds number is

from which the terminal velocity is

Care should be taken if the resulting terminal velocity corresponds to a shearrate that is considerably less than the one at which the power law parameters weredetermined. In such case the power law extrapolation overestimates the apparentviscosity. As a result, the predicted terminal velocity might be underestimated asdemonstrated by Roodhart [18J.

A convenient empirical approach to present the results of settling experimentswas suggested by Shah [19]. He found that the plot of the dimensionless groupVC~-II(NRe.p)2 versus the particle Reynolds number, N~e.p i~ a unique ~urve for agiven flow parameter, n'; and presented the curves both graphically and In the formof simple correlating equations.

(6.17)

A particle whose density is fl.p =pp - pf higher than the corresponding fluiddensity is accelerated by the gravitational force. With increasing velocity, however,the drag force on the sphere acting against the gravitational force is also increased.The falling particle soon reaches an equilibrium when the two forces are balanced:

7l'd~fl.pg37l'fJ,dpuo= 6 '

(6.24)

then Eq. 6.22 can be rewritten as

CDN~e.p = 24f(n).Terminal Velocity

(6.23)

If the power law Reynolds number is defined accordin~ to

d"u2-"pfN' _r_P _Reip = K

(6.16)CD = (0.63+ ~)2vNRe,p

gives sufficient accuracy for a very wide range of Reynolds numbers, 0 < NRe.p <2 X 105•

(6.22)[dn+1 fl. ] linp pg

u =o 18Kf(n)

At higher Reynolds numbers the inertial forces become important and the aboveproduct increases. For engineering computations Dellavalle's [16] correlation

141Solids transportNon-laminar flow and solids transport140

- - - - ------==== ===~================ .._.._--=======-------

Adding 10 Ibm bauxite (pp = 3700 kg/m") to 1 gallon liquid results in fluid volume,VI = 1 gallon = 0.00379 m3, and particle volume, Vp = 10 Ibm/ Pp = 10 x 0.456/

Solution

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 wasobtained adding 10 Ibm of proppant per gallon of fluid (10 ppga).

Example 6.5 Representative Settling Velocity Involving Slurryand Wall Effects

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

(6.31)2dp(J)=-.

w

where

(6.30)f II' = 1 - 0.652&0 + 0.1475,,/ - 0.131(1)4- 0.0644w5 + ...

(6.29)fw=UOw.UO

A treatable form of f II' derived theoretically by Faxen (see the detailed treatmentby Happel and Brenner [21]) for a particle located at a distance y = w/4 from thewall is

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

6.2.4 WallEffects

where F is still given by Eq. 6.12, understanding that the moving velocity is now ur/).Thus, a terminal velocity calculation should involve Eqs. 6.12, 6.16, 6.27 and 6.28,

(6.28)( UO)2( 1-4> ) (4F) 1 2-1CD,r/J = ur/J 1+ 4>1/3 d~T( (2PjUr/J) ,

(6.27)N _ (ur/J) ex [ 54>] (dpUr/JPf)Re,p.r/J - Uo P 3(1- 4» JL '

Dallavalle equation (Eq. 6.16) is still valid in a slurry characterized by the voidfraction of the solids, 4>,if both the Reynolds number and the drag coefficient ismodified according to

143SOlids transport

....--_._-_ ...-_-----------_ ..__ .....__ .....-._-_ .._ .•.--

The presence of other particles causes an additional decrease of the terminal settlingvelocity. According to the interesting observation of Bamea and Mizrahi [20] the

6.2.3 Effect of Slurry Concentration

The solution of Eq. 6.26 can be obtained numerically for different distances from thewall. The results are plotted in Figure 6.4. The calculated terminal velocity at thecenterline (y = w/2) is only Ui] = 3.4 X 11)-6 m/s, an unrealistically low value. Theterminal velocity at y = w/4 (uo = 8.6 X 10-5 m/s) can be accepted as a reasonablerepresentative value. 0

Solution

Construct a curve representing the terminal velocity of a 20/40 mesh bauxite proppant(dp = 6.4 x 10-4 m, Pp = 3700 kg/nr') in a borate crosslinked polymer characterizedby PI = 1010 kg/m", n = 0.68, K = 33.5 Pa- s". Assume fracture width w = 0.008 mand average linear velocity uavg = 33.5 m/s,

Example 6.4 Settling VelocityProfile in the Facture

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

Figure 6.4 Settling velocity for Example 6.4

~ 0.00008

.ss 0.00006

0.00004

0.00002

0

0 y w/2

Wall Centerline

Non-laminar flow and solids transport142

-----.~--

1. Moody, L.W.: Friction Factors for Pipe Flow, Trans. ASME, 66,671, 1944.2. Colebrook, C.E.: Turbulent Flow in Pipes with Particular Reference to the Transition

Region between Smooth and Rough Pipe Laws, 1. Inst. Civil Eng., (London), 11, 133,1939.

3. Chen, N.H.: An Explicit Equation for Friction Factor in Pipe, Ind. Eng. Chem. Fund.,18, 296-297, 1979.

4. Nikuradze, J.: Stromungsgesetze in rauen Rohren, VDI Forschungsheft, Arb. Ing.-Wes.,No. 361, 1933.

5. Dodge, D.W. and Metzner, A.B.: Turbulent Flow of non-Newtonian Systems, A.!.Ch.E.lournal, 5, 189~204, 1959.

6. Hanks, R.W.: Principles of Slurry Pipeline Hydraulics, Encyclopedia of FluidMechanics, Cheremisinoff, N. P. (ed.), Vol. 5, pp. 237~240, Gulf, Houston, TX, 1986.

7. Szilas A.P., Bobok, E. Navratil, L.: Determination of Turbulent Pressure Loss of non­Newtonian oil Flow in Rough Pipes, Rheologica Acta, 20, 486-496, 1981.

8. Virk, P.S.: Drag Reduction Fundamentals, A.l.Ch.E. Journal, 21 (4), 625-656, 1975.9. Denn M.M., Process Fluid Mechanics, Prentice Hall, Englewood Cliffs, N.J., 1980.

10. Chakrabarti, S., Seidl, B., Vorwerk, J. and Brunn, P.O.: The Rheology of Hydrox­ypropylguar (HPG) Solutions and its Influence on the Flow through a Porous Mediumand Turbulent Tube Flow, Respectively (Part 1) Rheologica Acta, 30, 114-223, 1991.

11. Keck, R.G.: The Effects of Viscoelasticity on Friction Pressure of Fracturing Fluids,paper SPE 21860 presented at the Rocky Mountain Regional Meeting and Low­Permeability Reservoirs Symposium, Denver, Co, Apr. 15-27, 1991.

12. Warpinski, N.R.: Measurement of Width and Pressure in a Propagating HydraulicFracture, SPEJ, (Feb.), 46-54, 1985.

13. Daneshy, A.A.: Numerical Solution of Sand Transport in Hydraulic Fracturing, 1PT,(Nov.), 135-240, 1978.

14. Novotny, E.J.: Proppant Transport, paper SPE 6813, 1977.15. Babcock, R.E. Prokop, CL, Kehle, R.O.: Distribution of Propping Agents in Vertical

Fractures, Producers Monthly, (NOV.), 11-18, 1967.16. Dallavalle, 1.M.: Miocromeritics, 2nd ed., Pitman, London, 1948.

References

When several settling particles form a cluster, the settling rate might be considerablyhigher then for individual particles. Since a cluster acts more or less as a largeparticle and according to Stokes' law the terminal velocity is proportional to thesquare of the particle diameter, the increase of the settling velocity is not surprising.Clusters may consist of a few particles or more. In hydraulic fracturing the proppantcarrying fluid may be different from the fluid already-present in the fracture and thismay lead to the limiting case where the proppant and its surrounding fluid behaveas a single cluster, moving together in the pool of the other fluid. The phenomenon,called convection (Cleary and Fonseca [22]), does not lend itself to easy description.Since no qualitative description is yet available, at present the role of the concept israther to illustrate the "overwhelming complexity of the underlying physics" than toobtain reliable design guidelines.

6.2.5 Agglomeration Effects

145References

----_--_ ..._-----

(-0.63 + 4.8 ) 2

V131.2x (17627+2.44 x 1Q6U~)016

From the definition of the drag coefficient for a particle settling in a slurry Eq. 6.28,

C . = (U¢)2 (.2.=..t_) [4(31T/-(adpU¢)] 1 2-1D.", Uo 1+ .p1/3 d~1T (7_Pfu¢)

we obtain the new approximation of the terminal settling velocity,

u¢ = (CD.t/>Pfu5) (1+ rpl/3) (_:!_P__)1 - rp 24fLa

= 1.27 x 10-11 x (17627 -:- 2.44 x 106u;)016

x (0.63 + 4.8 ) 2

V 131.2 x (17627 + 2.44 x 106u~)O 16

The solution of the above equation is Uq, = 0.00013 m/s.To include the wall effect, we use Eqs. 6.29 to 6.31 as follows:

2dp 0.00064w= - =2--- =0.16

w 0.008 '

f w = 1 - 0.6526w + 0.1475w" - 0.131w4 - 0.0644w5 = 0.899.

u¢.w = fwut/> = 0.899 x 0.00013 = 0.000117 m/s.

Thus, a reasonable representative value of the terminal settling velocity is 0.12 mm/s, 0

(17627 + 2.44 x 106u~)O.16·

and the Reynolds number is obtained from Eq, 6.27,

NRe.p.¢ = G:) exp [- 3(15~ ¢)] (dp::Pf)= 131.2 x (17627 + 2.44 x 1Q6U!)O.16.

Applying the Dallavalle correlation, Eq. 6.16, the resulting drag coefficient is

CD.q, = (0.63+ ~)2Re.p.¢

33.52

3700 = 0.00123 rrr', Therefore, the void fraction of solids is rp = Vp/Wf + Vp) =0.00123/(0.00123 + 0.00379) = 0.245.

For a terminal velocity, uq" the apparent viscosity is calculated (in accordance withEq. 6.26) from

Non-laminar flow and solids transport144

---------- ...-.-.-----.-----~----

Foams are gas-liquid dispersions with the liquid as the continuous and the gas asthe dispersed phases. The in situ volumetric phase relation is usually characterizedby the quality, I", defined as the ratio of the gas volume to the total foam volume(Cameron and Prud'homme [1]). Fracturing fluids with qualities lower than 50-55%have been referred to as energized. Very high-quality foams, above 93-97%, havethe tendency to invert into mist when the liquid becomes the internal and the gas theexternal phase.It is convenient to make an explicit distinction between microflow, where the char­

acteristic size of the space confining the flow is commensurable with the bubble size(e.g., flow in porous media and in traditional laboratory viscometers) and macroflow,where the bubble size can be neglected with respect to the characteristic size of theflow path (e.g., flow in a fracture or well). For the purpose of macroflow descriptionthe foam is described as a homogenous fluid with a "rheology" depending not onlyon the shear rate but on additional parameters, first of all on the relative gas content.(Note that the homogeneous fluid concept presumes a common linear velocity forboth phases at any given location.) It is well established now that the apparentviscosity increases with increasing foam quality and decreases with increasing shearrate (Assar and Burley [2]).

7.1 FoamRheology

In hydraulic fracturing foamed polymer solutions are used widely. They are attractivepropp ant-carrying fluids because of their excellent solids transport carrying propertywhile they may considerably decrease formation damage due to fracturing, espe­cially for water sensitive formations. This chapter concentrates on some aspects offoam flow.

ADVANCED TOPICS OFRHEOLOGY ANDFLUID MECHANICS

717. Acharya, A.: Particle Transport in Viscousand ViscoelasticFracturing Fluids,SPEPE,

(March), 104-110, 1986.18. Roodhart, L.P.: Proppant Settling in Non-Newtonian Fracturing Fluids, Paper SPE

13905, 1985.19. Shah, S.N: Proppant Settling Correlationsfor non-NewtonianFluids under Static and

Dynamic Conditions,SPEJ, (April), 164-170, 1982.20. Barnea, E. and Mizrahi, J.: The Chern.Eng. Journal,S, 111-189, 1973.21. Happel, J. and Brenner, H.: Low Reynolds Number Hydrodynamics, Prentice-Hall,

EnglewoodCliffs, 1965.22. Geary, M.P. and Fonseca,A., Jr.: ProppantConvectionand Encapsulationin Hydraulic

Fracturing:Practical Implicationsof Computerand LaboratorySimulations,SPE paper24825, 1992.

146 Non-laminarflow andsolids transport

~- .__ .... . --.------.----~------------------------:--.

Shear stress vs. shear rate data for foam flow of different qualities, pressures in pipesof different diameters (after f7])

Figure 7.1

100 100010

• r=48 to 50%(p=517 to 539 kg/m")

e r=57106O%(Jr"421 10447 kg/m")

.. r=6710 7oo",(p=324 to 355 kg/m")

(ii' ""~ 6 0

10 66

!I: 6....

(7.3)r (Y)"; = !y'vE + KVE -;;reduces to a yield power law with yield stress t;= t,.VEC and K = KVE£I-11 at anygiven specific volume expansion ratio and it is of the form of Eq. 7.2; thus it isjustified to name it volume equalized yield-power law.

Thus, Eq. 7.2 states that the volume equalized shear rate is a unique function of thevolume equalized shear stress.

In Chapter 5 we saw that a standard flow curve relates wall shear stress andnominal Newtonian shear rate. For foams formed from a given gas-liquid pair butat different qualities and pressures, such a plot looks scattered as seen on Figure 7_1.By virtue of Eq. 7.2 the wall shear rate versus nominal Newtonian shear rate plotcollapses into one curve if both coordinates are volume equalized. A volume equal­ized plot of the data of Figure 7.1 is shown in Figure 7.2. The volume equalizedflow curve can be described applying any of the known constitutive equations. Theparameter estimation procedure is exactly the same as for incompressible fluids,resulting in the parameters of the volume equalized model.

The three-constant constitutive equation

(7.2)

Newtonian flow and incompressible non-Newtonian flow along a flow path ofconstant cross section obey a certain invariance property: The friction factor (inother words the Reynolds number) is constant. If we demand the same invarianceproperty for the flow of a compressible non-Newtonian fluid, this restricts the form ofthe constitutive rheological equation. A constitutive equation, providing the requiredinvariance, is called volume equalized. The general form of shear rate versus shearstress functions, satisfying the principle of volume equalization, is

~=fVEG)·

149Foam rheology

The specific volume expansion ratio may vary between unity and a maximumdetermined by the ratio of gas density to the liquid density. It is convenient to referto a quantity divided by e as volume equalized.

The principle of volume equalization [5] is based on an observation concerningnon-Newtonian compressible flow. Incompressible Newtonian flow, compressible

(7.1)

In interpreting large-scale experiments, the rheological behavior of foams of differentqualities could be represented by one curve (Valk6 and Economides [5]) if both theshear stress and the shear rate were volume equalized.

The technique uses the specific volume expansion ratio, e, as the additional param­eter representing the volumetric relation of the gas and liquid phases. It is definedas the ratio of the liquid density (considered to be constant) to the foam density(varying along the flow path because of the change in pressure):

PI£= ~.

P

7.1.2 VolumeEqualizedConstitutive Equations

A frequent implicit assumption, when using quality-related correlations, is that thepressure (and hence the density of the gas phase, the compressibility of the bubbles)has no direct effect on the rheological behavior.

Several studies (Blauer et al. [3]; Reidenbach et al. [4]) start from the assump­tion that a rheological curve, determined at given conditions, undergoes continuousdeformation with small perturbations. Once a flow curve is described by a two- ora three-parameter rheological equation the most influential condition (the quality) ischanged and the transformation of the flow curve is described establishing empiricalcorrelations for the parameters of the rheological equation.

The above procedure of parametrization can be continued including other factors(e.g., by varying the texture via applying different foam generators and/or surfac­tant concentrations, fluid and gas composition, system pressure and temperature).Unfortunately, data reduction with respect to one flow curve, corresponding to aprescribed quality, is mathematically ill-conditioned. Several combinations of theparameters may reproduce the flow curve within the achievable experimental accu­racy. This is especially true for three-parameter models such as the yield-powerlaw. The small variations in the quality and quantity of additives (surfactants, claystabilizers and breakers) and in the other conditions (texture and shear history) maycause a shift in the flow curve which is then amplified in the numerical values ofthe parameters.

In a detailed investigation of water and HPG-solution foams at elevated pressuresReidenbach et al. [4] applied a yield-power law formulation. In order to incorporatethe non-Newtonian behavior of the base liquid into the constitutive equation thoseauthors postulated that the flow behavior index n was identical to that of the baseliquid while the yield stress and the consistency index varied with quality.

7.1.1 Quality Based Correlations

Advanced topics of rheology and fluid mechanics148

"""---.--- ------- ----------

3(1+ 3n)2

-=--u.vg l+n

(1+2n)(3 + 5n)OiKE =

(1 3)nf = 2.+1 : n KVEPtl-nD-nmn-2

Umax 1+3n

Table 7.1 Formulas for tube flow of foams

Enzendorfer (7] measured the pressure drop of N2 foam with base liquid 0.48 mass %HPG solution in water at room temperature (21°C) and elevated pressures. Figure 7.2

Example 7.1 Determination of the VolumeEqualized Power LawParameters

(7.7)r (y)n~= KVE ~

where the flow behavior index, n and the volume equalized consistency index, K VE,are constants for a given gas-liquid pair at a given temperature [5]. Table 7.1 showsthe relevant variables characterizing the foam flow in tubes where m is the mass fluxand PI is the liquid density.

Because of the closed analogy between incompressible rheological models andtheir volume equalized counterparts, one can readily modify the above relations forother geometries. A similar solution is available for the volume equalized Binghamplastic model [5].

In most cases the volume equalized power law containing only two parameters issufficient to describe the rheology of a given foam:

7.1.3 Volume Equalized Power Law

variables are determined by numerically solving the system, the change of pressurealong the flow path can be calculated as we will see in the section on mechanicalenergy balance.

The additional flow invariants - the distance of the plug from the wall and bothflatness measures - are exactly the same as in the incompressible case discussed inthe previous chapter.

151Foam rheology

------.---.-.---~----- ..,--- ,---

For a typical flow problem, where the geometry, D, L, the properties of the fluid,PI, KVE, r y,VE, n, and the mass flux, m, are specified, one arrives at a system oftwo equations (Eqs. 7.5 and 7.6) containing two unknowns: f and ¢. Once these

and (not surprisingly, if we remember where the principle of volume equalizingcomes from) the friction factor has the same property (Valko and Economides [6]):

f = 2n+1ptnK vED-nmn-2(1- ¢)-l

[(1+n)(1+2n)(1+3n) ]n

x n(l _ ¢)«2n2 + 3n + 1)+ (2n2 +2n)¢ + (2n2)¢2) (7.6)

(7.5)A.::;:: .y,VE£' ::;::2ry,vEPI'¥ f' 2 'rw m

(7.4).(y) ~, f [k~£ (~ iy' '; - "v£) 1"" dy',Note that the volume equalized wall stress is constant and the result of the integra­

tion does not change along the flow path. The velocity profile is linearly "stretched"with increasing e.

Let m denote the mass flow rate per unit cross sectional area of the pipe, m ::;::uavgP ::;::const. Proceeding the same way as in Chapter 5, we can derive the usualrelations between pressure gradient and flow rate. Since neither the pressure gradientnor the volumetric flow rate is constant along the flow path it is more convenient tolook for relations containing only flow invariants. The yield stress to wall stress ratio(and hence the distance of the plug from the wall) is constant along the flow path:

For the no-slip condition the velocity profile can be obtained from the integral

Figure 7.2 Volume equalized log-log plot of the rheological data for HPG 40 foams (after [15])

10 100

• r~105O%(p"517 to 539 kglm')

e r;571060%(1)''421 10447 kglm')

.. r=8710 70%(1"'324 to 355 kglm')

l00rr==========~------------.Advanced topics of rheology and fluid mechanics150

Mechanical energy losses per unit mass of flow are determined by the rheology andit is possible to write an accounting balance. The balance along the flow path resultsin the desired relation between end pressures and the mass flow rate.

7.2.1 Basic Concepts

7.2 Accounting for Mechanical Energyindicating that the experimental friction factors of reference [4J depend only onthe product (puavg), but not on the individual values of P and u.vg• The correlatingparameters for foams of an 0.48 % HPG fluid (rewritten from Table 6 of reference [4]into SI units) are m = 1.23 and A' = 0.109 [kg/(mz. s)]O.77.

Winkler et al. [9] found that (a) non-laminar foam flow can be well describedin the framework of the volume equalized principle and (b) large-scale turbulentflow of HPG foams shows typical drag reduction phenomena. They applied the VirkMDRA (in the form of Eq. 6.9) to obtain friction factors. The corresponding wall

i.e., a three times larger value. 0(7.9)f = 2 x 8m x A' x (puavg)m-2,

f :::2 x S" x A' x (pu,vg)m-2 = 2 x SI.23 x 0.109 X 2000-0.77 = 0.0081,where the exponents were found to satisfy the additional constraints x =m - 1 ande = m. Rewriting the above correlation into friction factor form we obtain

It is interesting to compare the friction factor with the one calculated from the widelyused correlation, Eq. 7.9, which gives

(7.8)

x = In[ln(NRe)l = 2.217InU)::: 28.135 + (-29.379 + (S.2405 - 0.86227x)x)x = -5.S9

f = exp( -5.91) = 0.002S.Dt;.p = A' xDe (8Uavg)m4L Pf D '

Applying the explicit approximation to the Virk MDRA (Eq. 6.9) the friction factor is

(1+3n ) _ (1 + 3 x 0.4) 0.4 1626 _ 9660NRe= -- NRe- X - .4n 0.4

Since the flow regime is not laminar, the wall Reynolds Dumber has to be calculated:

A unified view of the different flow regimes for foams was established by Blaueret al. [3] who assumed that the rheology of foams obeyed the Bingham plasticmodel with yield stress and plastic viscosity depending on quality. According totheir method the effective viscosity determined with respect to a given shear ratecan be used to obtain a generalized Reynolds number which, in tum, determines thefriction factor. The turbulent behavior is described by the Newtonian theory. Themethod gives higher than realistic friction pressures for foamed polymer solutionssince no drag reduction is taken into account.

The description of the laminar and the turbulent flow is separated in the workof Reidenbach et al. [4]. Their turbulent flow analysis is based on the extensionof the Melton and Malone [10] procedure, incorporating an additional dependenceon density:

23-n onp7-1mn-2 22.66 x 0.0550.34 X 1000-0.66 X 20001.66N - = = 1626.

Re >: (1+3n)" (1+3 X 0.34)°·34 .KV£ -n- 2.5 x 0.34

7.1.4 TurbulentFlow of Foam

From Table 7.1 the (volume equalized) generalized Reynolds number is

Solution

The procedure suggested by Winkler et al. [9] consists of fitting a straight line to thelog-log data. The flow behavior index is equal to the slope of the line, n = 0.34, andthe pipe consistency index is equal to the intercept, K p. VE = 2.S6 Pa .s" . Applying therelation between the pipe consistency index and the true consistency index, (Eq. 5.56),we obtain

A water based 0.48 (mass) % Hydroxypropylguar solution is foamed by Nz gas. Deter­mine the friction factor if the mass flux is m = 2000 kg/(m2s), the base liquid density isPI = 1000 kg/rn", the tube diameter is D = 0.055 m, and the volume equalized powerlaw parameters are n = 0.34 and KVE = 2.5 Pa- s",

Solution

Example 7.2 Friction Factor for Turbulent Foam Flow

Reynolds number was calculated from the volume equalized power law accordingto Table 7.1. This calculation method needs no additional parameters except for thetwo parameters of the volume equalized power law.

shows a log-log plot of the volume equalized wall stress versus volume equalizednominal Newtonian shear rate for his experiments covering the pressure range 3 to7 MPa, quality range 0.48 to 0.7 and density range 324 to 539 kg/rrr' (Enzendorferet al. [8]). Determine the volume equalized power law parameters describing the exper­imental results. 0

153Accounting for mechanical energyAdvanced topics of rheology and fluid mechanics152

~.---- ..---------------------------- --------

m = qp~ = 1000 kgl(m2 • s).D2-

4

The mass flux is

a = (2. _ 2.) / (..!.. - ..!..) = 4.32 x 103m2/s2PI P3 PI P3

b = (PI _ P3) (_1_) = 9.48x 10-4rn3/kg.Pi P3 PI - P3(7.14)1 a

- = - +b,p P

where a and b are constants. Such relation holds, e.g., if the gas obeys the virialequation of state (truncated after the second term), the liquid is incompressible andthe change in the amount of the dissolved gas can be neglected [5].

The firststep is to describethe volumetricbehaviorof the foamat the given temperature.We know the foam density at two pressure levels, and hence the two parameters ofEq. 7.14 can be readily determined:

(7.13)UavgdUavg dP 21 U;vg._!'--~ - - - -- dt = O.

ctKE P DBefore progressing further one has to describe the volumetric behavior of the foam.

Once entering the tube, the mass ratio of the gas to liquid is constant along the flowpath. The only variable affecting the specific volume of the foam is the pressure.While the level of description might vary from simple to very sophisticated, onecan assume that the specific volume of the foam along the flow path is describedsufficiently by the relation

Solution

Foams are compressible and the neglecting of the kinetic energy change may resultin a considerable error, especially at lower pressures. To simplify the treatmentconsider isothermal foam flow in a horizontal tube. By virtue of the principle ofvolume equalization both the friction factor and the kinetic energy correction factorare constant along the flow path and, thus, the following differential form of Eq. 7.10can be derived:

Determine the outlet pressure, P2, from an L == 100m long, D = 0.055 m diameterhorizontal tubing, if the inlet pressure is PI = 1MPa and the inlet Howrate of thefoam is q = 0.0125 m3/s. Assume isothermalflow.

The following additional information is available: The base liquid density is PI =1000 kg/rrr' The foam density was determined at two pressures: at the inlet pressure

. 3 IPI = 190kglm3 and at another pressure, P3 = 2 MPa, P3 = 320 kg/m . The va umeequalizedpower law parameters are n = 0.4 and Kvs = 1.6 Pa . s".

7.2.3 Foam Flow

Example 7.3 Pressure Drop Calculation for Foam Flow

If the fluid is incompressible, then the density is constant and the isothermal mechan­ical energy balance contains only the constant pressure gradient, the potential energychange and the pressure change. We can use all the rheological relations substitutingthe pressure drop, ll.p, by the potential difference, ll.p - pgll.z.

For vertical flow a similar but somewhat more complicated solution is avail­able [5]. Equation 7.15 can be solved for one of the pressures, if the other pressureis specified, or for m, if the two pressures are known.

7.2.2 Incompressible Flow

1KI = 2jbm2

-1K2=--

21aKE1 a

K3 == 21ctKE - 2jb2m2'

(7.12)LW1= -1-2-'

:zPUavg

(Uavg)3«se == -( 3) , (7.11)

U avg

characterizes the flatness of the velocity profile and the friction factor is defined interms of the wall stress by

where

(7.15)(7.10)

Under these assumptions the integral of the mechanical energy balance from inlet1 to outlet 2 (being at a distance L) yields [6]

The Bernoulli equation (for flow in a circular pipe) can be written as [11]

g(z} - Z2) - 4 [U;Vg.1 _ U;Vg.2] _ r dp _ r2 2ju~vg dt = 0,ctKE.I ctKE.2 I, P l, D

where Z, l, U, p, p, D, I, «xs are the vertical coordinate, length coordinate, linearvelocity, pressure, density, diameter, friction factor and kinetic energy correctionfactor, respectively. As we saw already, the kinetic energy correction factor

155Accounting for mechanical energyAdvanced topics of rheology and fluid mechanics154

·_--_ ..•._-_ .....---

The plot of the observed nominal Newtonian shear rate versus reciprocal diameterof the pipe at a fixed wall stress would yield a straight line. The slope divided by8Tw is the slip coefficient, j3 (characteristic for the given wall stress.) Note that forthe given wall stress this is an extrapolation to the infinite diameter pipe (where thedistortion caused by slip is negligible) and the optimal estimate of the true shearrate is nothing else but the intercept. In practice the intercept is not used; instead the

(7.19)(8uavg) _ (8Uavg) 1~- - ~- + 8j3r,,-.D obs D true D

where j3 is the slip coefficient (which itself might depend slightly on the wall stress).This is a reasonable assumption for fluids without macroscopically observable struc­tures. Rewriting Eq. 7.17 in terms of the slip coefficient leads to

(7.18)Us = j3T,....

The key assumption is that the slip velocity depends only on the wall stress;

7.3.2 Slip Correction

Mooney method [13J

where Us is the slip velocity, i.e., the velocity of the fluid at the wall.A procedure called slip correction (Cohen [12]) is intended to identify and cut

off the second term on the right hand side and hence reveal the corrected part of thenominal Newtonian shear rate. To do this some assumptions on the slip velocity areinescapable.

(7.17)( 8uavg) = (8Uavg) + (8Us) ,D obs D true D

The first part is the one related to shear flow, the second part is due to slip at thewall. To put Eq. 7.16 in terms of nominal shear rates we divide by the cross sectionalarea and multiply by 8/D. The resulting equation is

(7.16)

difference). The flow equations of Chapter 5 are readily applicable. The flow curvesobserved in pipes of different lengths and diameters should collapse into one curve .For some fluids this does not happen, indicating the violation of one or more of themain assumptions. The reason for the entrance and wall effects may be different.For crosslinked gels and especially for foams the wall effects are more important.Most likely the different composition of the near wall part of the fluid is responsiblefor the wall effect. Whatever the microscopic reason is, what we see is the shift ofthe flow curves along the shear rate axis with decreasing diameter of the tube. Thephenomenon can be well treated assuming an apparent slip velocity at the wall. Thenthe observed flow rate is the sum of two flow rates:

157Rheometry

The straightforward way to measure rheological properties is to establish tube flowand measure both the volumetric flow rate and the pressure drop (or potential

7.3. 1 Pipe Viscometry

7.3 Rheometry

16NRe.Vt = f = 601.

It is interesting to know whether the kinetic correction factor plays any significantrole in the above calculation. The reader can verify that if the kinetic correction factoris taken as unity, the calculated outlet pressure is 60% higher. The discrepancy is about100%. if the kinetic energy change is totally neglected, i.e., the kinetic energy correctionfactor is taken as a practically infinite number in the above calculations.

Correctly accounting for the kinetic energy change decreases the calculated outletpressure. Indeed, the quality at the inlet is about 82% and it increases to approximately98% at the outlet. Consequently, the average velocity at the outlet is much higher thanat the inlet (approximately 10 times) and the increase of the kinetic energy is manifestedby additional pressure loss. 0

1060.0195 X (l06 - pz) + (-30.3) In-P2

+ (-8.75 x 104)In 4.32 x 103 +9.48 x 10-4 x 106 = 100 .4.32 x 103+ 9.48x 10-4 x pz 0.055

Th~ solution of the above equation (found by a numerical method) is P2 = 0.974 x10' Pa. It is readiIy verified that the flow is laminar since the generalized Reynoldsnumber is

1Kl = 2fbm2 = 0.0195 [l/Pa]

-1Kz = -- = -30.3

2faKE

KIa 43 = -2f - --,-.-, = -8.75 x 10 .

OiKE 2fb-m-

After substitution, Eq, 7.15 becomes

The coefficients in Eq. (7.15) are

OiKE =

. ( 1+3 04) 0.4= 2°·4T! 0.:' x 1.6 X (03)1-0.40.55-0.4(103),,-2 = 0.0266

(1+ 2n)(3 + 5n)3(1 + 3n)2 = 0.620.

The friction factor and the kinetic correction factor is calculated according to Table 7.1:

. (1 + 3n)"f - 2nTI k t=tr: .n-'- -n- VEP! m-

Advanced topics of rheology and fluid mechanics156

oo 200 400 600 800 1000120014001600

aulD [s-11

Figure 7.3 Wall stress versus observed nominal Newtonian flow rate in different pipes(after [7])

10

40 V 1-0

V./>'

VV o [em]

J' --0--1.2

fl~ -<>-0.95---0-0.8-<>-0.6-<>-0.405

50

The observed wall stress versus nominal Newtonian shear rate curves are shown inFigure 7.3. The points corresponding to different diameters do not lie on one curve,indicating the presence of a wall effect. The nominal shear rate corresponding to agiven wall stress is higher in smaller pipes, which is typical for the (apparent) slipphenomenon.

Solution

Enzendorfer [7] measured the pressure drop of Nz foam with base liquid 0.48 mass %HPG solution in water at room temperature (21DC) and elevated pressures. Considerhis measurements (Table 7.2) at pressure p =5 MPa and quality I"= 0.585. (Since thefoam density is p = 447 kg/nr' and the liquid density is 1000 kg/m", the correspondingspecific volume expansion ratio is e = 1000/447 =2.24.)

Determine whether (apparent) slip at the wall should be taken into account. If theanswer is yes, decide which slip correction method is suitable. Determine the slip modeland correct the measurements for slip. Plot the corrected flow curve.

Rheometry 159

Table 7.2 Observed rheological data [7]

D [em]

1.2 0.95 0.8 0.6 OA05u/d 1"w u/d 1"", u/d 1",. uJd rw uJd rw

[I/SJ [Pal [lIS] [pa] [lIS] [pa] [lIS] [pa] [IISJ [PaJ

9.5 6.3 19.2 8.6 32.1 11.0 76.1 14.4 247.4 18.619.0 10.2 38.3 13.3 64.2 15.5 152.2 19.8 494.7 26.128.5 13.2 57.5 15.7 96.3 18.8 228.2 23.4 742.1 32.638.0 15 76.7 17.1 128.4 20.8 304.3 26.1 989.5 37.747.5 16.8 95.8 19.1 160.5 22.8 380.4 28.8 1236.8 42.557.1 17.7 115.0 20.6 192.6 24.4 456.5 30.9 1484.2 45.6

Example 7.4 Slip Correction of Foam Rheological Data

-----------------._--", •..__._--

According to the above relation a plot of the observed nominal Newtonian shearrate at a fixed wall stress versus the reciprocal diameter of the pipe squared yieldsa straight line. The slope divided by BTw is the modified slip coefficient, f3c (charac­teristic for the given wail stress).

Again, this is an extrapolation to the infinite diameter pipe (where the distortioncaused by slip is negligible) and the optimal estimate of the true shear rate is nothingelse but the intercept. In practice the intercept is not used; instead the same procedureis repeated for other wall stresses. It is anticipated that the wall stress, the modifiedslip coefficient should be constant or varying linearly. Once a suitable straight lineis determined the modified slip coefficient for any wall stress can be computed andthe original measurements can be corrected according to Eq. 7.21-

For slurries, the Oldroyd-Jastrzebski slip correction has been found the onlysuitable method (Hanks [15]). According to recent evidence the same method maybe used to correct for the apparent slip of foams (Enzendorfer et al. [8]).

(7.21)( BUavg) = (Buavg) + B,BcTw~.D obs D true D

where f3c is the modified slip coefficient (possibly slightly depending further on thewall stress.) Rewriting Eq. 7.17 in terms of the modified slip coefficient leads to

(3.20)

For fluids with macroscopic structure such as slurries or foams the apparent slip isthe result of a more complex interaction between the wall and the fluid and, hence,also the diameter of the pipe itself affects the apparent slip. The basic assumption isthat the slip velocity depends on the wall stress and on the diameter according to

Oldroyd-Jastrzebski method [14J

same procedure is repeated for other wall stresses, and a suitable smooth curve isdrawn through the slip coefficient versus wall stress points. The curve then gives aslip coefficient for any wall stress. Once the slip coefficient is known as a functionof the wall stress, the original measurements can be corrected by subtracting thesecond term in Eq, 7.19. .

Technically, slip correction should be made only if pipes of at least three differentdiameters are utilized and the flow curves cover (at least partly) the same wall stressinterval. Since usually the measurement points do not correspond exactly to thesame wall stress, interpolation may be necessary. Possible negative intercepts, largescatter around the straight line and hectic variation of the slip coefficient with thewall stress may indicate that the data do not support the basic assumption concerningthe dependence of the slip velocity on the wall stress.

158 Advanced topics of rheology and fluid mechanics

~--."- ....-----------------------------

The most widespread instrument to measure fluid flow properties is the rotating­cup viscometer. The evaluation of the data is more difficult than in the case ofpipe viscometry. The rea] shear rate at the bob cannot be revealed by first orderderivation of the experimental data. A Rabinowitch-Mooney type analysis wouldrequire higher-order derivatives. In addition, the presence or absence of slip ismore hidden than in the case of pipe flow curves. These problems may be over­come partly by minimizing the gap between the bob and cup, but in the caseof slurries and foams we are limited by an additional constraint: The gap sizeshould be significantly larger than the characteristic size of the particle or bubble.In view of the difficulties above it is not surprising that the application of rota­tional viscometry in hydraulic fracturing is restricted at most for routine quality

The corrected data points are shown in Figure 7.7. The points collapse into one curvewhich is the "true" pipe flow curve of the studied foam [8]. (Note that the correcteddata points, together with other measurements, were used in Example 7.1 to derive thevolume equalized power law parameters.) 0

m2b = 7 X 10-7__Pa·s

m2a = 8_6 x 10-6 __Pa·s

(8uavs) _ (~;) [a+brw]D observed

(8Uavg)

D corrected

the infinitely large diameter [8]. The slopes of the straight lines divided by 8t',.. are themodified slip coefficients, fic. A subsequent plot of the modified slip coefficient versusthe wall stress (Figure 7.6) shows a linear trend. The equation of the straight line isused to correct the measured nominal shear rates:

O~---- __~ ~ __-L __~ ~-~

o 10 20 30

TW [Pal

Figure 7.6 Modified slip coefficient vs. wall stress for Example 7.4 [8]

•••

Pc= a +b.wa = 8.6 10-6 m-2/(Pa.s)]

b=7.010-7 m-2/(pa·s)]

4Xla5rr==================~--------~

161Rheometry

-_.__ ..__ ._-_ .._--------------------

0~~~=3.o 10000 20000 30000 40000 50000 60000 70000

I/D2 [m-2]

Figure 7.5 Oldroyd-Jastrzebski plot of the data for Example 7.4 [8J

700",!pal

600 0 300 28C. 26

500 v0

'I +~ 400 X.,Cl.......:;)CO

100 150

l/D [m-1]

Figure 7.4 Mooney plot of the data (after [71)

200 25050

700

600 ;.(Pal0 30

500 0 28C. 26V 24

'I 400 0 22

~ .. 20X 17

Cl 300 ., 15'<, - 13::>co 200

100

0

Selecting different wall stress levels (from 12 Pa to 26 Pa) we can read off thenominal shear rates corresponding to different pipe diameters. The points are plottedversus the reciprocal pipe diameter in Figure 7.4. From this Mooney plot we cansee that the optimal estimate of the shear rate in an infinitely large pipe (i.e., theintercept of the straight lines with the vertical axis) is negative for virtually all wallstresses involved. Moreover, there is an inversion phenomenon: The optimal estimatesof the true shear rates increase with decreasing stress, which is physically impos­sible. It is obvious that the data do not support the slip model on which Eq. 7.18is based.

The same data plotted with the reciprocal diameter squared as the abscissa areshown in Figure 7.5. The Oldroyd-Jastrzebskimethod gives consistent estimates for

Advanced topics of rheology and fluid mechanics160

------_.- ..._--_-_ .._--- .._-------

11. Denn M.M.:Process Fluid Mechanics, PrenticeHall, EnglewoodCliffs, N.J., 1980.12. Cohen,Y.: Apparent Slip Flow of Polymer Solutions, in Encyclopedia of Fluid

Mechanics,Cheremisinoff,N.P. (ed.) Vol.7, Gulf, Houston,TX, 1986.13. Mooney,M.: ExplicitFormulasfor Slip and Fluidity,J. Rheology, 2(2), 210-222, 1931.14. Jastrzebski,Z.D.: Entrance Effects and WallEffects in an ExtrusionRheometerDuring

the Flow of ConcentratedSuspensions,Ind. Eng. Chem. Fund., 6 (3), 445-453, 1967.15. Hanks, R.W.: Principles of Slurry Pipeline Hydraulics, in. Encyclopedia of Fluid

Mechanics, Cheremisinoff,N.P. (ed.), Vol. 5, 237-240, Gulf, Houston,TX, 1986.

163References

1. Cameron,1.R. and R.K. Prud'homme, Fracturing-Fluid Flow Behavior, in RecentAdvances in Hydraulic Fracturing, SPEMonographSeries,Gilded, J.L, Holditch, S.A.,Nierode,D.E and Veatch,R.W. Jr. (eds), SPE RichardsonTX, 177-209, 1989.

2. Assar G.R., BurleyR.W.: Hydrodynamicsof FoamFlow in Pipes,CapillaryTubes, andPorous Media, in: Encyclopedia of Fluid Mechanics, Cheremisinoff,N.P. (ed.) Vol.3,26-42, Gulf, Houston,TX, 1986.

3. Blauer, R.E., Mitchell,B.l. and Kohlhaas,C.A.: Determinationof Laminar, Turbulent,and TransitionalFoam Flow Losses in Pipes, Paper SPE 4885, 1974.

4. Reidenbach,V.G., Harris, P.C, Lee, Y.N. and Lord,D.L.: RheologicalStudy of FoamFracturingFluidsUsingNitrogenand CarbonDioxide,SPE Production Engineering, 1,39-41, 1986.

5. Valk6P. and EconomidesM.J.: VolumeEqualizedConstitutiveEquations for FoamedPolymerSolutions,J. Rheology, 36, 1033-55, 1992.

6. Valk6P. and EconomidesMJ.: Accounting for mechanical energy in steady-statelaminar foam flow,Proc. 4th Eur. Rheol. Cant, Sevilla 49, Sept., 1994.

7. Enzendorfer,C.: Foam Rheology,MS Thesis, IDP, MiningUniversityLeoben,Leoben,Austria, 1994.

8. Enzendorfer,C.,Harris, R.A, Valko,P., Economides,M.J., Fokker,P.A.,Davies, D.O.:Pipe viscometryof foams,J. Rheology, 39 (2),345-358, 1995.

9. Winkler,W., Valk6,P. and Economides,MJ.: Laminar and Drag Reduced PolymericFoam Flow,J. Rheology, 38, 111-127, 1994.

10. Melton,L.L. andMalone,W.T.: FluidMechanicsResearchandEngineeringApplicationin non-NewtonianFluid System,SPEl, (March), 56,1964 .

References

control. Fundamental rheological characterization should rather rely on pipe flowmeasurements.

(8u/D)trw Is-11Figure 7.7 Correctedflowcurvefor Example7.4 [8]

o D_1.2 ....o 0 • .15 CJrI~ D_.8 emV 0 • .8 ....<> D_.405_

~r.=====~----~-----------,Advanced topics of rheology and fluid mechanics162

stating that the volume of the injected fluid, Vi, equals the sum of the created fracturevolume V, and the volume of fluid, VL, "lost" across the fracture faces into theformation. The latter is often termed as the leak-off volume. Equation 8.1 neglectsthe compressibility of the fluid and does not make a distinction between the volumeof the fracture and the volume of the fluid contained within, because secondaryeffects (such as a possible small unwetted zone at the tip) have little significance inmaterial balance calculations considered in this chapter.

In practice, the injected volume is that pumped down the well and Eq. 8.1 wouldimply that the created fracture volume consists of the volumes of two individualfracture wings. For modeling purposes, however, it is more convenient to deal with

(8.1)

In creating a hydraulic fracture, large volumes of fracturing fluids are injected. Thesefluids are expected to generate an appropriate fracture volume while a portion of thefluids is lost (or "leaks off') into the porous medium through the faces of the fracturethat is being created.

In practice, fracturing fluids are injected in various stages, differing from eachother in terms of chemical composition (and thus, rheological and leak-off properties)and/or functionality. In propped hydraulic fracturing, some of the fluid stages are usedas proppant carriers.

In this chapter a material balance is employed describing the link between theinjected volume, the created fracture. volume at the end of pumping and the leak-offvolume. Mechanisms intended for the description of leak-off are presented in detail.

While material balance itself is not intended as a predictive or modeling tool, itis an essential element of any fracture propagation model. In addition it provides ameans of consistency check for modeling results even if the details of the calculationsare not available. As such, the concepts of this chapter are critical to the understandingof the actual models presented in the subsequent chapters.

An overall material balance for a hydraulically induced fracture can be written as

8.1 The Conservation of Mass and its Relation toFracture Dimensions

MATERIAL BALANCE8

--------------------_. __...._----_.- ------" ..-~--"-"'-~'--"----"-----="'-----"-'-'-----

AppendixA contains details and results of the test problemof the Warpinskiet al. [1]comparisonstudy. Use the combination of Eqs. 8.5 and 8.6 to check the consistencyof the results providedby the modelers.

TablesAl and A.2 show the input data. TableA3 contains the results of 34differentsets of resultsprovidedby participants.The results of TableA3 correspondtofixed height models.According to the problem specification(in addition to others) thefollowingdata were given: injected volume, Vi = it.= 794.9 m3,and fracture height,hf = 51.8 m. (For unexplainedreasons, some of the participants did not use the valueof hf = 51.8 m, but assumed another height. Use the height provided by the modelerin the consistencycheck.)Figure S.l Schematicof fracturedimensionsandinjectionrate

distribution

Example 8.1 Simple Consistency Check of Model Results

(8.4)

With Eqs. 8.2 and 8.3 the efficiency, fracture surface and average width dependon each other if the injected volume is specified.

To remain consistent, we denote by i the injection rate entering one of the wings(Figure 8.1). Very often the injection rate, i, is constant and hence, the injectedvolume up to a given injection time, t, is given by Vi = it. (Usually the total injectionrate, 2i, is specified in engineering practice.) In the constant injection rate case,instead of Eqs, 8.1 and 8.2 we can use

The words fracture half-length and fracture length are used interchangeably. If aconstant hf cannot be assumed, the fracture length is the largest distance betweenthe well and a point on the fracture tip.

It is also convenient to make a clear distinction between the values of the abovevariables at any time, t, and at the end of pumping, i.e. at time te' We will use thesubscript e to emphasize that a given value corresponds to teo

The above variables and relations constitute an essential and basic commonlanguage with which hydraulic fracturing workers should communicate.

As in other disciplines, basic unassailable principles should override the descrip­tion of individual phenomena. Warpinski et al. [1] noted that "In recent years, therehas been a proliferation of fracturing simulators used in the oil industry. This prolif­eration was intensified by the availability of personal computers and the need forfast running design simulators for use in the field. Applying these models as 'blackboxes', without knowing the underlying assumptions, may lead to erroneous conclu­sions, especially for the unconfined fracture growth."

Equations 8.1 to 8.6, though seemingly trivial, may provide a useful means to filterout major inconsistencies of computed results, even if the details of the calculationsare not known. To illustrate this point, we will use the comparison of hydraulic­fracture models presented in the above cited study by Warpinski et al. [1].

(83)V=Aw.

(8.6)A

xf=-·hf

(8.2)V

T/= Vi'

and is between zero and unity (or between zero and one hundred, if expressed inpercent). The fracture surface, A, is half of the surface area of the fluid body ofone wing, i.e. it is the area of one face of one wing. From the point of view of thesurface area, the fracture wing is envisioned as a body limited by two large parallelfaces. Since the width is small relative to the other dimensions, we do not make anydistinction between the summed area of the faces and the total area. The averagewidth, w, is defined by the relation

respectively. .In many cases there is evidence that the created fracture remains in a well defined

lithological layer, and the fracture is characterized by a constant height, hf. In thiscase the length of the fracture, xf, is simply

(8.5)Aw

T}=-.-,It

andonly one of the wings, especially because the distance from the well to fracture tipis conventionally selected as the key variable. Therefore, in this book the variablesrefer to one fracture wing of a two-wing fracture. In particular, Vi is the injectedvolume into one fracture wing, VL is the volume of fluid entering the formationthrough the two created fracture surfaces of one wing and V is the volume of fluidcontained in one fracture wing. These variables correspond to a given instant in time.

The strongly related characteristics of a hydraulically induced fracture are thefluid efficiency, the fracture surface and the average width. The efficiency, T/, is thefraction representing that part of the fluid remaining in the fracture:

167The conservation of massMaterial balance166

._-_.. ".__ ...._--_ ........---_ .._-- .._._--. -_ ...---~--.--- ......- ..------------------ ..---- ......

(8.8)VL ..Ji-=2CL t+Sp,AL

where V L is the fluid volume that passes through the surface AL during the timeperiod from time zero to time t. The integration constant, Sp, is called the spurt losscoefficient and is measured in meters. It can be considered as the width of the fluidbody passing through the surface instantaneously at the very beginning of the leaksffprocess. The two coefficients, Cv and Sp, can be determined from laboratory tests.

where CL is the leakoff coefficient, measured in units of m/sl/2 and t is the timeelapsed since the start of the leakoff process. The Carter equation is postulated fromempirical observations as shown in Example 8.2. The integrated form of Eq. 8.7 is

(8.7)

The polymer content of the fracturing fluid is partly intended to impede the lossof fluid. The phenomenon is envisioned as a continuous build-up of a thin layer(the filter cake) which manifests an ever increasing resistance to flow through thefracture face. In reality, the actual leakoff is determined by a coupled system, ofwhich the filter-cake is only one element. (The other elements will be consideredin more detail in the next section.) A fruitful formalism dating back to Howard andFast [2] is to consider the combined effect of the different phenomena as a materialproperty. According to this concept, the leakoff velocity, VL, is given by the Carterequation (See Howard and Fast [2]):

B.2.1 Carter Equation I

The power of material balance can be realized only if we are able to describe the termVL.There are two main schools of thought concerning leakoff. The first considers thephenomenon as a material property of the fluid/rock system. The second considersleakoff as a consequence of flow mechanisms into the porous medium and uses acorresponding mathematical description.

8.2 Fluid Leakoff and Spurt Loss as Material Properties

where subscript c denotes a calculated value. The calculated efficiency is shown in thelast column of Table 8.1.

Some discrepancy between the modeler-provided and the calculated values maybe attributed to the finite number of digits. A larger discrepancy means, however,that certain modelers either do not apply rudimentary criteria or do not use similarterminology. For example, the deviation in Case 1 is difficult to explain. An efficiencygreater than 100% as in Case 3 indicates that a fracture with volume larger than theinjected volume was created. In Cases 12 and 13 a fracture only half as large as thatindicated by the modeler's efficiency was created. The above results underline thedanger of relying on a computer code without cross checking as Warpinski et at. [1Jpointed out. 0

Fluid leakoff and spurt loss as materialproperties 169168 Material balance

Solution

In Table 8.1, we present the x], IVand 1) values provided by the modelers, distinguishingthem by the subscript m. Of course, these three quantities are not independent, once V;and h , are specified.

The combination of Eqs. 8.5 and 8.6 yields the following expression for the effi-ciency:

hf.mXfmWm1)e = V;

Table 8.1 Characteristics provided by the modelers (m) andcalculated efficiency (1),.)

hj.m xJ~m Wm 11m 11em m m % %

51.8 774.8 0.0154 85.5 77.62 51.8 1480 0.0073 72.3 70.83 62.2 787.6 0.0185 93.0 114.24 51.8 810.5 0.0157 83.1 83.25 51.8 1374 0.0081 72.2 72.86 51.8 697.4 0.0188 85.4 85.47 51.8 1159 0.0102 76.6 76.88 51.8 830.3 0.0155 84.0 83.99 51.8 1231. 0.0094 75.0 75.4

10' 61.0 755.9 86.011' 61.0 1267. 77.012 51.8 410.6 0.0152 81.9 40.813 51.8 618.4 0.0091 73.0 36.914 51.8 1401. 0.0081 73.8 74.215 51.8 674.2 0.0196 85.9 86.016 51.8 827.8 0.0152 82.5 82.217 51.8 1215. 0.0094 74.4 74.418 51.8 1178. 0.0098 75.0 75.519 51.8 1084. 0.0110 75.0 77.920 51.8 774.8 0.0152 6L8 77.021 51.8 1411. 0.0071 73.6 65.422 62.2 766.9 0.0190 93.0 114.323 51.8 639.5 0.0208 86.4 86.824 51.8 1255. 0.0091 74.3 74.825 51.8 551.1 0.0246 88.3 88.526 51.8 1035. 0.0117 79.0 78.827 51.8 652.9 0.0206 89.0 87.628 51.8 1020. 0.0119 79.0 79.429 51.8 1233. 0.0097 76.9 77.630 51.8 619.0 0.0213 86.0 86.131 51.8 702.3 0.0186 85.2 85.232 5L8 1114. 0.0105 76.5 76.633 51.8 1035. 0.0116 78.0 78.134 51.8 961.6 0.0128 81.7 80.2

'The necessaryinformationhasnot beenprovidedby the modeler

---------=--------_.---._------_._---_.

(8.12)w

"1/= ,2KCL'.ji +w

shows that the term 2KCL.fi can be considered as the "leakoff width". The K factorplays an important role in some design procedures discussed in Chapter 9.

Another form of the same relation,

(8.11)K=

and its value depends on the history of the evolution of the fracture surface, or ratheron the distribution of the opening-time. In particular, if (1) the spurt loss coefficient iszero, and (2) all surfaces are opened at the beginning of injection, then K is exactly 2.For the no-spurt-loss case this is an absolute maximum of the factor.

The relation of the K factor, efficiency and average width is obtained from Eqs. 8.2,8.3 and 8.9:

(8.10)

where the variable K is the opening-time distribution factor. It is defined by thematerial balance itself, i.e.

(8.9)

A hydraulic fracturing operation may last from tens of minutes up to several hours.Points of the fracture face near to the well are opened at the beginning of pumpingwhile the points at the fracture tip are "younger". Application of Eq. 8.8 or of itsdifferential form, 8.7, necessitates the tracking of the opening-time of the differentfracture face elements. If only the overall material balance is considered, it is naturalto rewrite Eq. 8.1 using the formalism of Eq, 8.8:

8.2.2 Formal Material Balance. The Opening TimeDistribution Factor

An additional note concerning the time t is necessary here. Equations 8.7 and 8.8mean that the given surface element "remembers" when it has been opened to fluidloss and has its own "zero" time which might be different from location to locationon a fracture surface.

A closer look at Figure 8.2 reveals that the spurt loss is a matter of conveniencerather than a physical reality. It is the consequence of our Willingness to describe thepoints by a straight line and our indifference toward the finer details of what actuallyhappens at early time. 0

the slope, m, and the intercept, b, of the straight line are m = 6.9 x 10-5 m/sl/2 andb = 2.4 X 10-3 m.

Comparing Eq. 8.8 with the obtained m and b shows thatmCL = - = 3.5 X 10-5 rn,/SI/2(= 8.8 X 10-4 ft/rninlf2) and2 .

Sp = b = 2.4 X 10-3 m(= 0.1 in.).

171Fluid leakoff and spurt loss as material properties

~---' .._----------

Figure 8.2 Filtrate volume through a core (Example 8.2). Slope provides the leakoff coef­ficient, intercept provides the spurt-loss coefficient

Square root time, tl12 (Sll2)

605040302010

y =0.0024 +0.000069x

E 0.007-.I<:

"':::.J 0.006>tVo 0.005~:::Jtn 0.004..'2:::J... 0.003Q)CoQ)E 0.002:::J(5> 0.001'00...J 0

0

5.26.77.38.69.710.611.412.513.2

124102030405060

miu

Table 8.2 Measurements offiltrate volume forExample 8.2

First we compute the square root of time for every point and plot the filtrate volumedivided by the cross sectional area, VLIAL, VS. the square root of time. From Figure 8.2,

Solution

Fracturing fluid, pressurized to a representative pressure, flows through a core samplewith cross sectional area of 20 cm2. The filtrate (loss of fluid) is recorded during onehour as shown in Table 8.2. Determine the two parameters of Eq. 8.8.

Example 8.2 Determination of the Leakoff Parameters fromLaboratory Data

Material balance170

'The necessary information has not been providedby the modeler

(8.19)2CL.Jm~= .w+2Sp

Equation 8.18 gives the fracture surface if both the width and the time are specified.It will be used extensively in Chapter 9.

where

(8.18)(w +2s )i [ 2~ ]A(t) = 2 p exp(~2) erfc (~) + r;; - 1 .4CL~ .,;n

To obtain an analytical solution for the constant injection rate case, Carter solved asimplified version of the material balance, neglecting the fact that the width increasesduring the fracture growth. If w is constant during the entire pumping period, i.e. thefracture has its final width already in the first instant of injection, the solution is [2]:

Hence,

Act Ad Ad,m2 Ac2m-

1 80.3 75.0 1.072 82.6 85.1 0.973 69.5 65.7 1.064 46.9 41.4 1.135 75.9 76.8 0.996 68.4 68.5 LOO7 83.9 85.1 0.998'9 53.8 85.1 0.6310 78.3 73.3 1.0711 81.9 82.8 0.9912 39.6 51.7 0.7713 45.5 40.4 1.1314 71.1 71.6 0.9915 60.2 60.6 0.9916 83.1 82.8 1.0017 70.0 95.2 0.74

(8.17). . l' CL dA dA dw1=2 r.-=-dr+(w+2Sp)-+A-.o .,; t - r dt dt dt

Table 8.3 Fracture Surface Estimation fromthe Data of Table A 4

(8.16). dA2Sp-.

dt

Carter [2J formulated the material balance in terms offlow rates. He argued that ifat time t the injection rate entering one wing of the fracture is i, it should be equalto the sum of the different leakoff rates plus the growth rate of the fracture volume.

and the actual shape of the fracture face is not relevant. The results are given inTable 8.3.

and the creation of new surface brings about an additional loss due to spurt loss

2CL.,fiit;where f3 = ,

Wm

wmi (z 2f3]Ac2 = -,- exp({3) erfc(fJ) + r;; - 14CL~ vn(8.15)

dA dww-+A~dt dr '

whatever the shape of the fracture face is. From Eqs. 8.18 and 8.19(where the factor two now comes from the two fracture faces.) Not all the fluidinjected leaks off and hence the fracture grows. The growth rate of the volume is

(8.14)1A(I)CL l' CL (dA)2 dA-2 -- - dr

o .Jt - rCA) - 0 ~ dr '

From Eq. 8.5 the first estimate of the fracture surface is simply

Solution

Assume .that we know the history of the fracture surface growth, i.e. the functionA(r) and/or its inverse function, rCA). Then the leakoff flow rate through the twofracture faces is the summation of the different flow rates along the surface elementsof different age:

The opening-time is denoted by T and every surface element has its own T. If theactual time is denoted by t, the leakoff flow rate, corresponding to the given surfaceelement is

In Example 8.1 we investigated the solutions to a specific design problem where theheight was specified. Table A.4 of the Appendix gives the results for variable heightmodels (using three different layers). Estimate the fracture surface from Eq. 8.5 (Ad)and from the Carter equation II (Acz) assuming for the latter that the average width(given by the modeler) is constant during the whole injection period. Compare the tworesults using the following data: i = 0.0662 m2/s (25 bpm per wing), I,= 12000 s(200 min) and CL = 9.84 X 10-6 m/slJ2(0.00025 ft/minl/2).

(8.13)

Example 8.3 Fracture Surface Calculation for Height Growth Models(Consistency Check II)

8.3 The Constant Width Approximation(Carter Equation II)

173The constant width approximation (Carter equation /I)Material balance172

-_.,_ ....._--_-- ..._-,--- -_...._-_ .._--- ..__ ._--

Figure 8.3 Log-logplotof fracturesurfacecalculatedfromthe CarterequationII vs. time(8.29)O'.Jif(O')

go(a) = f(~ + a) ,

Injection time

The integral on the right-hand side can be given in closed form as

(8.28)

Substituting Eqs. 8.26 and 8.27 into Eq. 8.24 gives

(8.27)From Eqs. 8.25 and 8.26

(8.26)

It is convenient to consider dimensionless variables:(8.21)

Nolte [3,4] also introduced a new function(8.25)

8.4.1 The Consequences of the Power Law Assumptionwhere the opening-time is given (by virtue of Eq. 8.20) as

(8.24)1 r 1 (it' 1 )go(O')= Ae Jo ..jt; r.Jt _ 1: dt cIA,

Substituting Eq. 8.23 into Eq. 8.21 yieldswith exponent a being constant during the injection period. In other words heassumed a particular form of the solution of the mathematical model which is stillnot even specified, because the actual fracture surface evolution is determined (alongwith the material balance) by additional phenomena such as elasticity and fluid flow,as we will see in the following chapters.

(8.23)lA'i"CLVLe =2 ;;--:; drda,

o r ...;t-1:

(8.20)

While the fracture surface increases from zero to Ae, the volume of fluid leaking offis the integration of Eq. 8.22 with respect to the surface

(8.22)i t, CLdVL = cIA ;;--:; dt.

r ...;t - 1:

If we plot the fracture surface computed from the Carter equation II vs. time usinglog-log coordinates, the result is always similar to the one shown in Figure 8.3. Atearly times the slope of the curve is unity, and at later times it decreases to !.

Probably motivated by this fact, Nolte [3] postulated a basic assumption leading toa remarkably simple form of the material balance. He also considered the constantinjection rate case and assumed that the fracture surface evolves according to apower law,

8.4 The Power LawApproximation to SurfaceGrowth

where the subscript e refers to the end of pumping. The reader may wonder why weintroduce a new symbol, go(a). Isn't it exactly the opening-time distribution factor kat the end of pumping? The answer is yes, but with some restrictions. The functiongo(O') can be determined by an exact mathematical method because it involves theassumptions that (1) the surface grows according to the power law Eq. 8.20, (2) thefluid leaks off according to Carter equation I and (3) the spurt loss is zero.

In order to derive a closed form of the function go(O'), consider an elementarysurface, cIA, which is opened at time r. The volume of fluid lost through the elemen­tary surface since its opening until time t, is given by Eq. 8.13:

If the two differentways to estimate the fracture surface yield the same result, i.e.the quotient is 1, there is no question of consistency(howevera "too good" agreementmay indicate that the given model effectivelyused the Carter equation II even if itclaimed to be a real pseudo-Sf)or 3D model.)A quotient larger than unity indicatesthat the leakoffwas less than reasonable.0

175The power law approximation to surface growthMaterial balance174

. .0·-·.·__

for the unknown o.The solution can be obtained by any numerical root-finding method.The results are shown in Table 8.4.

Having anticipated the exponent between half and unity, the diversity of the calcu­lated exponents raises some serious questions. As was noted in [6], the results do not

(1 - Tlm)iteJ(~ = .

2AcCLA(The same value could be obtained directly from Eq. 8.11.) Finally, a suitable expo­

nent, a., is determined solving the nonlinear equation

Whatever the shape of the fracture face is, the factor K can be computed from (seeEq.8.10)

indicating that for two extremely different surface growth histories (Ci=0.5 and a =1), the opening-time distribution factor differs by less than 20%.

The values given in Eq. 8.31 are often referred to as lower and upper bounds,respectively. The reason is that Nolte [3J, and many others impressed by hispioneering work, postulate that a and TJ are strongly related and that for a dominantfluid loss (TJ == 0) the value of a will be approximately ~ and for a negligible fluid loss(TJ == 0) the exponent a will approach 1. It is difficult to argue about assumptions onthe value of an exponent the existence of which is also an assumption. Nevertheless,we note that once we assume that the fracture surface grows with constant exponent,we cannot relate this exponent to the fluid efficiency, because the fluid efficiency isnot constant during the fracture propagation. It always starts from unity at time zeroand decreases afterwards. For very long pumping times, the fluid efficiency alwaysapproaches zero.

(8.31)go(~) = ~ == 1.57 and go(l) = ~ == 1.33,

As in Example 8.2, first we estimate the fracture surface from Eq. 8.5,

Solutiongiven by Valko and Economides [6] is completely identical to Eq. 8.29.There are two remarkable facts concerning the above result. First, go(Ci) is really

a function of the exponent only. Second,

(8.30)

Consider the five-layer results of the comparative study, given in Table A.5 of theAppendix. Compute the opening-time distribution factor, k. Assuming that the fracturesurface has evolved according to a power law, estimate the exponent, <X. Comment onthe results from the point of view of consistency.22aa[fCa)]2

go(a) = (1+2a)f(2a)'

Example 8.4 ConsistencyCheck In

where fCO') is the Euler gamma function. Figure 8.4 shows the plot of the functiongoCa). Not surprisingly, the function is two when a = 0, i.e. go(a) reaches the abso­lute maximum of K. Indeed, if the exponent is zero, the whole fracture surface hasto be opened at the start of injection and maximum fluid volume is leaked off.

Notice that because of the special properties of the Gamma function (Abramowitzand Stegun [5]), the form

Figure 8.4 The plot of the go function

a

at the end of the injection.The exponent a has been expIicity related to the theological behavior of the

fluid by many authors (see Section 9.6). Therefore, it is often considered knownin design calculations. In spite of this, to our knowledge Eq. 8.32 has never beenused for design purposes, probably because the analytical form (Eq. 8.30) has notbeen known. Instead, an interpolation technique described in Section 8.4.2 has beenpreferred in the literature.

1.50.5

(8.33)0.60.40.2

which becomes

(8.32)

If the first two assumptions of this section are accepted (i.e. power law surfacegrowth and Carter I Ieakoff), and the exponent a is assumed to be known, the materialbalance for any time instant, during injection, can be written in the form

177Thepower law approximation to surface growthMaterial balance176

----- ------ --- ----- ------------. -_.__ ..._._ .. ------------------------

(8.38)

n-l Aj-Aj-1ib.t+ (2Sp +4CL~)An-l - 2CL~ 2: .

j=l ./n - j + 0.25An =--------------------------~~~-------------

Wn + 2Sp + 4Ccv'fV(8.35)

one can use different approximations for the last term. Assuming (1) that the valueof K is given by

{~ if7]-l

K= ;'

3' if7]-O

where we have introduced the constant ~, which is between zero and one. If~= 1,we take the leakoff rate at the end of the time interval, i.e. we underestimate it. If~= 0.5, we take the leakoff rate at the middle of the time interval, which is a betterchoice. The optimal choice is, however, ~=0.25 since then the actualleakoff rate isexactly the average leakoff rate for the first time interval. Our numerical procedure,therefore, will be based on

(8.34)

Returning to the formal material balance (Eq. 8.9) rewritten for the constant injectioncase:

8.4.2 TheCombination of the Power Law Assumptionwith Interpolation

The simple result of Eq. 8.18 should hold exactly if the leakoff obeyed the Carterequation I and the width opened into its final value at the very first moment. Sincethe latter cannot be true we may wish to investigate the question: Can Eq. 8.18 stillwork as a bound on the surface growth?

To answer this question, we develop a simple numerical procedure. We assumethat the injection rate is constant and discretize the time with constant time step, b.t.The current time has subscript n{tn = nb.t) and we allow both the surface and theaverage width to vary with time. The finite difference form of Eq. 8.17 is

mean that any of the computer codes are wrong. Since the fracture surface does not grow~ccording to a power law in these numerical models, some discrepancy from the antic­~pated.interva.I may be justified. The existence of large discrepancies (including phys­ically impossible K values and corresponding negative exponents), however, suggeststhat the common language has not been found yet. 0

8.5 Numerical Material Balance

We note again that it is difficult to accept these assumptions because the exponentis considered to be constant during propagation while the fluid efficiency decreasesmonotonically. Indeed, how does the fracture surface know which exponent a it hasto grow with, if this exponent is determined by the final value of fluid efficiency,which depends very much on the time when the injection is stopped? Nevertheless,Eq. 8.36 is used extensively in design calculations (Meng [7]). It is a useful equationbecause in "design mode", when the fracture surface is specified and the width iscalculated from principles beyond the material balance, it yields a simple quadraticequation from which the necessary time of injection (first its square root) can beeasily obtained. Since a suitable value of the fluid efficiency is involved, Eq. 8.35is used simultaneously with Eq. 8.5 embedded into an iteration loop. The procedurewill be illustrated in Section 9.2.3.

(8.36)it r: 8- =w +Ccv t[37] + (1 - 1);1"].A

and (2) that K varies linearly between these bound with the final 7],the global materialbalance is often used in the form

Numerical material balance 179178 Material balance

Table 8.4 Opening-time distributionfactor and estimated powerlaw exponent for the dataof Table A.4

Kc etc

1 0.91 3.062 1.52 0.593 1.03 2.174 0.24 55.225 1.38 0.886 1.39 0.867 1.33 1.01S"9 3.06 -0.48

10 0.81 4.0011 1.26 1.2012 1.06 2.0513 0.59 8.4114 1.56 0.5215 1.41 0.8116'17"18'19"20 2.25 -0.17

'The necessary informationhas no! beenprovidedby the modeler

._- ----_ ... --._ "'--_ ..'---"'---'-.'--. __ .

(8.41)l hf CLVL = 2.6.x~t -- dh' + 2~x[hJ(t + 6.t) - hJ(t)]Sp,

a ..;t=rand the surface opening-time, r, is a function of the lateral location, x,=the verticallocation, h'. (In writing Eqs, 8.40 and 8.41 we emphasize only that independentvariable which is important in the given term.)

where¢ Numericalwith n, steps Exactn, = 128 n, = 256 n, = 512 n, = 1024

0 44.1 43.6 44.2 44.3 44.30.25 43.4 42.9 43.7 43.70.5 42.7 42.3 43.1 43.10.75 42.1 41.7 42.4 42.5

41.6 44.1 41.8 41.8

Table 8.5 Created fracture surface, Ac, in 1000 nr' for different exponentsof width growth, ¢. Numerical solution (Eq. 8.38) with differentfl. and the exact solution (Eq. 8.18)

The global material balance Eq. 8.1 is a simple bookkeeping for the. whole timeperiod and for the whole fracture. Equation 8.17 is already more. det~lled bec~useit can account for any time instant. If we think about every location m every umeinstant, we can formulate an even more detailed material balance. .

For generality, we assume that not only the flow rate, .q, and th~ cross sectionalarea, Ac, vary both with time and location, but also the height, h (FIgure 8.5). Then,an elementary volume (control volume) between location x and x.+ ~x changesduring the time interval between t and t + ~t according to the equation

volume flowing in-volume flowing out-volume leaking offeevolume increase

q(x)6.t - q(x + ~x)~t - VL = [Ac(t + ~t) - Ac(t)]~x, (8.40)

8.6 Differential Material Balance

Applying the numerical procedure of Eq. 8.37, we have to decide the number of timeintervals, n : We apply the simple technique, doubling the number of steps until theresult does not change up to three digits. The results are shown in Table 8.5. It happensthat 210 time steps are enough in all cases. We can check our result for the ¢= 0 casesince Eq. 8.18 provides the solution to this particular case. The identity of the resultsgives further confidence concerning the numerical results.

The important message of the example is that the "constant width during propa­gation" assumption provides an upper bound of the created surface if the final widthis known. (Our results do not imply that the surface should be below 44.3 m2 in thebase case of the comparative study because the final width from different models maybe different from We = 0.015 m. However, our results do imply that the quotients inTable 8.3 should be below unity, if the models really applied Carter's equation I.)

For completeness, we calculate the opening-time distribution

aJ]rnOl) _ Kc =D.n~+a)

Table 8.6 shows the results. As seen from the table, the constant width (¢ = 0)and the moderate width growth (4J = 0.25) result in K factors lying indeed betweenthe Nolte [3] bounds. (Furthermore, these results imply that a~y a in Table ,8.4 largerthan 0.9 contradicts common sense, if the modelers used indeed Carter s leakoffequation I.) 0

Solution

and the corresponding estimate of the power law exponent of the fracture surface growthfrom the numerical solution of the equation

The specific design problem of Example 8.1 corresponds to i= 0.0662 m3/s (25 bpmper wing), t,= 12000 s (200 min) and CL = 9.84 x 10-6 rn/s1/2 (0.00025 ft/minl!2).Assume that the final width is w. = 0.015 m. Calculate the fracture surface assumingdifferent width-growth exponents, 4J = 0, 0.25, 0.5, 0.75, and L

because with ¢= 0 it contains the constant-width problem as a special case, thesolution of which is known exactly. In the following we consider a problem withinput data similar to the specification of the comparison study [1] to arrive at usefulconclusions.

¢ A, K, 01,

in 1000 m2

0 44_3 1.37 0.920.25 43.7 1.48 0.670.5 43.1 1.59 0.470.75 42.5 1.72 0.28

41.8 L87 0.12Example 8.5 Material Balance with Power Law Width Growth

(8.39)

Table 8.6 Opening-time distribution factor K andestimated power law exponent offracture surface growth, a. for the dataof Table 8.5

where Ao = 0, and we have to know the law according to which the width grows.Of particular interest is a simple power-law-type width growth with exponent ¢,

Wj = We C~tr= We (:er181Differential material balance

Material balance180

-- ._ ...._ .. -- --_ .._-- ..__ .... ---------.---.~

Filtercake

Figure 8.6 Fracture filtercake, invaded zone, reservoir and respective pressure gradients for fluidleakoff

'/ '{ Invadedzone

where /!,Pface is the pressure drop across the fracture face dominated by the filter­cake, /!,Ppiz is the pressure drop across a polymer invaded zone and /!,Pres is ~hepressure drop in the reservoir. This concept is shown in Figure 8.6. In a senesof experimental works (Mayerhofer et at. [101; Zeilinger, et at. [11]) using typicalhydraulic fracturing fluids (e.g. borate and zirconate crosslinked fluids) and coresof permeability less than 5 x-iS m2 (5 md), no appreciable polymer invaded zonewas detected. So, the second term in the right-hand side of Eq. 8.47 can be ignored.

(8.47)

The great advantage of the Carter I fluid loss model is its exceptional simplicity,especially if the spurt-loss coefficient is considered to be zero. The price of simplicityis that the form of the leakoff volume vs. time curve is restricted. Flow in a porousmedium might be more complicated, and in certain cases a more detailed fluid-lossmodel might be necessary.

An alternative means to describe fracturing fluid leakoff is to decompose andmodel rigorously the controlling phenomena. First, the total pressure gradient frominside a created fracture to the reservoir, /!'P, at any time during the injection, canbe written as

8.7 Leakoff as Flow in the Porous Medium

The material balance, even in its differential form, is not a closed mathematicalmodel. It is rather a framework which has to be filled with content using physicalprinciples beyond the conservation of mass. This will be done in the next chapters.In particular, Eq. 8.43 with boundary conditions is the basis of the Nordgren-Kempmodel and virtually all pseudo-3D models.

183Leakoff as flow in the porous medium

Equation 8.45 is the so-called Stefan's boundary condition, somewhat overlooked byNordgren [8] but clarified by Kemp [9]. It shows that the flow rate at the tip is usedby creating new volume and by spurt loss occurring while creating new surface.

(8.46)dxf

uf(t) =-.dt

where the tip propagation velocity, uf, is defined as the growth-rate of the fracturelength

(8.45)at x = x f : q(x, t) = U f (t)Ac (x, t) + 2uf(t)hf(x, t)Sp'

which is the material balance for the wellbore connection.(2) The material balance at the tip is

(8.44)qtx, t) = iCt),at x = 0:

Considering the material balance, there are two boundary conditions. (Additionalboundary conditions may arise if other variables are introduced.)(1) At the wellbore, the flow rate in the fracture is the injected flow rate

(8.43)

where the opening-time function is simply the inverse of the length-growth functionx f(t). After taking the limits /!'t --'>- 0 and Lll --'>- 0 Eq. 8.42 becomes

aq + 2hfCL + aAc =o.ax ~ at

C8.42),:_q...:...CX_+_Lll_:_)------..:q...:...CA_:_)+ _2h_f_C_L+ AcCt + /!'t) - AcCt) = 0,

f!,x ...;t=r /!'t

Nordgren [8J considered an important special case where the height is constant.Then

Figure 8.S Local material balance variables

-4----x

h(x)

Ac(X)

----x+L1x(;lq(x)

Material balance182

(8.53)

The pressure drop in the reservoir can be tracked readily by employing a pressuretransient model for injection into a porous medium from an infinite conductivityfracture. For this purpose, known solutions are available in the petroleum engineeringliterature. The only additional problem we face is that the surface area is increasingduring fracture propagation. Therefore, for every time instant we have a differentfracture length which, in turn, affects the computation of the dimensionless time.

The general solution for the injection of fluid through a fracture of any specifiedfracture conductivity into a fissured (w =1= 1) or homogeneous (w = 1) reservoir hasbeen presented by Cinco-Ley and Meng [14]. Assuming segments of equal lengthwithxDj located at the jth segment, the equation can be written (in Laplace space) as

Pres.D(s) - !t ZhDi(S) LtD''''! [KO(XDi-x')ySj(S5 + KO(XDj +x'h/sf(s)] dx'i-I :AD,

8.7.2 Pressure Drop in the Reservoir

(8.52)(D. Pface..;t;) 1q= A-,lr/.lfRO ..ji

where the quantity in parentheses can be interpreted as a "pseudo" leakoff coefficient.In this approach, the time is measured from the start of the leakoff (or opening ofthe surface). If the surface is opened during a longer period, the time is measuredfrom the opening of the first element of the surface.

The value of Ro (expressed in m") can be obtained either in the laboratory (whereit will be a weak function of the selected characteristic time, te) or, preferably, in thefield from an injection test, where we can select the characteristic time to be equalto pumping time, t., as will be described in Chapter 9.

Figure 8.7 shows schematically the evolution of fracture area at different timeswhile fresh filter-cake is deposited on the older walls. At the fracture tip, the actualdeposition of the filter-cake is delayed which may cause considerable fluid loss. Thetransient leakoff model implies an average filter-cake resistance for each evolvingfracture area. In fact, we can rewrite Eqs. 8.49 and 8.51 into the form

(8.51)RD = fi.vr;

where K2 is the rate of pressure increase (given in Pals) and A is the retardation time(given in seconds). Except for very early time, the term itt, is much larger than thesecond term and therefore Eq. 8.50 becomes simply (Mayerhofer et al. [13])

Leakoffas flow in the porous medium 185

-------- ~....---- .... ------------------------------

Figure 8.7 Propagating fracture with growing filtercake

(8.50)

wh~re /.l f is the filtrate viscosity, Ro is the dimensioned, final and characteristicreslstanc~ of th~ filt:r-cake, :,hich is reached during a characteristic time t.. Usingthe _Kelv1O-VOIgt :l.scoelastlc model for the description of the flow through acontinuously depositing fracture filter-cake as depicted in Figure 8.7, Mayerhoferet al. [12J have shown that the dimensionless filter-cake resistance is

(8.49)

8.7.1 Filter-cake Pressure Drop

The filter-cake pressure term has been given by Mayerhofer et al. [12] as

lr/·I.Ro!).Pface= --;;:-RDq,

(8.48)D.p(t) = !).Pface(t) + !).Pres{t).

I:I0wever, this is not th~ case when using linear gels in higher-permeability forma­nons, where ~ polymer Invaded zone is likely to be formed. This would be observedfor exa~p~e 10 fracspack treatments. In this section, it will be assumed that filter­cake building fracturing fluids are used in relatively lower permeability formations,and t~us, only the filter-cake and the reservoir terms in Eq. 8.47 will be considered.Equation 8.47 reduces to

184 Material balance

1. Warpinski N.R., Mosochovidis, Z.A., Parker C.O. and Abon-Sayed, I.S.: ComparisonStudy of Hydraulic Fracturing Models: Test Case-GRI-Staged Field Experiment No. 3,SPE Production & Facilities, 9(1), 7-16,1994.

2. Howard G.C. and Fast, c.R.: Optimum Fluid Characteristics for Fracture Extension,Drilling and Production Prac., API, 261-270, 1957 (Appendix by E.D. Carter),

3. Nolte, K.G.: Determination of Proppant and Fluid Schedules from Fracturing PressureDecline, SPEPE, (July), 225-265, 1986 (originally paper SPE 8341, 1979).

References

Equation 8.63 can be used in a hydraulic fracture model. It allows determinationof the leakoff rate at time instant t; if the total pressure difference between thefracture and the reservoir is known, as well as the history of the leakoff process. Thedimensionless pressure solution, PD(tn - tj-1), has to be determined with respect toa dimensionless time which takes into account the actual fracture length at tn. Theabove leakoff model will be used in the next chapter.

(8.63)

and a simple rearrangement yields

(8.62)

Substituting Eqs. 8.61, 8.49 and 8.51 into Eq. 8.48 we obtain

JT:J.LfRo fin IL ~t:l.p(tn) = ~ -qn + ~ L./qj - qj-l)PD(tn - tj-l),n te JT: fb f j=l

8.7.3 Leakoff Rate from Combining the Resistances(Ehlig-Economides et al, [6J)

(8.61)

wing, qn:

In general Pres.D can be obtained from solving the system of Eqs. 8.53 and 8.54 orusing a type curve [14].

According to Duhamel's principle, the linearity of the diffusivity equation allowsthe application of the superposition theorem as a sequence of constant rates wherethe rate history is known and the only unknown is the actual leakoff rate from one

(8.60)Pres.D = ..j7iiD;f.

The inverted form of Eq. 8.59 is in wide use in petroleum engineering and issimply

187References

--------.----------------------------~----~---

(8.59)_ JT:Pres,D = 2s3(2 .

4>f is the fissure porosity, and clf is the fissure total compressibility.For w = 1 (homogeneous reservoir) and for kfb f ~ 00, the system of Eqs. 8.53

and 8.54 has a well known limiting solution

J.L(¢el)ma x} .kfb h2 IS the dimensionless matrix hydraulic diffusivity,

___ rna

j.J.(4)ci)r

TfmaD=

In Eqs. 8.53 to 8.58"

(k b ) kfbf. th dim .f f D = -k-- IS e ensionless fracture conductivity

f~f '

AfbhmaVbAfD = V ma = Afmahma is the dimensionless fracture network area,

kma

(8.58)

(8.57)(J)= ¢etf 4>fctf

¢erf + 4>macima = (¢el) /

and A as the interporosity flow coefficient

The Cinco.-Ley and Meng [14J solution follows the great tradition of double porositysystems pioneered by Warren and Root [15] using (l) as the dimensionless fissurestorativity:

(8.56)f(s)=w+ (l-w)Af(1- w)s+Af

(8.55)f(s) = (J)+ (1- W)AfD~ tanh (~),

while for pseudosteady-state flow it is

For transient interporosity flow,

(8.54)n .6..xL qfDi(S) = -.i=l S

Wr~ting this .equation for every fracture segment, a system of n equations is obtainedwhich con tams n + 1unknowns. The first n unknowns are the dimensionless flowrates per unit- of fracture length, qfDj(s), i = 1, ... , n, and the (n + 1)-th one isPres,D(S).One additional equation results if we recall that the flow leaving the fracturehas to flow through the filter-cake. Since this leakoff rate is used to create thedimensionless variable for the given time instant, we have

Material balance186

------_._-------------------------

9.1 Width Equations of the Early 20 Models

9.1.1 Perkins-Kern Width Equation

The elegantly simple approach of Perkins and Kern [1] envisions the ~acture ~sshown in Figure 4.2. The model assumes that the condition of plane. strain ~olds mevery vertical plane normal to the direction of propagation, but - unhk~ the ngorousplane strain case - the stress and strain state is not exactly the sam~ m subsequ~ntplanes. In other words, the model applies a quasi-~Iane ~train a:'sumptJon. Neglectingthe variation of pressure along the vertical coordinate, It considers the net pressure,.t». as a function of the lateral coordinate x: As ,:"e saw .in Chap~er 2, the constantpressure gives rise to an elliptical cross section With maximum WIdth

Engineering models for the propagation of a hydraulically induce~ .fracture comb~neelasticity, fluid flow, material balance and (in some cases) an additional propaga.tlOncriterion. In this chapter we derive the basic models and examine the assumptionsand simplifications of the individual approaches. . . .

Given a fluid injection history, a model should predict the evolution With timeof the fracture dimensions and the wellbore pressure. For design purposes, a roughdescription of the geometry might be sufficient, and hence, simple models, predictinglength and average width at the end of pumping, are very useful. Indee~, the le~gthis an important variable from the production point of view,. and so IS th~ Widthwhich provides the potential to establish conductivity b~ placmg. proppant into thefracture. Models which predict these two dimensions while the third one - fractureheight - is fixed are referred to as 2D models. Even when the fracture. height .is notfixed a priori, but is postulated to have a circular surface, the model IS consideredto be 2D.

9COUPLING OF ELASTICITY,FLOW AND MATERIALBALANCE

4. Nolte, K.G. Fracturing-Pressure Analysis. In Recent Advances in Hydraulic FracturingGildly, J.L et al. (ed.) Monograph Series, SPE, Richardson, TX (1989) Vol. 12 Chap. 14.

5. Abramowitz, M. and Stegun, I.A. (ed.).: Handbook of Mathematical Functions (9th ed.),Dover, NY, 1989.

6. Valko,P. and Economides, MJ.: Fracture Height Containment With Continuum DamageMechanics, paper SPE 26598, 1993.

7. Meng, Hai-Zui: The Optimization of Propped Fracture Treatments, in: Economides, M.J.and Nolte, K.G.: Reservoir Stimulation (2nd ed.), Prentice Hall, Englewood Cliffs, N.J,1989.

8. Nordgren, R.P.: Propagation of a Vertical Hydraulic Fracture, SPEJ, (Aug.), 306-314,1972; Trans. AIME, 253, 1972.

9. Kemp, L.F.: Study of Nordgren's Equation of Hydraulic Fracturing, SPE ProductionEngineering, (Aug.), 311-314, 1990.

10. Mayerhofer, M.1., Economides, M.1. and Nolte, K.G.: Experimental Study of FracturingFluid Loss, paper CIM/AOSTRA 91-92 presented at the Annual Technical Conferenceof the Petroleum Society of CIM and AOSTRA, Banff, April 21-24.

11. Zeilinger, S., Mayerfoher, M.1. and Economides, M.J.: A Comparison of the Fluid-LossProperties of Borate-, Zirconate-Crosslinked and Non-Crosslinked Fracturing Fluids,paper SPE 23435, 1991.

12. Mayerhofer, MJ., Economides, M.J. and Nolte, K.G.: An Experimental and Funda­mental Interpretation of Fracturing Filtercake Fluid Loss, paper SPE 22873, 1991.

13. Mayerhofer, M.1., Economides, M.1. and Ehlig-Economides, C.A: Pressure TransientAnalysis of Fracture Calibration Tests, Paper SPE 26527 presented at 68th AnnualTechnical Conference and Exhibition, Houston, Texas, 3-6 October 1993.

14. Cinco-Ley, H. and Meng, H.Z.: Pressure Transient Analysis of Wells with FiniteConductivity Vertical Fracture in Double Porosity Reservoirs, Paper SPE 18172presented at the 63rd Annual Technical Conference and Exhibition held in Houston,TX, October 2-5, 1988.

15. Warren, J.E. and Root, P.1.: The Behavior of Naturally Fractured Reservoirs, SPEJ,(Sept.), 245-55, 1963; Trans. AIME, 228, 1963.

16. Ehlig-Economides, CA., Fan, Y. and Econornides, M.J.: Interpretation Model for Frac­ture Calibration Tests in Naturally Fractured Reservoirs, Paper SPE 28690 presentedat the International Petroleum Conference & Exhibition, Veracruz, Mexico, October10-13,1994.

1B8 Material balance

(9.15)

(9.14)

(9.13)

(9.12)

The shape factor contains tt/4 because the vertical shape is an ellipse. Also it containsanother factor which accounts for the lateral variation of the width. Since the solution

(9.7)w= YWw,o,

Equation 9.6 is the Perkins-Kern width equation. Knowing the maximum width atthe wellbore, we can calculate the average width, multiplying it by a constant shapefactor, y,

and by combining Eqs. 9.14 and 9.1, the net pressure is given by

= (80)1/4 (E'4J1-P) 115 t1/5 = 1.52 (E'4J-Li2) 1/5 t1/5.Pn,w 2 h6 h6

7r f I

(9.6)_ (512)1/4 (J1-iXf) 1/4 (J-LiXf) 1/4W 0 - - -- :::3.57--w, n E' E'

Taking the fourth root, maximum wellbore width is given byliS (2 ) 1/5 ('2 ) 1/5= (2560) _!...!::_ tl/5 = 3.04 _!__!!:_ t1/5,

~o ~ E~ p~

(9.5)

and hence, by combining Eqs. 9.1 and 9.4,

4 512J-Lixfw - --'--~w.o - 7rE'

(9.4)

from which the length growth is obtained as

To obtain a solution, one has to specify the net pressure at any location. Perkinsand Kern [11 postulated that the net pressure is zero at the tip. Integrating Eq. 9.3between the wellbore and the tip yields

4 4 32E'3 uixIPn,w - 0 = h4

7r I

(

3) 1/4 ( . 5 4) 1/4 ( . 5h4 ) 1/4. _ 5127r J1-lxfhf J1-IXf Iit = wXfhl = -- --- = 2.24 --- ,625 E' E'

dp SJ1-iE'3dx = -7rhj~'

Thus, in the no-leakoff Perkins-Kern model, fracture length grows with the ~power of time and the maximum width (and average width) grows with the! powerof time. More importantly, the pressure is an increasing function of time, growingwith exponent !.

x = ( 625 ) 1/5 (i3 E' ) 1/5 t4/5 =0.524 (i3 E' ) 1/5 t4/5,f 5127r3 J1-hj J1-h}

The maximum fracture width at the wellbore can be expressed by combining Eqs, 9.6and 9.13 as

It is enlightening to couple the Perkins-Kern width equation with a simple materialbalance valid for the constant-injection-rate/no-leakoff case. Then,

(9.3)

(9.11)

(9,H)

(9.9)

(9.S)

191

Combining Eqs, 9,6, 9,7 and 9.10, the average width is given by

14 7rY = 7r45 = '5 = 0.628.and hence

(9.2)t::.p 64ML = 7IW~h/

where J1-is the viscosity of the fluid and q is the flow rate at the lateral coordinate x.To simplify the treatment, Perkins and Kern [1] approximated the flow rate with theinjection rate (q ~ i) where i (as before) is for one wing. In reality, q < i, not onlybecause part of the fluid leaks off but also because the increase of width "consumes"another part of the fluid. In fact, if there is anything that is more or less constantalong the fracture, it is not the flow rate but rather the flow velocity. Nevertheless, ifthe Simplification is accepted and Eq. 9.1 is substituted into Eq. 9.2, the followingdifferential equation is obtained for the pressure:

the lateral component of the shape factor is

where hf is the constant fracture height and E' is the plane strain modulus. Themaximum width, wo, is a function of the lateral coordinate. At the wellbore it isdenoted by Ww,o.

From Chapter 5, the pressure drop of a Newtonian fluid in a limiting ellipticalcross section is given by

~=Jl- x,Ww,o xI

(9.1) of Eq. 9.3 implies2hfPnwo=-p

Width equations of the early 2D modelsCoupling of elasticity, flow and material balance190

,,--" .._.... --_ _-- ..- ---._--_._-_.- ...•._-- -----------------

(9.28)1/6 ( 3 ') 1/6 ( '3E' ) 1/6= (~) i E t2/3 ::::0.539 _,_ t2/3.xf 211r3 JLh} J.Lh}

from which the length is calculated as

(9.27))

1/4 ( . 6 3) 1/4 ( . 6 h3 ) 1/4. _ (211r3 fJ,lxfhf _ fJ,IXf fIt= wxfh f = 16 ----p- - 2.43 £'

therefore the two equations give nearly the same average width for a fracture withequal vertical and horizontal dimension. In Chapter 4, it was establish.ed that for2xf < hf the horizontal plane strain assumption (KGD) is m?re appropnate and for2xf > hf the vertical plane strain assumption (PKN) is phys~caUym?re a~ceptabl~.Equation 9.26 shows that the "transition" between the models IS essentially smooth.

As we did for the Perkins-Kern width equation, now we couple the Geertsma-deKlerk width equation with a simple material balance valid for the constant-injection­rate/no-leakoff case. Then,

(9.26)

If we compare Eq.9.25 with Eq. 9.11, we can calculate the ratio of theGeertsma-de Klerk width to the Perkins-Kern width:

WGK = (21 X 625)1/4 (2xf) 1/4 = 0.95 (2xf)I/4;WPK 32 x 512 hf hf

(9.25)(

, ) 1/4 ( . 2) 1/4 ( . 2) 1/4_ 211r~ fJ,IXf IJ.IXfW = 16 E'hf = 2.53 £'hf

and the average width is simply

(9.24)ttY = - = 0.785,

4

It is convenient to refer to Eq. 9.23 as the GDK equation. If,however, the conceptof horizontal plane strain approximation is to be emphasized, it is usual to speakabout the KGD model or KGD view of the fracture. In this case the shape factorhas no vertical component Its horizontal component is simply it /4 because of theelliptical shape of the constant-pressure fracture. Thus, the shape factor is given by

(9.23)(

") 1/4 ( ''') 1/4. _ (336) 1/4 fJ,IXj = fJ,IXjWw - 3.22 E'h

tt Phf f

therefore the wellbore width is given by

(9.22)336fJ,ix} .Ww = ,

7rE'hf

From Eqs. 9.18 and 9.21,

193Width equations of the early 2D models

(9.21)

(9.20)C~loX! :3 ctx) = ~~.'

Combining Eqs. 9.17 to 9.20, we obtain

(3) the average value of 1/w3 can be obtained from its value at the wellbore multipliedby a constant which is postulated to be 7j7r (with explanation later); thus,

(9.19)Pn.tip = 0,

(2) the net pressure at the fracture tip is zero

(9.18)4Xf Pn.wWW= E' ,

Now suppose that(1) the pressure equals the wellbore pressure almost everywhere and hence, inEq, 9.16, the average pressure can be substituted by the wellbore pressure:

(9.17)12fJ,ixf ( 1 loX! 1 )Pn.w- Pn.tip = -h-- ~ 3" dx .f xf 0 w

(Note that for the KGD model the width at Ww = w,,·.o.) Assume again that theflow rate is everywhere equal to the injection rate, i, Since now the flow channel isrectangular, the total pressure drop from the wellbore to the fracture tip (assuming aNewtonian fluid) can be calculated from (see Table 5.2)

(9.16)

The first model of hydraulic fracturing, elaborated by Khristianovich and Zheltov[2,3], was based on another geometric picture, shown in Figure 4.3. Those authorsassumed that the fracture opens with the same width at any vertical coordinate withinthe fixed height, hf. The underlying physical hypothesis is that the fracture facesslide freely at the top and bottom of the layer. The resulting fracture cross sectionis a rectangle. The width is a function of the coordinate .r. It is determined from theplane strain assumption, now applied in (every) horizontal plane. The Khristianovichand Zheltov [2,3] model contained another interesting assumption: the existence ofa non-wetted zone near the tip.

Geertsma and de Klerk [4] accepted the main assumptions of Khristianovich andZheltov [2,3] and - using some innovative mathematical techniques - reduced itinto an explicit formula. First, we provide a simplified derivation.

Recalling the plane strain solution of Chapter 2, a constant net pressure, Pn' wouldcause a fracture width at the wellbore given by

4x/Pnr-r :»:

9.1.2 Geertsma-deKlerk Width Equation

Coupling of elasticity, flow and material balance192

-_ ....•_---_ ...._-------------

A favorable consequence of the above is that it can be generalized for a radi~llypropagating fracture, i.e. xI = h! /2 = R. In this case the same average ~acture WIdthwill be calculated from (1) a PK width equation with given xI, (2) a honzontal planestrain width equation with h! = Zx] and (3) a radial width equation with R=X!.

(9.34)

(9.38)(. 2) 1/4 ( . 2) 1/4

- - 21/4 X 224 j.LIXI = 2.66 j.LIX!w - . E'h! E'lt]

(9.33). npn.w . n(p - O"min)Xo = x! Sin = x! sm _'-- _

2(Pn,w - Pn.tip) 2p

Starting from Eqs. 9.31 and 9.32, Geertsma and deKIerk made a series of approxi­mations. They described the width at the wellbore by

and

9.1.3 Radial Width Equation

A closer look at Eq. 9.26 may convince us that ahorizontal plane strain (KGD-typ~)width equation could be postulated with the requirement that the constant factor mEq. 9.26 be unity. Such a width equation would have the form

(9.32)

q1 = (x} - x~)1/2

q2 = (x} - x2)1/2

_ { (x6 - x2)1/2, if x :5 Xoq3 - (2 2)1/2

X -xo ' ifx>xo

where

We do not believe that this assumption has strong basis. We gave our versionof the "derivation" only because we wished to show that the Geertsma-de KIerkwidth equation is not inherently connected with the "zero absolute pressure at thetip" assumption. Of course the only authentic derivat.ion of the Geerts~a-de KIerkequation is the one given by those authors, and the interested reader IS referred totheir original paper [4].

(9.31)

(9.37)

which was derived by the "visual examination of a plot of the left-hand-side" ofEq. 9.36. The plot was created by substituting Eq, 9.31 into the left-hand side andcalculating the integral numerically.

Our factor of 7/rr in Eq. 9.20 can be "explained" if we assume that the length ofthe unwetted zone at the tip is always 0.0877x], because then (using the ellipticalshape)

(9.36)

From Eqs. 9.34 and 9.35, they obtained our Eq. 9.16. Also, our Eq. 9.20 was notused by those authors, who used instead

(9.30)

(9.35)R_ 2_xoP- amin ::::;- P- .

n xf

(9.29)(5376) 1/6 ( i3 j.L ) 1/6 ( i3u. ) 1/6

W = -- -- tI/3 = 2.36 __ ti/3.w 7(3 E'h3 E'h3

! fSubstituting Eqs. 9.28 and 9.29 into Eq. 9.21 the net pressure is obtained as

Pn,w = (~~) 1/3 (E'2j.L)1/3rI/3 = 1.09(E'2j.L)1/3t-i/3.

Thus, in the no-leakoff Geertsma-de Klerk model the length grows with the ~power of time; the wellbore width (and average width) grows with the! power oftime. The net wellbore pressure behavior deserves further attention.

Equation 9.30 shows that the net pressure decreases with time. This is a wellknown result of the model. In massive hydraulic fracturing, however, the net pressureis more often increasing with time. Even more startling is the (less well known) otherconsequence of Eq. 9.30: The net pressure does not depend on the injection rate. Thiscontradicts the daily experience of fracturing workers.

Some notes regarding our Eq. 9.23 are necessary here. We gave a possible "deriva­tion" emphasizing the "symmetry" with respect to the Perkins-Kern model. The threeassumptions leading to Eqs. 9.18 to 9.20 were not stated by Geertsma and deKIerk.On the contrary, they postulated that the tip net pressure equals minus one timesthe minimum far-field stress, i.e. there is vacuum near the tip. They considered a"zipper" crack with piecewise constant pressure having the value p in the intervalfrom x = 0 to a certain value x = Xo, and the value zero from x =Xo to the tip,x = x!. The width as a function of the location for that case was derived in ourExample 2.6 and takes the form

which is a good approximation of Eq. 9.31 for Xo -4- xf. Also they stated thatEq. 9.33 can be approximated by

The wellbore width can be calculated now from Eq. 9.22 and 9.28 as

195Width equations of the early 20 modelsCoupling of elasticity, flow and material balance194

---~------~------------------,---,- -- ----- ------------- "-------_

0.00025 ft/minl/2o in170 It8.89 x 106 psi200 cp50/2 = 25 bpm200 min

9.84 x 10-6 m/sl/2Om51.8 m6.13 x 1010 Pa0.2 Pa-s0.0662 m3/s12000 s

Table 9.1 Input data for Example 9.1

Determine the created fracture length, maximum wellbore width, average width andfluid efficiency for the data used in the comparative study by Warpinski et al. [9} fora pumping time of 200 min (simulation rnode.) The input data (changed into 5I units)are given in Table 9.1. Notice that the injection rate, i, is defined for one wing. (Theplane strain modulus, E', was obtained from E' =£/(1- \}2), where E is the Young'smodulus and l! is the Poisson ratio as given in Table 2.1).

Repeat the calculation in design mode if the target length is 1000 m (3280 ft).

Example9.1 PKN-C Simulationand Design

(9.43)£'

PII,\>" = 2hf Ww.O·

we obtain a closed system of equations from which either the length (simulationmode) or the time (design mode) can be easily determined using a numerical root­finding method. The corresponding net wellbore pressure is determined from

(9.42)2CL.fiiiwhere {3= ,

w+2Sp

(w + 2S )i [., 2{3]xf = ., {J exp({3-)erfc({3)+ '- - 14Cinhj ..;7[

and the constant in Eq. 9.12 becomes 2.05 instead of 2.24.For reasons that are not delineated here, several other shape factors are also in

use by certain authors, e.g. 3n"/16 (see Economides et al. [7], p. 434) or 7[2/16 (seeNierode [81, p. 400).

The letter "C" in the name PKN-C denotes that we use the Carter II solution ofthe material balance. Expressing the fracture length from Eq. 8.18

(9.41)

where the constant factor, 3.27, is derived from an interesting limiting result ofNordgren [6], who himself has never advocated its use in such context.

(In the petroleum engineering literature, Eq. 9.40 is often written in terms of(1 - v)/G [= 2/E'] and total injection rate for two wings. Then the factor 3.27 hasto be reduced by 41/4 and it becomes 2.31.)

Using the shape factor as given by Eq. 9.10, the average width is obtained from

197Algebraic (20) models as used in design

(9.40)( fLiX ) 1/4Ww,O = 3.27 E/

A considerable part of the petroleum engineering literature considers Eq. 9.6 some­what inaccurate and uses instead an "improved" constant:

9.2.1 PKN-C

At this point we have sufficient information to present some widely used designmodels. It is assumed, that hf, E', i, u, CL and Sp are known, and two problemsare considered. In design mode, a target length xf is given and the pumping time, t.;is determined from the model; final net wellbore pressure, wellbore width, averagewidth and fluid efficiency are useful byproducts of the calculations. In simulationmode, the pumping time is given, and the length is determined from the model; thesame byproducts are generated.

The governing idea behind these algebraic models is to assume the"validity of awidth equation such as Eq. 9.11 (even if the leakoff cannot be neglected) and combineit with a suitable form of the material balance. Remember that the derivation of thesesimple width equations involves the assumption of uniform flow rate everywhere inthe fracture. As a fracture changes width and/or loses fluid through the fracture faces,the assumption of constant flow rate along the lateral coordinate is not correct. Oncewe accept this theoretical inconvenience (which seems to affect the numerical valuescalculated from the models, but not their characteristic behavior) the treatment isrigorous and the results are very useful.

9.2 Algebraic (20) Models as Used in Design

(Notice that the same average width means smaller volume because the area of thecircle is less than the area of the square.) Depending on the author's preference inapplying analogy, different constants are used in the literature. Geertsma [5] gives1.32 and 1.62, derived from some additional considerations.

The constant-injection!no-Ieakoff case for a radial model results in the following:The radius, R, varies with the ~ power of the injection time, and hence, the fracturesurface area grows with the ~ power of time. The width grows with the ~ power oftime (since the volume is a linear function of time). The pressure is proportional tothe minus ~ power of time, thus giving the same characteristic pressure behavior asthe KGD modeL Here, however, the decreasing pressure seems to be more logicalbecause radial fracture growth implies the absence of growth barriers in the verticaldirection, i.e, less resistance to growth.

(9.39)_ (fLiR) 1/4w=2.24 E!

The corresponding radial width equation is

Coupling of elasticity, flow and material balance196

'l= 84.7%Pn.w = 5.30 x lOS Pa (77 psi)

t,= 1.88 X 10' s (313 min)Ww = 2.59 X 10-2 m (1.02 in)IV = 2.03 X 10-2 m (0.800 in)

Table 9..5 KGD-C Design

'l= 85.7%P'.W = 4.58 x lOS Pa (67 psi)

Xf = 748 m (2450 ft)Ww =2.24 X 10-2 m (0.881 in)IV= 1.76 X 10-2 m (0.692 in)

Table 9.4 KGD-C Simulation

The results are given in Table 9.4. The design calculation for xf = 1000 m yieldsTable 9.5. 0

The KGD width is larger than the PKN width. Consequently, the KGD lengthis smaller (and the desired pumping time larger) than the respective PKN values.This explains why the KGD model is used from time to time far outside its region

2 x 9.84 x 1O-6.jrr x 12000f3 = ( 0.0662 x 0.2 x x} ) 1/4.

2.53 6.13 x 1010 x 51.8

where

~2 1/4

(0.0662 x 0.2 x Xf )

2.53 10 0.0662 2x = 6.13 x 10 x 51.8 [ex ({32)erfc({3)+ __p_ _ 1] .f 4(9.84 x 10-6)2 x tt x 51.8 p ..fii

Equations 9.25 and 9.42 can be reduced to one equation, eliminating all variables butxr, namely

Solution

Repeat the simulation and design calculation of Example 9.1 with the KGD-C model.

Example 9.2 KGD-C Simulation and Design

(9.44)

The procedure for design and simulation with the Geertsma-de Klerk width equationcombined with the Carter II material balance equation is, of course, similar to the onedescribed for the PKN-C model. Traditionally. this model is called KGD and we addthe letter C to indicate the special form of the material balance. Instead of Eqs. 9.40and 9.41, Eqs. 9.23 and 9.25 are used, respectively. Equation 9.43 is replaced by

£'Pn.w= -Ww·

4xf

9.2.2 KGD-C

199Algebraic (2D) models as used in design

'l =76.5%Pa ,,., = 7.42 X 106 Pa (1080 psi)

/, = 8060 s (134 min)Ww.O = 1.25 X 10-2 m (0.494 in)w = 7.88 X 10-3 m (0.310 in)

Table 9.3 PKN-C design for data of Table 9.1

'l= 74.1%P'.IV = 7.98 X 106 Pa (1160 psi)

XJ = 1340 m (4400 ft)ww,o == 1.35 X 10-2 m (0.531 in)IV= 8.48 X 10-3 m (0.334 in)

Table 9.2 PKN-C simulation with data in Table 9.1

The results are given in Table 9.3. 0

{3== 2 x 9.84 X 10-6 v'7fXt .7.88 X 10-3

where

7.88 X 10-3 x 0.0662 . [ 2{3 ]1000 = 4(9.84 X 10-6)2 x :rr x 51.8 exp(,82)erfc({3) + ..fii - 1 ,

After substitution of the average width, Eq, 9.42 takes the form

( )

1/4IV= 2.05 0.0662 x 0.2 x 1000 = 7.88 X 10-3 m.

6.13 x 1010

The solution (to three significant digits) isxf = 1340 m (4400 ft), which can be obtainedfrom any suitable numerical method. Using Eqs. 9.40, 9.41, 8.5 and 9.43 the necessarybyproducts of the calculation are easily obtained. The results of the calculation arereported in Table 9.2 in a format used throughout this chapter to ease comparison withother models.

The calculations in design mode are very similar to the ones in simulation mode.We substitute the given xf == 1000 minto Eq. 9.41 and calculate the average widthaccording to

2 x 9.84 x 1O-6..jrr x 12000{3= 1/4'

2.05 (0.0662 x 0.2 x Xf)6.13 X 1010

where

(0.0662 x 0.2 x xf) 1/4

2.05 6.13 x 1010 0.0662 [2 2{3]Xf = 4{9.84 X 10-6)2 x tt x 51.8 exp(p )erfc({3) + ..fii - 1 •

Equations 9.41 and 9.42 can be reduced to one equation, eliminating all variables butxf' Introducing the values given in Table 9.1, the equation takes the form

Solution

Coupling of elasticity, flow and material balance198

lJ = 73.5%Pn.w = 7.97 X 106 Pa (1.16 x 1aJ psi)

Xf = 1330 m (4370 ft)W"'.o = 1.35 X 10-2 m (0.530 in)W = 8.46 X 10-3 m (0.333 in)

Table 9.7 PKN-Ct Simulation

Note that the results are very similar to the ones obtained with the PKN-C andPKN-N models. The only reason why we favor the a-type material balance is that it

~ = O.8fi['(0.8) = 1.415.goes> r(3/2 + 0.8)

The results given in Table 9.7 are hardly distinguishable from the ones obtained inExample 9.1: 0

where

XI = 1(4

2 OS(°00662 x 0.2 ~ XI) + 2 x 9.84 x 1O-6..j12000 x go(~). 6.13 X 101

120000.06625l.8

Equations 9.41 and 9.46 can be reduced to one equation, eliminating all variables butXI' Thus,

Solution

Repeat the simulation calculation of Example 9.1 with the PKN-a model.

Example 9.4 PKN-a Simulation

We saw in Section 9.1.1 that a = ~for the no leakoff case. It is reasonable to assumethat the exponent remains the same in the presence of leakoff.

(Note that when applying Eq. 9.46 for any selected time, t, "neither w: nor thefracture has to know" how long the injection will last. Equation 9.46 satisfies theprinciple that the future cannot influence the past, and it can be used to calculate thelength at any time during injection without referring to the width or efficiency at theend of pumping. Unfortunately, the C- and N-type material balances, i.e. Eqs. 9.42and 9.45 do not satisfy this basic requirement.)

Xi = [ a..jirf(a) ] .iii+ 2Sp + 2CL..fi f(3/2 + a)

(9.46)

The analytical solution of the material balance assuming power law length growth(Eq. 8.33) can also be combined with any width equation. Instead of Eq. 9.42, we use

it

9.2.4 PKN-a and KGD-a

201Algebraic (2D) models as used in design

'I= 76.2%Pn."· = 7.42 X 106 Pa (l080 psi)

t, = 8080 s (135 min)W = 7.88 X 10-3 m (0.310 in)wlV.o = 1.25 X 10-2 m (0.494 in)

Table 9.6 PKN-N Design

The results are given in Table 9.6. They are almost identical to the ones obtained inExample 9.1. 0

and hence, Eq. 9.45 takes the form

0.0662 x t -31000 x 51.8 = 7.88 x 10

6 010-Jr [8 1000 x 51.8 x 7.88 x 10-3+ .13x1 t:3 0.0662 x I

(1000 x 51.8 x 7.88 x 10-3) ]+ 1- J[ •

0.0662 x I

5 (0.0662 x 0.2 x 1000) 1/4 _

W = 2.0 = 7.88 x 10-' m.6.13 x 1010

From Eq, 9.41 the average width is given by

Solution

Repeat the design calculation of Example 9.1 with the PKN-N model.

Example 9.3 PKN-N Design

where 1J = hfxfw/it.

(9,45)

we use

The Carter IImaterial balance can be readily replaced by the "interpolation betweenthe Nolte bounds" procedure (Eq. 8.36) shown in Chapter 8. Instead of Eq. 9.42,

9.2.3 PKN-Nand KGD-N

of primary validity; engineering intuition and/or other practical considerations mayrequire a "larger than PKN" width, and the KGD model provides it. The predictednet pressure for the KGD case is an order of magnitude less than the PKN case,indicating that such large KGD fracture is nearly a "perpetuum mobile" needingalmost no energy to propagate.

Coupling of elasticity, flow and material balance200

·...-----..~---------

0.52.87 Pa· 5°.5 0.060 lbf· s°.5/fI2

nK

Table 9.S Power law parameters

Equations 9.53,9.41 and 9.42 can be reduced to the following, eliminating all variablesbut XI:

Solution

Repeat the simulation calculation of Example 9.1with the PKN-C model and th.en w~tbthe KGD-C model, but instead of constant viscosity, assume a power law behavior WIththe parameters of Eq. 5.5 given in Table 9.8.

Example 9.5 PKN-C and KGD-C Simulation Involving Power LawFracturing Fluid

/(2n+2) (.n 2) 1/(2n+2)Ww = 1l.11/(2n+2) X 3.24nl(2n+2) [1:2nr K1/(2n+2) ~fi,

(9.55)

(9.54)f-Le= 2n;1 K c:2nr (~hf r-1 •and the corresponding KGD width equation is

For the KGD model, the results of Section 5.2.2 imply

(9.53)(

'n h1-n ) 1/(2n+2)x K1/(2n+2) I f xf

E'

(9.52)=K [1+ (rr - 1)n]n (25)n-l (_/ )n-lrrn 2 ww,ohf

where we have used y = n/5 to relate average fracture width to the maximumfracture width at the wellbore. .

Substituting Eq. 9.52 into Eq. 9.40 yields the PKN maximum width equation interms of the power law parameters as. [1+ 214n]n/(2n+2)

WW,o = 9.151/(2n+2) x 3.98n/(2n+2) n'

Eq. 9.51 takes the form

_ [1+ (n _ l)n]n (rr2)n-l (_i )n-lf-Le - K 2 -2hnn W f

203Algebraic (2D) models as used in design

-/

where both the average linear velocity, uavg, and maximum width of the ellipse, wo,are functions of the location. In accordance with the Perkins- Kern model, the' flowrate is considered constant and equal to the injection rate. Since the average crosssectional area available for flow is Whj. we substitute uavg = i/ (wh f) to obtain thelinear velocity in a representative (average) cross sectional ellipse. The maximumwidth of an average elliptical cross section is expressed through the average fracturewidth using the geometric relation Wo = 4W/n. (Note that Wo does not correspondto the wellbore but to an "average" location.) After trivial algebraic manipulations,

(9.51)JLe = K [1 + (n -1)n]n (2rruavg)n-l ,1m wohf

There are several ways to incorporate non-Newtonian behavior into the widthequations. A convenient procedure is to add one additional equation connecting theequivalent Newtonian viscosity with the flow rate. Assuming power law behavior,from Table 5.9 the equivalent Newtonian viscosity is given by

9.2.6 Non-Newtonian BehaVior

(9.50)R2nW

where n = -.-.It

where the geometry factor relating the average width to the wellbore maximum widthis postulated mostly by analogy. In this book, we accept the value y = Is, given byGeertsma [5].

The R-N model is the same, except for Eq. 9.48, which is replaced by

(9.49)

and

(9.48)

(W + 2S )i [ 2f3 ]4CIIJ exp(f32) erfc(f3)+ -;-- 1 ,R=2CL.J1ii

where f3 = --­W+2Sp

(9.47)(f-LiR) 1/4W=2.24 - ,E'

The R-C model can be formulated as follows:

9.2.5 Radial Model

lacks the logical contradiction of incorporating an end-value into the description ofa dynamic process.

Coupling of elasticity, flow and material balance202

The inherent drawback of the algebraic 2D models is that they cannot account forthe variation in flow rate with the lateral coordinate. Nordgren'S [6J material balance

, r:-:" Aj -Aj-lz6.t =2CLV t:.t L.J _. +~+ 25p(An - An-I) +AnWn - An-I Wn-1, (9.56)

j=i n J

9.4 Differential 2D Models

Dividing the injection period into 2, 4, 8, 16, ... equal intervals, we check the conver­gence of the length. The change from 128 to 256 time steps does not cause any changesout to three significant digits. Therefore, we accept the results calculated with 256 timesteps. The results given in Table 9.11 are hardly distinguishable from those obtained inExample 9.1. 0

By now the reader should not be surprised that several different combinations of theelastic, flow and material balance submodels exist. In this section, we consider aninteresting case when the width is given by the PKN width equation (Eq. 9.41), butinstead of Eq. 9.42, the more detailed material balance of Section 8.5 is considered.Equation 8.37,

Solution9.3 Numerical Material Balance (NMB) with WidthGrowth

Repeat the simulation calculation of Example 9.1 with the PKN-NMB model.

Example 9.6 PKN-NMB Simulation

for x-: (All other variables are known because they correspond to values of previoustime steps.) Once Xn is known, calculate the new average width, Wn = alx~/4 andcontinue with the next time step. (Clearly, a similar algorithm can be derived for theKGD and the radial width equations.)

The results shown in Table 9.9 indicate a somewhat shorter and wider fracture(compared to the one of Example 9.1). That is the result of increased viscosity. A newitem appearing in the table is the equivalent Newtonian viscosity, calculated for anelliptical cross section which corresponds to average conditions at the end of pumping.Similar calculations for the KGD-C simulation problem yield the results shown inTable 9.10. 0

(9.58);r [1 +214 05] 0.5/(2xO,5+21W = _9.151/(ZxO.5+2) x 3.9S0.5/{2xO.s+2) . x.5 0.5

(0 066205 - 81-05 ) 1/(2xOj-2)X 2.871/(2x05+2)· x ::>1. x

6.13 x 1010 x 51.8 f

and

where(2) At time t«, solve the nonlinear equation

alx~/4 + (4CL~ + zs,», =fJ = 2 x 9.84 x 1O-6J;rx 12000w '

w x 0,0662 [ 2fJ JXI = 4(9.84 x 10-6)2 x J'[ x 51.8 exp(,81)erfc(fJ) + ..fii - 1 ,

Table 9.10 KGD-C Simulation (Power law fracturing fluid, parame­ters in Table 9.8)

(9.57)

is still valid, but now Wn, the average width at time tn, is not known a priori, buthas to be determined from the PKN width equation (Eq. 9.41). Therefore we havea system of two equations with two unknowns. The system can be reduced to onenonlinear equation, which can be solved numerically. Omitting the details of thederivation, the PKN-NMB algorithm is as follows:(1) Determine the constant al in the width equation

_ (J,Li) 1/4 1/4 1/4W = 2.05 E' xf = alx f .

w = 2.27 x 10-2 m (0,892 in)IJ= 88.6%Pn,,, = 7040 X 10l Pa (107 psi)

Xf = 600 m (1970 ft)fl.. = 0.858 Pa .s (858 cp)W",= 2.88 x 10-2 m (1.14 in)

17 =72.6%p•. 'N = 7.95 x 1(1" Pa (1150 psi)

XI = 1320 m (4330 ft)W",.O = 1.34 X 10-2 m (0.529 in)w = 8.44 X 10-3 m (0.332 in)

w = 9.86 X 10-3 m (0.388 in)17 =76.9%p; .w = 9.28 x 106 Pa (1350 psi)

XI = 1200 m (3930 ft)fl., =00409 Pa- s (409 cp)W,,·.O = 1.57 X 10-2 m (0.618 in)

Table 9.11 PKN-NMB Simulation (Data in Table 9.1)Table 9.9 PKN-C Simulation (Power law fracturing fluid, parameters inTable 9.8)

205Differential 2D modelsCoupling of elasticity, flow and material balance204

(9.73)Wo = 0 and xf = 0,at t =0:

The propagation velocity is finite. Hence, in the Nordgren model, the first derivativeof the width (and of the pressure) is infinite (negative). The solution has a singularityat the tip.

To understand singularity, the reader may depict a hypothetical width profile ofthe simple form w = a(xf - x)I/3, where c is a constant. For that profile, the width iszero at the tip, its derivative with respect to the lateral coordinate is (minus) infinityand the derivative of the function w3 = a3(x f - x) is a finite value (in this caseequal to _a3). The solution of the Nordgren equation behaves similarly near the tipat every time instant: The flow rate is zero at the tip while the propagation velocityis finite.

The system of Eqs. 9.67,9.68,9.71 and 9.72, augmented with the initial conditions

(9.72)E'w~ (owo) E' (aW6)vr = -32/.lhf & = -96/.lhf & .

and

(9.71)Wo =0

Whichever form is used, it should be clear that the propagation velocity is an addi­tional variable that is obtained from the system of equations. To obtain a closedsystem, another boundary condition is needed which determines the propagationvelocity, either directly or indirectly.

Nordgren considered the special case when (1) there is no spurt loss and (2) thenet pressure is zero at x = x f, and hence, the width at the moving fluid front iszero. If these two assumptions are accepted, the two boundary conditions at the tipreduce to

(9.70)at x =xr

where the tip propagation velocity, U[» is the growth rate of the fracture length, i.e.Uf = dxf Idt. This equation shows that the flow rate at the tip equals the sum ofvolume growth rate at the tip plus the rate of spurt due to creating new surface.Another useful form of Eq. 9.69 is obtained by rearrangement:

rrE'w6 (awo)uf = - 32JLhf(rrwo + 8Sp) & .

(9.69)

where the injection rate may vary with time.The moving boundary (Stefan's) condition formulated by Kemp [10] is given by

(9.68)aW6 512/.l.-=---1,ax TCE'

at x = 0:

The wellbore boundary condition, Eq. 8.44 now becomes

207Differential2D models

(9.67)

(9.66)TCE' (a2wri) _ 2hfCL TChf (awo)

4 x 128/.l ax2 -..;t=r +4 at 'the rearranged form of which is the well known Nordgren [6] equation:

_£_ (o2wti) = 8hfeL +h (awo)128/.l ax2 TC..;t=r f at .

Su?stit:uting ~qs. 9.61 and 9.65 into Eq. 9.59, the following partial differentialequation IS obtained for the width:

(9.65)

(9.64)

(9.63)

(9.62)

and hence,

q = _T(W6E' (awo) _ _ TCE' (awti)128/.l ax - 4 x 128/.l & '

The result is

and then substituting in the derivative form of Eq. 9.1:

apn E' (awo)&=2hf & .

Similarly, the flow rate at the given location can also be expressed through thewidth, firstly rearranging Eq. 9.2 into the form

= _T(Wghf (oPn)q 64/.l ax '

(9.61)

where Wo is a function of the lateral location, x. If the cross sectional area varieswith time, this is due to the width change and hence,

aAc = TChf (awo) .at 4 at

(9.60)

As~u?Iing the quasi-plane strain condition introduced by Perkins and Kern [1], theelliptic cross sectional area, Ac, of the constant height fracture is given by

TChfAC=4wo,

(9.59)

9.4.1 Nordgren Equation

oq 2hfCL oAc-a + ~+-=ox "It - r at

and will be used below to account for lateral flow rate description.

was derived in Section 8.6 as

Coupling of elasticity, flow and material balance206

----_ ------------------ ------- - --------_--- ----------

A differential 2D model using the horizontal plane strain assumption would requirethe following ingredients.

9.4.2 Differential Horizontal Plane Strain Model

In a constant-injection-rate case (i/io = 1), the system has only one solution as waspresented by Nordgren [6] in the form of plots including x fD vs. to and WwO vs. to­(The 11 vs. to curve did not make it into the original paper, causing some difficultiesin later engineering use of the solution, e.g. see Appendix G in Nierode [8]). Sincethere is an uncertainty of how Eq. 9.79 was used (if at all) in the numerical schemeof Nordgren, we do not reproduce the original plots here. Similar plots, however,will be provided in Chapter 10.

(9.81)

a zero width at the tip. However, together with the basic assumption of the PK viewof the fracture, it means that the net pressure is zero at the tip. We know that theenergy needed to create a fracture is only partly used to increase the elastic energyof the surrounding formation. Another part is dissipated during fracture propagation.One part of the energy dissipation is considered correctly in ali models: This is theviscous dissipation connected with the flow of the fracturing fluid. The other partof the energy dissipated is connected with the irreversible creation of new fracturesurface. Within the PK view of the fracture, the zero net pressure assumption alsomeans that the energy dissipated during the creation of a new surface is totallyneglected. In other words, the zero net pressure assumption means that once thefluid reaches the tip it can flow further without any resistance; thus it is opening therock as "a knife cuts butter".

Current evidence of higher fracturing pressures has made it clear that the zero netpressure assumption cannot be valid in general. In most cases, the fracture propaga­tion is retarded compared to the "knife cuts butter" case. Within the PK view of thefracture, and especially in the Nordgren - Kemp model, any accounting for the dissi­pation of energy during the creation of a new fracture surface can be accomplishedby, and only by, dropping the assumption of zero net pressure and using anotherboundary condition, resulting in a positive net pressure at the tip. Chapter 10 isdevoted to a more detailed discussion of this issue. Now we return to the originalmodel of Nordgren and Kemp, including Eq. 9.80.

From a strictly mathematical point of view, we need an initial condition. In thiscase it is rather trivial, indicating that the fracture is of zero length and width at timezero.

The resulting system can be solved numerically. Once the solution is known, thefluid efficiency (in fact the dimensionless volume) can be calculated as

209Differential 2D models

The (sometimes tragi-comic) history of hydraulic fracture modeling can be betterunderstood in the light of Eq. 9.80. At first sight it may look rather natural to assume

(9.80)at XD = xfO: Woo=O.

The other boundary condition states that the width at the tip is zero,

(9.79)

One boundary condition assures that the injected fluid rate gets into the' fracture

aw~oat XD = 0: -- = -- (9.78)aXD io

At the other end, two boundary conditions have to be satisfied. One of them is thedimensionless form of Eq. 9.69 with Sp = O. It states that the propagation "rate"of the fracture (the tip velocity) equals the quotient of the flow rate and the crosssection, i.e. the linear velocity of the fluid;

(9.77)

The model works well even with variable injection rate. Then, of course, one has toselect a nominal injection rate, io, in order to define the constants in Eq. 9.76. Thedimensionless partial differential equation is

a2w~o 1 awooaxb = ,Jto - 7:D + atD .

(9.76)

[

.2 ] 2/32 IJLC2 = rr 16CEhfE'[

.5 ] 1/31 JLc -rr1 - 128cfE'hj

c = [ 32i2JL J 1/33 CtE'hf

where the constants are selected in accordance with Nordgren's suggestion as

(9.75)

constitute the Nordgren and Kemp (NK) model of hydraulic fracturing. The importantproperty of the NK model is that it can be written in a dimensionless form where~h~solution of the dimensionless model is unique. In practical terms, this means thatIt IS ~no~gh to solve th~ system once (by an appropriate numerical method). If thesolUtl0~ IS repr.esen.ted In the form of graphs or tables or any suitable approximatealgebraic equation, 11 can be used for any (constant) injection rate and formation andfluid data.

The dimensionless variables are defined as:

(9.74)

and with the implicit definition of the fracture surface opening function,

for every x: x = Xf[T(X)],

Coupling of elasticity, flow and material balance208

~--"-'" ----

Simulation is one way to apply a given model. It is used when we know the param­eters and wish to predict the evolution of fracture dimensions with time. The inverse

where w", the average width of the fracture in the time interval, is not known apriori, but has to be determined from the PKN width equation, and similarly q", the

9.6 Pressure Decline Analysis(9.87)

In Section 8.7 we introduced an alternative means to describe fracturing fluid Ieakoff.The models of Sections 9.1 to 9.4 of this chapter can then be reformulated bysubstituting the simplistic Carter I leakoff with a more detailed description of theleakoff rate. As an example, consider the derivation of Section 9.2, but substitutingthe Carter I assumption with the Ieakoff rate obtained explicitly in Section 8.7.

The injection rate is considered constant. We assume that the PKN width equation(Eq. 9.41) describes the width at any time instant. The material balance in the n-thtime interval is given by

can be solved for xfn. Note that Eq. 9.90 is highly nonlinear, not only becausethe unknown xfn is raised to the power !, but also because the change from realtime to dimensionless time involves the actual estimate of xf n- Nevertheless, anyreliable root-finding method can solve the equation, especially because a good startingestimate (or lower bound) of xfn is known to be Xfn-l. Once xf" is found, qn isobtained as either side of Eq. 9.90. Notice that the dimensionless pressure function,characteristic for the given formation, has to be known. Pressure transient models,long established in petroleum engineering for both homogeneous and heterogeneousformations, are readily available. In using them, the transition from real time todimensionless time will involve the actual xf n v valid only for the given time step.The parameters of the leakoff model, namely Pr - the average reservoir pressure,f.l - the reservoir fluid viscosity, kfb - the formation bulk permeability, Mf - thefiltrate fluid viscosity and Ro - the characteristic filter-cake resistance (determinedwith respect to the selected reference value of time equal to the injection time, te),have to be known as well.

9.5 Models With Detailed Leakoff Description

(9.90)

and we have to keep both the pressure and the width as variables of the model. As aresult, the system consists of a partial differential equation and an integral equation(with two integrations in the latter).

The wellbore boundary condition, Eq. 8.44, and the Stefan boundary conditionat the tip, Eq. 8.45, remain valid. In this case, the width at the tip will be zeroautomatically. In spite of this, the system is not yet·determined. Again, one additionaltip boundary condition is needed, involving the tip velocity. A strict formulation ofthe problem in mathematical terms is not available. Limited numerical results andsemi-analytical approximations are the subject of current research (Leonach [11]).

(9.86)

where

(9.85)

(9.89)

(9.88)_ (Mi) 1/4 1/4 1/4W = 2.05 E' xf = QIXf '

(E'3Mi)1/4 1/4 1/4

pwn = 1.63 hj xf = a2xf .

(2) At time t« we wish to determine xr«. Since we know all the variables up totime tn-I, the equation

(9.84)q =_%h f (aPn) .12M ax

The coupling between width and pressure is not so simple as in the Nordgren model.Instead, we have an integral (see Eqs. 2.32 and 2.33):

(9.83)<lAc _ hf·(aw)at - at·The flow rate expression is derived from the assumption of slot-flow (see Table 5.1)

with time derivative

leakoff rate, is also unknown a priori, but it is determined by Eq. 8.63. Therefore,we have a system of equations for every time step.

Omitting the details of the derivation, the algorithm is as follows. Suppose weknow all the variables up to time tn-I.

(1) First the constants QI and Q2 are determined in the width and pressure equations:

(9.82)

The material balance is still Eq. 9.59, but the cross section is defined as

211Pressure decline analysisCoupling of elasticity, flow and material balance210

Dimensionlessshut-in lime.ll.to

Figure 9.1 Nolte's g function for ex= ~(PKN Newtonian fluid case)

7.64 10953 82o

vV V--

V~L

Vv

/V

V

7

6

5

Ii; ~......~~<l 30;

2

(9.95)

where F[a, b; c; z] is the hypergeometric function. For definition and properties seeAbramowitz and Stegun [20], Chapter 15. The hypergeometric function is a tabulatedfunction, e.g. similar to the sine function, and it is built into up-to-date mathematicalsoftware.exactly the same way as the function sine is built into the programminglanguage FORTRAN. Using Mathematica (Wolfram [21]) one can simply call it bythe name HypergeometricZi-L, For the value of ex =~(basic PKN case), the g(extD, ex)function is shown in Figure 9.1.

The volume of the fracture after the end of pumping is given by subtracting thespurt loss volume and the leakoff volume from the injected volume, Vi. Thus,

(9.94)4a~ + 2../1 + ~tD x F[~, ex; 1+ex; (1 + lHD)-lJ

g(6.tD, ex) = 1+ 2a

Nolte [12,13] gave the closed form of the g(6.tD, ex) function for two distinct valuesof ex, namely ~ and 1. In Valko and Economides [19] a closed form solution is givenfor any ex as

(9.93)lo1 (ll+AtD 1 ) .g(~tD, ex)= dtD dAD.o Alia J AI/"

D to - D

Clearly at ~tD = 0, Eq. 9.91 reduces to the original definition of go(ex). The func­tion, g(~tD, ex) should be a monotonically increasing function of the dimensionlessshut-in time (since the fluid leaks off and does not return into the fracture). Followingthe reasoning of Section 8.4.1, it can be shown that the g(6.tD, ex) function is theintegral .

213Pressure decline analysis

(9.92)t-t6.tD = __ e.te

9.6.1 Nolte's Pressure Decline Analysis (Power Law Assumption)

In Secti?n 8.4 we introduced Nolte's power law hypothesis, which augments theassumptions of Carter (constant leakoff coefficient plus spurt loss) with an a prioriknowledge o~ the f~rm of the solution of the coupled system. According to the powerlaw hypothesis, dunng the constant-rate injection period the fracture surface evolveswith power ex of the time, see Eq, 8.20. The ratio of the leak-off volume to theproduct 2CLA~0eis denoted by go(a). The subscript 0 indicates that the ratio istak~n at t = teo Assuming that the fracture surface remains constant after the injectionperiod, Nolte [12,13J extended the definition of the ratio to the whole shut-in period~~ ,

g(t1tD, ex) = VL(t.+At), (9.91)2CLAe0e

wher~ the subscript e refers to the end of pumping and Sr is the time elapsed afterstopping the pumps (shut-in time). The subscript L in VL denotes "Ieakoff", thusVL(t,+At) stands for the volume lost during the injection period plus the volume lostduring the shut-in period up to the time i,+ 6.t, but including only leakoff. (Thetotal volume lost might be larger if there is a non-zero spurt loss.) The dimensionlessshut-in time is simply

problem is calle~ par~meter estimation, when we do not know some of the param­eters, but try to identify them from known data. Unfortunately, the time evolutionof the fracture dimensions in the formation is almost impossible to observe, evenWith the most up-Io-date seismic systems. However, the pressure at the wellbore~an be observed rather accurately. Here we consider a very special model identifica­non p~oblem, termed pressure decline analysis. It is the valuable engineering meansto denve the parameters of the leak-off process from observation of the pressurebehavior after the pumps are stopped.

~sume that up to the end-of-injection time, te, the injection rate, i, is constant~nd IS .zero afterwards. The pressure in the wellbore is declining because the fluidIS leaking off, and consequently, the fracture faces are approaching each other. Aconsequence of the narrower width is that the elastic force trying to close backthe fracture (i.e. the induced stress) decreases. This effect is seen in the decline ofthe :vellbore pressure. Since the whole process is controlled by the leak-off, pressuredeclme a~al~sis has been a primary source of information on the leak-off parameters.If t.helimited descriptive capabilities of the Carter I Ieakoff model (see Eq. 8.8) are

sufficient in a certain application, the pressure decline analysis procedure becomeselegant and simple. It culminates in a straight-line fit, a familiar and reliable meansof traditional engineering. In the following we give a brief derivation of the method,known as Nolte's analysis [12,131, omitting the many minor modifications andimprovements [14-18] but emphasizing a neglected question: the determination ofthe spurt loss coefficient together with the Ieakoff coefficient.

Coupling of elasticity, flow and material balance212

-------_.,,,---_.---_ _-_ _---------------

(9.100)2£'

Cf.PKN = --.irhf

Similarly, for the horizontal plane strain assumption (KGD-geometry) we applyEq. 2.68 to calculate the volume of one wing by substituting C with the half length

and thus, we obtain

showing that the plot of wellbore pressure vs. the g(tltD, Ct) values results in astraight line if the fracture area, Ac' the proportionality constant, CI> and the l~akoffcoefficient, CL, do not vary with time. Only when the fracture finally closes Will thepressure behavior depart from the linear trend. . . .

Equation 9.105 is the basis of Nolte's pressure decline analysis. The techniquerequires the plot of the wellbore pressure vs. the values of the g~function, as firstsuggested by Castillo [17]. The g-function values are genera.ted ~lth .the exponent,Ct, considered valid for the given model and rheology. A straight line IS fitted to theobserved points.

For the straight line fit, that portion of the data should be selected which is after thepumps are stopped but before the fracture faces touch again. To restrict the straightline fit to this period of time, the closure point, i.e. both the time of closure .and thecorresponding value of the pressure (pc), have to be selected, at least .approxlmately.In many cases, the closure point is manifested by a clear ~hange m the pr~ssurebehavior, and a plot of the pressure decline curve vs. any SUitable ~ransform~tlo.nofthe shut-in time (including the g-function, square root or even straight shut-In time)can be used for this purpose. However, a better practice is to obtain the closurepressure from independent information (see Chapter 3).. .

The slope of the straight line is denoted by m and the intercept by b. ~Irstwe concentrate on the slope, which is related to the unknown leak-off coefficient

(9.99)

ttp (hf)2W::::: ~ = 2 x xf n 2 x _1_ = 1rpnhf,

hfxf £' hfxf 2£'

Equation 9.98 suggests that the pressure variation after the pumps stopped isstrongly connected to the leakoff coefficient. To obtain a useful relationship, weneed a closer look at the proportionality constant, Cf.

The formulas to calculate Cf are easily derived if we accept that during closurethere is no lateral flow in the fracture and the pressure is constant along the fractureat a given time instant. For the vertical plane strain assumption (PKN-geometry) weapply Eq. 2.68 to calculate the volume of one wing of the fracture. This is done bysubstituting C with half of the height and 8 with the half-length, and then multiplyingthe result by two because both wings of the vertical line crack are present Theaverage width is the volume divided by the area and hence,

(9.105)

CfV· !.)Pw= Pc + ~ - 2cfSp - (2CfCLyte) x g(tltD,Ct

= b +m x g(tltD, Ct),

(9.98)

(9.104)37r£'Cf.Rad = 16R'

Once the proportionality constant, CJ, is related to the geometry of the fracture,we can interpret the observed pressure decline in terms of the fluid-loss parameters.Indeed, adding the closure pressure, Pc- to both sides ofEq. 9.98 allows us to expressthe weJlbore pressure as

thus we obtain

Hence, the time variation of the width is determined by the g(tltD, Ct) function, thelength of the injection period and the leakoff coefficient, but is not affected by thefracture area.

The fracture closure process (decrease of average width) cannot be observeddirectly. However, from Chapter 2 we know that the net pressure is directly propor­tional to the average width, Pn =C fW, simply due to the fact that the formation isdescribed by linear elasticity theory. (The coefficient Cf is measured in Pa/m andit plays a similar role as the constant in Hook's law. In tbe petroleum engineeringliterature, its inverse, l/cf, is called fracture compliance.) Because of the linearitybetween net pressure and width, we can rewrite Eq. 9.97 in terms of the net pressure:

cfV;Pn =T-2cfSp - (2cfCL0e) x g(tltD, ce).

(9.103)

Finally, for a radial geometry, Eq. 2.69 shows that

V 16RpnW------·- lR27r - 37r£' '

2

(9.97)

(9.102)£'

Cf.KGD =-.irXf

and hence, we obtainCombining Eqs. 9.91 and 9.96 results in

(9.96)

(9.101)

and .3 with the height. The volume divided by the area is the average width:where Ae is the fracture surface (one wing, one face) at the end of pumping. Theactual fracture volume is, by definition, the product of the constant fracture surfacewith the time-varying average width, thus

215Pressure decline analysisCoupling of elasticity, flow and material balance214

---------.----------._---_ ..._-----_ ....._--- .._---- ..._---- ---

Since the radial geometry can be a satisfactory approximation for the majority of"minifrac" tests, an interesting sequence of parameter estimation can be suggested(Shlyapobersky et al. [22]). Once the slope and intercept of the straight line are

(9.114)3 3E'ViRnsp = 16(b - PC)

Once we restrict the Carter fluid loss model, specifying that the spurt loss coefficientis zero, the intercept of the g-plot provides an estimate of the characteristic length.Indeed, setting the right hand sides of Eqs. 9.111 and 9.112 equal to zero, xf canbe expressed, and setting the right hand side of Eq. 9.113 to zero, R can be easilyestimated. Such a procedure (including even the compressibility of the fluid in thewellbore) was first suggested by Shlyapobersky et al. [22J.

Perhaps the most useful result is obtained for the radial case. The no-spurt-lossestimate of the radius, Rnsp, of a radial fracture is obtained as

9.6.2 TheNo-spurt-lose Assumption (Shlyapobersky method)

(9.113)Vi 8R(pc - b)

Sp = 2nR2 + 37CE'

Before making use of the intercept, the selection of a closure pressure, Pc- cannotbe- avoided. Once the closure pressure is considered known, Eqs. 9.J.ll to 9.113provide a restrictive relationship between the extent of the created fracture and thespurt-loss coefficient. At this point two possibilities arise. One possibility is to sacri­fice the spurt-loss coefficient, i.e. to assume that Sp = O. The other possibility is toderive additional condition( s) from considering the fracture propagation period.

and for the radial geometry,.Note that the names "PKN geometry" and "KGD geometry" are used, but actually

neither the PKN nor the GDK (KGD) width equations have been utilized in thederivation. In the expression of the proportionality constant, Cf' only basic linearelasticity relations are included. In other words, not the PKN model, but only someof its underlying assumptions, are used, and the same is true for· the KGD model.If anything, from the propagation models themselves, it is only the exponent a thatis used (to generate the g-~nction values). The exponent should be ~, ~ and ~for the PKN, KGD and radial models, respectively. To apply the technique if thefluid is non-Newtonian, a somewhat different a should be selected. Nolte et al. [16]advocate the use of (2n + 2)/(2n + 3) for the PKN geometry, (n + l)/(n + 2) forthe KGD geometry and (4n + 4)/(3n + 6) for radial geometry, where n is the powerlaw exponent of the rheological equation of the fluid. Fortunately, the estimate ofthe leakoff coefficient is not very sensitive to a small variation in a.

Equation 9.107 deserves further attention. It shows that assuming the verticalplane strain assumption (PKN geometry), the estimated leakoff coefficient does notdepend on unknown quantities since the time of pumping is known, the fractureheight can often be considered equal to the length of the perforations in the welland the plane strain modulus can be assumed with some confidence. To estimate theleakoff coefficient, there is no need to know the viscosity of the fluid or the createdfracture area. Neither does the exact value of the closure pressure have to be known.The elegance and usefulness of the Nolte technique - at least for the PKN geometryand when only the leakoff coefficient is considered - lies in the independence ofthe estirna_tedleakoff coefficient on such uncertain variables as the closure pressure,and especially, the created fracture length.

For the other two models considered, the procedure results in an estimate of theleakoff coefficient which is strongly dependent on the characteristic dimension (xfor R). Unfortunately, the characteristic dimension is not known without some modelcalculations involving the still unknown leakoff coefficient.

(9.112)Vi nXf(pc - b)Sp = 2"Cfhf + 2E' ;

(9.109)8R

CL = (-m).37r.,fi;E'

for the KGD-geometry,

(9.111)(9.108)

(9.110)S _ ~ _ b- Pcp - 2A, Zc]

Thus, for the PKN-geometry the spurt-loss coefficient is given by(9.lD7)

7ChfCL = --(-m).

4.,fi;E'

Similarly, for the KGD-geometry the result is

CL = 7Cxf (-m)2.,fi;E' '

and finally, for the radial geometry the leakoff coefficient is given by

Assuming that the g-plot shows a straight line, the Nolte analysis still gives thesame answer for the leakoff coefficient even if there is a non-zero spurt loss. Theeffect of the spurt loss is concentrated in the intercept of this straight line with theg = 0 axis. From Eq, 9.105, the spurt loss coefficient can be expressed in terms ofthe intercept, b, as

(9.106)CL = (-m)

2.J"i;cf·

Substituting the relevant expression for the proportionality constant (Eq. 9.100), forthe PKN-geometry we obtain

according to Eq. 9.105 by

217Pressure decline' analysisCoupling of elasticity, flow and material balance216

'---'~"-'---' ---- ---- _._-_ ---_._.__ ._._---_- _---------_.

where the straight-line slope and intercept are derived from a g-plot generated witha = ~.The similar estimate for a radial model is

Vl/3 t2/3(b2 + 1.377 x m2 - Pc )8/3Sp.mb=0.152 'e £'2J-t2/3

It is not necessary to assume that the spurt-loss coefficient is zero, but then theestimation procedure should include some independent information.

The additional information cannot come from the shut-in period only, because thatpart is described by the straight line on the g-plot and any reasonable combination ofSp and X I (or R) satisfying 9.110 will result in the same pressure decline curve. Theindependent additional information has to be connected with the injection period.

Here we suggest two alternative approaches. The first one determines the spurtloss from material balance connecting the end of injection with the start of shut-in.The second one matches the propagation pressure directly before the end of injection.Both of these methods are more model dependent than the original Nolte method inthe sense that they need the particular form of the width equation.

To illustrate the techniques, from now until the end of this chapter we will assumethat Eqs. 9.40 and 9.41, i.e. the PKN width equation, hold during fracture propaga­tion.

(Material balance approach) At time t., when the pumps stop, there is aninstantaneous relocation of the fluid, but the average width should remain thesame if we assume that the fracture does not propagate further. Thus, the materialbalance provides an additional condition, i.e. the necessary equation to determinespurt loss.

(9.118)(J-tV2£'3) -4 0.785hf(pC - b)

S".mb = 1.46 ~5-'- (b + 1.415 x m - pc) + E' .hfte

Accepting this estimate of the spurt loss to.gether ,:i~h t~e esti~ated leakoffcoefficient has the following consequence. Dunng the mjecnon period, the p~propagation model will arrive at length Xfrnb- which, combined with the two ~uldloss parameters, will force the pressure decline curve to follow the path determinedby the two parameters of the straight line, m and b. If the observed data fo~lows astraight line on the g-plot, the combined simulation of the .fracture propagation andpressure decline will result in a match of the pressure dechne cun:e. .

(We note that a similar spurt loss estimate for the KGD model IS given by

V?5t~.5(bl + 1.478 x ml - pc)2Sp.mb = 0.771 J-t0.5£'1.5h~5

£'sf.L°.5V?S(Pc - b) (9.119)+ 1.02 - 2'h~·Jt~.5(bl + 1.478 x ml - pc)

9.6.3 Material Balance and Propagation Pressure Estimates of theSpurt Loss

(9.117)XI mb = ( W )4 (£'.) = 0.342 ( hi\) (b+ 1.415 x m _ pc)4,. 2.05 tu J-tV .E

where the subscript "mb" stands for material balance estimate. Introducing the lengthestimate into Eq. 9.111 provides the optimal estimate of spurt loss for the PKNmodel as

Cf.PKN

The average width before the end of injection should be the same, and hence,substituting Eq. 9.116 into Eq. 9.41, we can express the estimated length as

(9.116)2£'

pW.isi - Pc (b+ 1.415 x m - PC)nhlWist = =

Note that pw,iSi is not a physically observed pressure. It is the value ~orres~ondingto zero shut-in time but read off from the straight line. One can consider this valueas the optimal estimate of the real instantaneous shut-in pressur~. This value cannotbe read directly from the wellbore pressure record because the instantaneous pumpshut-down creates large fluctuations in the pressure in a very short time period.

The average width after the end of injection can be obtained from pw.isi accord­ing to

(9.115)Pw.isi = b + m x go(~) = b + m x 1.415.

From the g-plot straight-line fit first the instantaneous shut-in pressure, Pw.isi isdetermined as

known, first Rnsp is calculated from Eq. 9.114 and then CL from Eq. 9.109, substi­tuting the no-spurt-loss estimate of the radius on the right-hand side. The procedureneeds a minimum number of input parameters, namely m, b, Vi, i, E' and Pc. Noticethat a knowledge of the viscosity of the fluid is not necessary.

A similar technique can be easily derived for the KGD geometry, but fractureheight should be included in the list of input parameters. The no-spurt-loss assump­tion results in the estimate of the fracture length also for the PKN geometry, butthis value is not used for obtaining the leakoff coefficient because the fracture lengthdoes not enter the right-hand side of Eq. 9.107.

Assuming a zero spurt-loss coefficient brings about serious consequences withrespect to the propagation model. Once the assumption of no-spurt-loss is accepted,the final extent of the fracture stems from the pressure decline data alone, and forthe description of the fracture propagation one has to select a model which providesthe identified fracture extent (or a somewhat smaller extent if some growth of thefracture is allowed after the pumps stop). Thus, the Shlyapobersky method leadsalmost invariably to the discarding of the fracture propagation models describedin this chapter, since once the spurt loss is set to zero and the leakoff coefficientis known from the pressure decline analysis, no additional degree of freedom isleft in these models. The answer to this challenge in the Shlyapobersky method isthe introduction of a new degree of freedom into the known models of hydraulicfracturing, utilizing the apparent fracture toughness concept [22]- The concept willbe discussed in Chapter 10.

219Pressure decline analysisCoupling of elasticity, flow and material balance218

---_.- .._---_ ....._---------

£' 6.13 X 1010 Pa 8.89 x 106 psihf 51.8 m 170 ftPc 39.30 x 106 Pa 5700 psin 0.56K 0.622 Pa· S°.56 0.013 lbl- S0.56jft2

Table 9_12 Formation and fluid data [9,23] forExample 9.7 .

The end of the injection period is at t.= 2220 s (37 min). The injected volume, V;,is calculated as the time integral of the injection rate. The average injection rate isi= V;/te = 0.0564 m3/s.

Solution

The comparative study reported by Warpinski et al. [9] is based on a field experimentnamed "OR! Staged Field Experiment No.3". By courtesy of Prof. S. A. Holditch andcoworkers [23), we present here the data corresponding to the injection test No.2carried out on March 16, 1989. Table 9.12 contains formation and fluid rheology dataused for the analysis.

InTable 9.13, we show the injection rate (notice that i corresponds to one wing) andthe calculated bottom hole pressure (obtained by adding a known hydrostatic pressuredifference to a value measured at the surface). The time is relative to the start ofinjection. The data were collected using a modem fracturing data acquisition system(Crockett et al. [24]).

Two items of Table 9.13 are set with italics numbers. Time t = 37 min is the lastinstant of the injection period and time t = 180 min is the assumed closure point.

Use the data of Tables 9.12 and 9.13 to determine the fluid-loss parameters andprovide a consistent model of the fracture propagation, if possible. To simplify the datatreatment use the Newtonian fluid model with constant viscosity, estimated from thepower law parameters assuming a reasonable average width.

Example 9.7 Pressure DeclineAnalysis, SFEJ Injection Test No.2

(9.126)

A useful byproduct of the technique is the estimate of the created radius

(EIVl/3 1/3)

Rpr = 1.95 ;;/: (Ppr - PC)-4/3,

where the subscript "pr" indicates an estimate from the propagation pressure condi­tion.)If the two approaches give similar results, i.e. the spurt-loss coefficients determined

from Eq. 9.118 and from Eq. 9.123 are almost the same, we have a consistent modeldescribing fracture propagation which can be used in design calculations. Since inreality the fracture may propagate differently from what a model predicts, it cannotbe excluded that the spurt-loss coefficients determined from Eq. 9.118 and fromEq. 9.123 are significantly different. Moreover, negative spurt-loss coefficient mightbe also determined from the data. Any discrepancy means that the assumptionsunderlying the method have to be revised.

221Pressure decline analysis

-------_ ..•.•--_ ...._-_ •... -----_ .._---

(9.125)

(9.124)1.02(pc - b)E'o.sfLo.5VO.5 0772(p _ P )zVO.5t05Sp, pr = I +' pr C I ese-» - pc)2h~.5t~·5 E'J.5h~5fL0.5

and for the radial model by

(9.123)3 '3 2 .S _ .58E fLVi O.785hf(Pc - b)

p,pr - ( 4 5 + I 'Pw.pr-pc)h/te E

which assures the desired propagation pressure at the end of the injection.(For the KGD model, the alternative spurt-loss estimate is given by

(9.122)('3 )4 E fLV;

(Pw.pr - Pc) = 7.15 -4-- Xf·h/te

Eliminating the unknown fracture length from Eq. 9.111 and 9.122, the alternativeestimate of the spurt-loss coefficient is

where ·the subscript "mb" stands for material balance.)(Propagation pressure approach) If there is anything left we might be concerned

about, it is the observed pressure during the injection period. Indeed, the materialbalance technique provides a match for the pressure decline period, but there is noguarantee that the simulated treatment pressure during the injection period will evenresemble the observed values.

To deal with this challenge, we have to characterize the pressure during propaga­tion. In sharp contrast to the scenario after the pumps stop, the situation before thepumps stop is rather steady. In other words the wellbore treating pressure is almostconstant and it is not difficult to pick a representative treating pressure directly beforethe end of pumping. The pressure at the wellbore observed directly before the pumpsstop is called the propagation pressure, Pw.pr. The propagation pressure approach forestimating spurt loss assumes that this value is known.

From Eqs. 9.40, and 9.43 the relation between the propagation pressure and frac­ture length can be written as

(9.121)(EIVl/3j1.1/3)

Rmb = 1.45 ;;(3 (b2 + 1.377 x m2 - »c )4/3,

(9.120)1/3Vl/3( b)+ 1.23 1/3 fL i Pc - ,

t; (b2 + 1.377 x mZ - Pc )4/3

wher~ the straight-line slope and intercept are derived from a g-plot generated witha = 9' A useful byproduct of the technique is the estimate of the created radius

Coupling of elasticity, flow and material balance220

The shut-in period lasts from t = 38 min until the time the pressure decreases belowthe closure pressure, t = 180 min. Table 9.14 shows the shut-in time, 6t, the corre­sponding dimensionless shut-in time, 6tD, the g(6tD, a) values with ex= 3' and thewellbore pressure.

The plot of the wellbore bottomhole pressure vs. the g(6tD' ~) function is shown inFigure 9.2. The straight line was determined by the method of least squares. The slope is

r i p", t i Pw I i Pk'min mlls MPa min mlls MPa min m'/s MPa

0 0 27.61 38 0 43.54 114 0 40.321 0.0069 27.61 40 0 43.02 116 0 40.292 0.0230 41.33 42 0 42.72 118 0 40.273 0.0263 41.17 44 0 42.50 120 0 40.244 0.0339 41.41 46 0 42.31 122 0 40.215 0.0386 41.62 48 0 42.15 124 0 40.196 0.0386 41.71 50 0 42.00 126 0 40.167 0.0452 42.06 52 0 41.88 128 0 40.148 0.0511 42.25 54 0 41.76 130 0 40.129 0.0488 42.22 56 0 41.60 132 0 40.1010 0.0519 42.33 58 0 41.75 134 0 40.0611 0.0577 42.58 60 0 41,70 136 0 40.0412 0.0637 43.03 62 0 41.62 138 0 40.0213 0.0642 43.19 64 0 41.55 140 0 40.0014 0.0642 43.33 66 0 41.47 142 0 39.9715 0.0641 43.48 68 0 41.39 144 0 39.9516 0.0640 43.58 70 0 41.32 146 0 39.9317 0.0642 43.71 72 0 41.28 148 0 39.9018 0.0640 43.84 74 0 41.21 150 0 39.8819 0.0640 43.98 76 0 41.16 152 0 39.8720 0.0642 44.07 78 0 41.10 154 0 39.8321 0.0642 44.14 80 0 41.04 156 0 39.8222 0.0642 44.20 82 0 40.98 158 0 39.8023 0.0642 44.26 84 0 40.94 160 0 39.7924 0.0642 44.31 86 0 40.86 162 0 39.7725 0.0640 44.37 88 0 40.64 164 0 39.5626 0.0640 44.48 90 0 40.73 166 0 39.5327 0.0640 44.51 92 0 40.71 168 0 39.4728 0.0641 44.57 94 0 40.68 170 0 39.4529 0.0640 44.60 96 0 40.63 172 0 39.413Q 0.0641 44.64 98 0 40.59 174 0 39.3931 0.0642 44.67 100 0 40.53 176 0 39.3732 0.0641 44.71 102 0 40.42 178 0 39.3333 0.0635 44.73 104 0 40.35 180 0 393034 0.0636 44.76 106 0 40.30 182 0 39.2735 0.0638 44.80 108 0 40.41 184 0 39.2536 0.0635 44.82 110 0 40.38 186 0 39.2337 0.0637 44.86 112 0 40.35 188 0 39.21

222 Coupling of elasticity, flow and material balancePressure decline analysis 223

Table 9.14 Injection test g-plot data for Example 9.7

6/ 61D g(6ID, a) pw 6/ 6tD g(61D, a) p w

min a = 4/5 MPa min a = 4/5 MPa

1 0.02703 1.460 43.54 73 1.973 3.175 40.38

3 0.08108 1.541 43.02 75 2.027 3.209 40.35

5 0.1351 1.616 42.72 77 2.081 3.242 40.32

7 0.1892 1.686 42.50 79 2.135 3.276 40.29

9 0.2432 1.752 42.31 81 2.189 3.309 40.27

11 0.2973 1.816 42.15 83 2.243 3.341 40.24

13 0.3514 1.876 42.00 85 2.297 3.373 40.21

15 0.4054 1.935 41.88 87 2.351 3.405 40.19

17 0.4595 1.992 41.76 89 2.405 3.437 40.16

19 0.5135 2.046 41.60 91 2.459 3.468 40.14

21 0.5676 2.100 41.75 93 2.514 3.500 40.12

23 0.6216 2.152 41.70 95 2.568 3.530 40.10

25 0.6757 2.202 41.62 97 2.622 3.561 40.06

27 0.7297 2.251 41.55 99 2.676 3.591 40.04

29 0.7838 2.300 41.47 101 2.73 3.621 40.02

31 0.8378 2.347 41.39 103 2.784 3.651 40.00

33 0.8919 2.393 41.32 105 2.838 3.681 39.97

35 0.9459 2.438 41.28 107 2.892 3.710 39.95

37 1.000 2.483 41.21 109 2.946 3.739 39.93

39 1.054 2.526 41.16 111 3.000 3.768 39.90

41 1.108 2.569 41.10 113 3.054 3.797 39.88

43 1.162 2.611 41.04 115 3.108 3.825 39.87

45 1.216 2.652 40.98 117 3.162 3.853 39.83

47 1.270 2.693 40.94 119 3.216 3.881 39.82

49 1.324 2.733 40.86 121 3.270 3.909 39.80

51 1.378 2.773 40.64 123 3.324 3.937 39.79

53 1.432 2.812 40.73 125 3.378 3.964 39.77

55 1.486 2.850 40.71 127 3.432 3.991 39.56

57 1.541 2.888 40.68 129 3.486 4.018 39.53

59 1.595 2.926 40.63 131 3.541 4.045 39.47

61 1.649 2.962 40.59 133 3.595 4.072 39.45

63 1.703 2.999 40.53 135 3.649 4.098 39.41

65 1.757 3.035 40.42 137 3.703 4.125 39.39

67 1.811 3.070 40.35 139 3.757 4.151 39.37

69 1.865 3.106 40.30 141 3.811 4.177 39.33

71 1.919 3.140 40.41 143 3.865 4.203 39.30

m = -1.208 x 106 Pa (-175 psi) and the intercept is b = 4.432 X 107 Pa (6428 psi).Note that the intercept is usually not on the plot, because it corresponds to g = 0 andthe actual data points have g values greater than 1.415. From Eq. 9.107, the leakoffcoefficient is

-Jrhfm Jr X 51.8 x 1.208 x 106C ----L - 4.j£;E' - 4..12220 x 6.13 x 1010

= 1.70 X 10-5 m/.Js = 4.32 X 10-4 ft/.rntiU.

Table 9.13 SFE3 Minifrac 2 data, March 16, 1989 [23] for Example 9.7

---_ ..- .._---

--------------------------------. ----

PKN-a simulation results for Example 9.7, IJ- =0.1 Pa-s (100 cp); (a) Bestfit of the shut-in period, CL = 1.70 X 10-5 m/Sl/2 (4.32 X 10-4 ft/rninl/2) andSp = 0.00197 m (0.0775 in.); (b) Simulation with zero spurt-loss, CL = 1.70 x10-5 m/SI/2 (4.32 X 10-4 ftJrninl/2) and Sp = 0

Figure 9.3

150100

TII'ne, t (min)

50o

Table 9.15 Equivalent Newtonian viscosi-ties for several average fracturewidths in Example 9.7

w f-l,.m in Pa·s cp

0.002 0.0787 0.0297 29.70.004 0.158 0.0546 54.60.006 0.236 0.0780 780.008 0.315 0.100 1000.010 0.394 0.122 1220.012 0.472 0.144 1440.014 0.551 0.164 1640.016 0.630 0.185 1850,018 0.709 0.205 2050.020 0.787 0.225 2250.022 0.866 0.245 2450.024 0.945 0.264 2640.026 1.02 0.283 2830.028 1.10 0.302 3020.030 1.18 0.321 321

Can we use the obtained leakoff coefficient without determining the spurt-losscoefficient? The answer is no. Arbitrarily setting a spurt-loss coefficient equal to zerowould contradict Eq. 9.111 and distort the match of the shut-in pressure curve.

To use the results of the straight-line analysis correctly, we have to estimate thespurt-loss coefficient from the intercept of the straight line, b, using Eqs. 9.118 and9.123. These equations involve the viscosity; hence we need a reasonable estimateof the equivalent Newtonian viscosity of the fluid under average shear conditions.Table 9.15 shows equivalent Newtonian viscosities, calculated from Eq. 9.52 assumingseveral average width values.

Figure 9.2 Straight-line fit of bottomhole pressure vs, g function

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 Using the PKN-a method, we simulate the injection test. For the shut-in periodEq, 9.97 is used with the fracture area obtained at the end of injection. The calculatedpressures are shown in Figure 9.3 as curve a. Clearly, the calculated pressure followsthe observed pressure decline with good accuracy. However, during the injection periodthe simulated pressures are much higher than observed. The available data contradictthe hypothesis that the fracture propagated as a PKN fracture with Carter-type leakoffmechanism and stopped propagating right at shut-in.

For comparison, the simulated pressure, assuming Sp = 0, is also shown in the figureas curve b. The zero spurt loss assumption does not correct the lack of match duringpropagation and corrupts the fit of the decline period.

The spurt loss coefficient can be adjusted to account for the fracture propagationpressure using Eq. 9.123. To use the equation, we need the propagation pressure, pw.pr

which is the wellbore pressure at t = 37 min and can be read from Table 9.14 to bepw.pr = 44.86 MPa. Substituting the known quantities into Eq. 9.123 yields

g(AtD.4/5)

40

(f-l,VfE'3) -4 0.785hf(pc - b)

Sp.mb = 1.46 --5--- (b+ 1.415 x m - Pc) + E'hfte

= (0.1 X 1252 x (6.l3 X 1010)3) [(44.32 _ 1.415 x 1.208 _ 39.3) x 106]:"'41.46 51.85 x 2220

+ 0.785 x 51.8 x (39.3 - 44.32) x 106 _ 0.00197 m (0.0775 in)6.13 x 1010Pc=39.3 MPa

43

First we assume u. = 0.1 Pa- s (100 cp) and estimate the spurt loss from Eq. 9.118.Substituting the known quantities

~r---------------------------------,225Pressure decline analysisCoupling of elasticity, flow and material balance224

· _'--. ----_._-_ .. ,----" ---_. -------_._., '. ,--_.... ._-------_.,".._-_._ ._----,-- ._

Time. t (min)

Figure 9.4 Further simulation results for Example 9.7 and for Example 9.8, u.=0.3 Pa-s (300 cp); (a) Material balance estiamte on the spurt loss, CL =1.70 X 10-5 m/sl/2 (4.32 x 10-4 ftlminl/2) and Sp = 0.0126 m (0.495 in.);(b) Propagation pressure estimate of the spurt-loss, CL = 1.70 x 10-5 rn/sl/2(4.32 X 10-4 ftlminl/2) and Sp = 0.00155 m (0.0611 in.); (c) Optimal three­parameter fit, CL '" 1.70 x 10-5 rn/S1/2 (4.32 X 10-4 ft/min1/2) and Sp =0.000476 m (0.0611 in.), E.ta =20 min

c= 39.3M~a

4S

44

<0ll. 43::E~Q, 42!::J..,..! 410..

40

390

The previous example showed the basic contradiction of pressure decline analysis:the inability to match propagation pressure and the pressure decline simultaneously.

If a zero spurt loss is assumed, neither the propagation pressure nor the decline'part will be reproduced in general.

If the spurt loss is obtained from the material balance consideration, the declineis described correctly but the propagation pressure may not be.

If the spurt loss is identified from the propagation pressure, the injection periodwill be matched correctly but the decline part will be shifted either up or down withrespect to the observations.

An additional degree of freedom is necessary to resolve the contradiction. Ofcourse, it is always possible to assume that the observed wellbore pressure is stillnot the real propagation pressure. There might be a significant pressure drop throughthe perforations. In addition, near the wellbore there might be a large pressure dropinduced by a tortuous flow path. If such a phenomenon can be identified and quan­tified, then the additional pressure drop can be subtracted from the observationscorresponding to the injection period.

For the SFE3 test and in most other cases, we have no evidence of significantperforation or near wellbore pressure drop, and the introduction of an auxiliarypressure drop would mean that we practically discard the propagation pressure infor­mation. For instance, in the case of the SFE3 injection test, one may attribute 1 MPa(145 psi) to "tortuousity", but there is no real basis for doing this and 0.5 MPa(73 psi) or 1.5 MPa (218 psi) could be also selected. We do not encourage thispractice and suggest the use of more physically sound techniques.

One physically sound solution to the problem is to relax the strict postulate thatthe fracture stops propagating exactly at the end of pumping. In this section, weinvestigate this extension of the theory.

The likely reason for overshooting the decline part of the pressure curve inExample 9.7, when the spurt loss is selected from the propagation pressure consid­eration (i.e. curve b in Figure 9.4) is that the fracture may propagate for a whileafter the pumps stop. Mathematically, the problem lies in the fact that the lengthobtained during the propagation period and the spurt-loss coefficient (which wasused to obtain the length) do not satisfy the intercept condition of Eq. 9.111.

20015010050

9.6.4 Resolving Contradictions

Both spurt-loss coefficients are positive, but different from each other. Figure 9.4shows that the material balance estimate provides an excellent fit of the pressure declineperiod (curve a) while the propagation pressure estimate provides a satisfactory descrip­tion of the injection period (curve b). The two estimates still contradict each other butthis contradiction can be easily resolved without rejecting the PKN model, as will beshown in the following. 0

3.57 x (6.13 x 1010)30.3 x 1252 0,785 x 51.8 x (39.3 - 44.32) x 106S - +---------~--~--------P.P' - [(44.86 - 39.3) x 106]451.85 x 2220 6.13 x 1010

= 0.00155 m (0.0611 in).

357 x (6.13 x 1010)30.1 x 1252 0.785 x 51.8 x (39.3 - 4432) x 106[(44.86 - 39.3) x 106J451.85 x 2220 + 6.13 x 1010

= -0.0017 m (-0.067 in).

Surprisingly, the spurt loss coefficient is negative. In other words, additional fluidmust enter the fracture to yield the observed propagation pressure. A negative spurt lossis, of course, physically impossible and it indicates that one of the basic assumptionsinherent in the PKN model (i.e. "unretarded tip propagation") is not supported by theobserved data.

In Chapter 10 we will provide a consistent interpretation of the SFE3 injection testassuming retarded tip propagation. However, in this chapter we deal with classicalmodels. For illustrative purposes, in the following we assume a much higher viscosity,/L = 0.3 Pa· s (300 cp). From Table 9.15 it is obvious that this average viscosity corre­sponds to an unrealistic fracture width. Nevertheless, the high viscosity allows us toshow the technique to estimate fluid loss parameters in the cases where the classicalPKN model may be applied.

Assuming /L = 0.3 Pa· s (300 cp) does not change the estimate of the leakoff coef­ficient. The material balance estimate of the spurt loss coefficient is obtained fromEq. 9.118 as

S _ 1 46 (0.3 x 1252 x (6.13 X 1010)3) 6-4p,m/J - • 51.85 x 2220 [(44.32 - 1.415 x 1.208 - 39.3) x 10 1

0.785 x 51.8 x (39.3 - 44.32) x 106 .+ 6.13 X 1010 = 0.0126 m (0.495 10).

The propagation pressure estimate of the spurt loss coefficient is obtained fromEq. 9.123 as

227Pressure decHne analysisCoupling of elasticity, flow and material balance226

----- ---- ------- ..~---,

Wpr = 0.00590 m (0.232 in.),

Wist = 0.00439 m (0.172 in.),

xf.pr = 247 m (813 ft),

!:"xf = 69.7 m (229 ft) and

Sp = O.0tXl476m (0.0187 in.).

Wist X (Xj.pr + 6.xf) + 2Spfuf =wprxf.pr,

_ (wpr)4 (6.13 X 1010 x 2220)xf·pr - 2.05 0.3 x 125 '

44.86 x 106 = 39.3 X 106+ 5 x 6.13 x 10108xwpr2 x zr x 51.

125 tt x 51.8 x (39.3 - 44.32) x 106s= +---__:_-:----:--::-:~--p 2 x (Xf.pr + !:"xf) x 51.8 4 x 6.13 X 1010

The solution of the system of equations is

(9.131)S_ Vj ;rhf(PC - b)

p- + .2(xf,pr + b...xf)hf 4E'

The system of Eqs. 9.127 to 9.131 contains five unknowns, x/.PT> Sx], Sp, Wpr andWisi which are completely determined through the five equations. The solution of thesystem can be found relatively easily, combining some algebraic Simplifications witha reliable root finding method.

The incremental length growth cannot be too large because "after-growth" mayhappen only in a very limited time period. The time available for after-growth maybe well estimated from the g-plot as the interval where the observed pressures signif­icantly deviate from the straight line. The estimated after-growth is realistic only ifthis limited time is enough to reach the estimated b.xI: To check this the tip propa­gation velocity has to be calculated. Once the spurt-loss coefficient is estimated, themodel can be solved in a "simulation mode". Using a small time difference directlybefore the end of pumping, we can calculate another length, and the fracture prop­agation velocity, uj, can be estimated from the length increment during the smalltime step. Assuming a linearly decreasing propagation rate after the pumps stop, onecan easily estimate the after-growth time from b.ta = b.xf / (uJl2). This time should

2 x 6.13 X 1010

The leakoff coefficient is exactly the same as in Example 9.7, i.e. CL = 1.7 X10-5 m/s1/2 (4.32 X 10-4 ft/minl/2). From Figure 9.7 it is seen that the propagationpressure estimate of the spurt loss yields a calculated pressure decline curve lyingabove the observations. Therefore, the introduction of a (preferably small) after-growthcan result in a consistent interpretation. The system of Eqs. 9.127 to 9.131 is writtensubstituting the known quantities as

(44.32 - 1.415 x 1.208 - 39.3) x 106 x 1r x 51.8WiST =

(9.130)5E'Wpr

Pw,pr = Pc + ---.2;rhf

Of course, the intercept must satisfy Eq. 9.111, but with the increased length whichis valid after the short "after-growth" period. Therefore,

Solution(9.129)X/.pr = (:~~r(:~J,

and Wpr should correspond to the observed propagation pressure, prr (throughEqs, 9.41 and 9.43) according to

Continue the pressure decline analysis of Example 9.7 using the method of Section 9.6.4but still assuming the artificially increased viscosity, M= 0.3 Pa- s (300 cp).

where Wpr is the average width at the end of the injection period. At the end ofpumping Eq. 9.41 should hold, thus

Example 9.8 Improved Pressure Decline Analysis of the SFE3 InjectionTest

(9.128)

This width corresponds to a length which is the propagation length plus the incre­mental length growth. From volume balance,

not contradict the g-plot and should be limited to the interval where the pressurepoints lie significantly above the straight line.

Obviously, the leakoff coefficient is still estimated from Eq. 9.107. It should beemphasized that the introduction of the after-growth can improve the fit only if thepropagation pressure estimate of the spurt loss yields a simulated pressure declinecurve which lies above the observations, since the incremental increase of the lengthdecreases the calculated pressure in this period. However, if the propagation pressurecannot be matched with a positive spurt loss or the relative position of the calculatedpressure curves is the opposite, there is no way to obtain a consistent match withinthe framework of the PKN model and retarded tip propagation should be assumedas will be discussed in Chapter 10.

(9.127)

Allowing for a limited "after-growth" of the fracture after the pumps stop cancorrect the problem. The additional degree of freedom suggested in this section isan incremental length growth, Sx], To simplify the derivations we assume that theafter-growth happens in a relatively short time period and the fitted straight linealready reflects the increased length.

The average width instantly after shut-in, Wist, is a hypothetical width obtainedfrom the straight line directly after shut in. It is still given by Eg. 9.116 as

_ (b + 1.415 x m - pc);rhfw··- --------------~~~

lSI - 2E' .

229Pressure decline analysisCoupling of elasticity, flow and material balance228

._-_.._-_._-._----_ _. --_ _.__ _-_ _--- ._----------------

The only unknowns in Eq. 9.133 are Ro and kfb since the qn values have to satisfy

qn = hjXfn(Pobs,n-l - Pobs,n) (9.134)cf6.t

for every tn > to and hence are given by the left hand sides of the already satisfiedequations. (The reservoir pressure, P" the fracturing fluid viscosity, u.f' the reservoirfluid viscosity, u, and the fracture height, hj, are assumed to be known.)(9.132)

(9.133)

9.6.5 Pressure Decline AnalYSis With Detailed Leakoff Description(Mayerhofer et al. Technique)

The basic method of Section 8.7 involving a detailed description of the flow in thereservoir can be applied for pressure decline analysis as well. Ideally, the modelshould describe both the injection and shut-in periods. For brevity, we restrict ourconsideration again to the PKN geometry.

Assume that we know all the parameters governing the flow of fluid into and inthe formation as listed in Section 8.7. In the injection period the model is solved asdescribed in Section 8.7. The result will be a history of flow rates and fracture lengths.The final fracture length is denoted by xfe and the final fracture volume by Ve.

In the shut-in period, the same equations are used, except for the following minorchanges. The injection rate is set to zero, and the length is' considered constant andset to the value xfe. Now the unknown variable is the width, and the net pressuredepends on the width according to p.; = Ciw, where the proportionality constant isgiven by Eq, 9.100.

At time In (where In > to) we wish to determine wn. Since we know all of thevariables up to time tn-I, the following equationhfxfe(wn-l-Wn)

6.t

_ j.t [ n-l ~(cfwn+ Pc- Pr)--k h -Qn-IPD(tn - tn-l)+:L (qj - qj-l)PO(tn - tj-l)7r jb f j=1

can be solved for wn. Once wn is known, qn is set equal to the value of either side.The fracture length, x fe,' is constant and is obtained as the end value of the fracturelength in the first cycle of calculations. (Of course, the after-growth concept can beintroduced into this model as well.)

To start the second cycle of calculations, we need a starting value of Wn-l. whichis the end value after the first cycle is accomplished. Also, all of the qj values are"inherited" from the first cycle.

The outlined method is able to track the evolution of length and width during'the injection period and the decrease of width during the shut-in period. The corre­sponding net pressures are easily obtained from the width values using Eq. 9.89in the injection period and Pn.n = cfwn in the shut-in period. (Of course, if theobserved propagation pressure cannot be matched with the simple Carter -type fluidloss model, as was the case in Example 9.7, neither can the detailed leakoff modelprovide a match.)

The exciting new feature of the detailed leakoff model is that it provides thewellbore pressure even after fracture closure. To show this point, it is enough torealize that after fracture closure the actual leak-off rate, qn, is zero but the previousnon-zero injection history still causes a change of pressure according to Eq. 8.61. Astime elapses, more and more new leakoff rates will be zero and Eq. 8.61 provides asmooth return of the wellbore pressure to the original reservoir pressure. This partof the pressure decline curve is an excellent source of information on the reservoirpermeability because the filter-cake resistance has no direct effect on these pressurepoints.

Knowing the observed pressure decline curve, one may adjust one or more param­eters (first of all the final resistance of the filter-cake, Ro, and the reservoir perme­ability, kfb) by curve fitting. This is a "history matching" procedure. The techniqueof Mayerhofer and co-workers (25,26] differs from a simple history matching in oneimportant respect. It accepts the observed pressures as given and inserts them intothe equation. Then Eq. 9.127 becomes

hfxfn(Pobs,n-1 - Pobs.n)

cf6.t

To simulatethe wholepressurecurvewe considerthree timeperiods.In the injectionperiod the PKN-a model is solved assuming the above-determinedleakoff and spurtloss coefficients.In the after-growthperiod the first line of Eq. 9.105 is used, but withvarying half-length. In a given time interval the increment of the fracture length isdetermined by the tip velocity. First the tip velocity equals the fracture propagationrate determined numerically at the end of the injection period. Then this velocity isdecreased linearly to zero. The only additionalparameter entering the calculations isthe time of after-growth, /),to.

For the present example, if the fracture propagation is simulated with the esti­mated leakoff and spurt-loss coefficients, the tip propagation velocity is obtained asuf = 0.09 m/s (18 ft/min). Assuming linearly decreasing propagation rate (constantdeceleration),the after-growthperiodwould last for more than 20 min. The long after­growth is not supportedby theg-plot but in this case we are less concernedbecause theassumedviscosity is artificialand the calculationswere carriedout only for illustrationof the technique. In spite of the artificial input data, the match of the pressure curveis quite satisfactoryas seen from curve c in Figure 9.4. Note that for the simulationofthe pressure curve we use only three parameters estimated from the observedpressurecurve: the leakoff coefficient,the spurt loss and the length of the time interval after thepumps stop, duringwhich the tip propagationvelocity decreasesto zero. 0

231Pressure decline analysisCoupling of elasticity, fJow and material balance230

.__ ...•__ ..,-_. ---_., -----------------_-- ._-----

11. Lenoach,B.: HydraulicFractureModellingBased on AnalyticalNear-TipSolutions, inComputer Methods and Advances in Geomechanics, Siriwardane, H.J. andZaman,M.M.(eds.), AA Balkerna, Rotterdam,1994.

12. Nolte, K.G.: Determinationof Fracture Parametersfrom Fracturing Pressure Decline,Paper SPE8341presentedat the SPE AnnualTechnicalConferenceand Exhibition,LasVegas,Sept. 23-26, 1979.

13. Nolte, KG.: Fracturing-PressureAnalysis, in Recent Advances in Hydraulic Fracturing,Gidley, J et al. (eds.)., Monograph Series, SPE, Richardson, Texas, Vol. 12, SPE,Richardson,TX, 1989.

14. Nolte, KG.: Determinationof Proppant and Fluid Schedulesfrom Fracturing PressureDecline,SPEPE, (July), 255-265, 1986.

15. Nolte, KG.: FractureDesignConsiderationsBasedon PressureAnalysis,SPEPE, (Feb.),23-30, 1988.

16. Nolte, K.G., Mack,M.G. and Lie W.L.: A SystematicMethod for ApplyingFracturingPressure Decline: Part 1., Paper SPE 25845 presented at the SPE Rocky MountainRegional/LowPermeabilityReservoirsSymposium,Denver, CO, April 12-14, 1993.

17. Castillo,J.L.: ModifiedPressureDecline AnalysisIncludingPressure-DependentLeak­off, paperSPE 16417presentedat the SPE RockyMountainRegional/LowPermeabilityReservoirsSymposium,Denver, CO, May 18-19, 1987.

18. Zhu, D and Hill AD.: The Effect of Temperatureon MinifracPressureDecline, PaperSPE 22874, 1991.

19. Valk6,P. andEconomides,MJ.: FractureHeightContainmentWithContinuumDamageMechanics,Paper SPE 26598, 1993.

20. Abramovitz,M. and Stegun, LA (ed.): Handbook of Mathematical Functions, Dover,New York, 1972.

21. Wolfram,S.: Mathematica: A System for Doing Mathematics by Computer, 2nd ed.,Addison-Wesley,NY, 1991.

22. ShlyapoberskyJ., Walhaug,W.W., Sheffield.R.E., Huckabee,P.T.: Field Dete.remina­tion of Fracturing Parameters for OverpressureCalibratedDesign of Hydraulic Frac­turing, Paper SPE 18195, 63rd Annual TechnicalConference and Exh. Houston, TX,October 2-5, 1988.

23. Holditch,SA.: Personal Communication,1993.24. Crockett,AR., Okusu,N.M. and Cleary,M.P.: A Complete Integrated Model for

Design and Real-TimeAnalysis of Hydraulic Fracture Operations, Paper SPE 15069,presentedat the 56th CaliforniaRegional Meeting,Oakland,CA, 1986. .

25. Mayerhofer,M.I., Economides,M.J. and Ehlig-Economides,CA.: Pressure TransientAnalysis of Fracture Calibration Tests, Paper SPE 26527 presented at 68th AnnualTechnicalConferenceand Exhibition, Houston,Texas,3-6 October, 1993.

26. Ehlig-Economides,CA., Fan Y. and Economides,M.J.: InterpretationModel for Frac­ture CalibrationTests in Naturally Fractured Reservoirs,Paper SPE 28690 presented atthe InternationalPetroleumConference& Exhibition,Veracruz,Mexico,October 10-13,1994.

27. Gringarten,AC. and Ramey, AJ., Jr.: Unsteady State Pressure Distributions Createdby aWellwith a Single-InfiniteConductivityVerticalFracture,SPEJ, (Aug.), 347-360,1974.

28. Cinco-Ley,H. and Meng, H.Z.: Pressure Transient Analysis' of Wells with FiniteConductivity Vertical Fracture in Double Porosity Reservoirs, Paper SPE 18172presented at the 63rd Annual Technical Conference and Exhibition of the SPE.heldin Houston,TX, October 2-5, 1988.

233References

~--.--.- .....-- ..---- ..•....-~-----------------

1. Perkins,T.K and Kern, L.R.: Width of Hydraulic Fractures,1PT, (Sept.), 937-949,1961; Trans. AIME, 222.

2. Khristianovitch,S.A and Zheltov, y'P.: Formation of Vertical Fractures by Means ofHighlyViscousFluids,Proc. World Pet. Cong., Rome,2,579, 1955.

3. Zheltov,Y.P. and Khristianovitch,S.A: On theMechanismof HydraulicFractureof anOil-bearingStratum,Izv. AN SSSR, OTN, (No 5), 3-41, 1955.

4. Geertsma,J. and de Klerk, F.: A Rapid Method of Predicting Width and Extent ofHydraulicallyInducedFractures,lPT, (Dec.), 1571-1581, 1969.

5. Geertsma,J.: Two-dimensionalFracture-PropagationModels, in Recent Advances inHydraulic Fracturing, Gidley, J.L. et at. (eds.), SPE Monograph 12, SPE, Richardson,TIC, 1989.

6. Nordgren, R.P.: Propagationof a VerticalHydraulicFracture, SPEJ, 306-314, (Aug.)1972; Trans. AlME, 253.

7. Economides,M.1., Hill, AD. and Enlig-Economides,C.A: Petroleum ProductionSystems, PrenticeHall, EnglewoodCliffs,NJ., 1994.

8. Nierode,D.E. Simple Calculation of Fracture Dimensions, in Recent Advances inHydraulic Fracturing, Gidley, J.L. et al. (eds.), SPE Monograph12, SPE, Richardson,TX,1989.

9. WarpinskiN.R., Moschovidis,Z.A, Parker C.D. and Abou-Sayed,I.S.: ComparisonStudyof HydraulicFracturingModels:TestCase - GRI-StagedFieldExperimentNo.3,SPE Production & Facilities, 9(1), 7-16, 1994.

10. Kemp, L.F.: Study of Nordgren's Equation of Hydraulic Fracturing, SPE ProductionEngineering, (Aug.),311-314, 1990.

References

The dimensionless pressures, PD, correspond to solutions of transient flow throughporous media such as Gringarten and Ramey's [27] infinite conductivity fracturesolution or Cinco-Ley and Meng's [28] solution for a naturally fractured reservoirwith a finite conductivity fracture. These solutions are available in several forms suchas tables, algorithms, or curves. The pressure decline analysis module computes thedimensionless time differences tn - tn-l, tn-l - tn-2, ... using the actual fracturelength (corresponding to time tn) and passes them to the module for calculating PD.The module can be selected according to our knowledge of the actual structure ofthe formation. Since the fracture length at time tn is known only at time tn, thecalculation of the dimensionless pressures has to be done again and again in everytime step during the propagation period, but it can be done for the rest of the periodonly once if the final fracture length is already known.

The obtained system of equations allows the best estimate of the value of Ro andk fb through a least squares fit.

The technique of Mayerhofer and co-workers employs standard methodologies ofmodem pressure transient analysis, including the log-log plot diagnostic depiction ofthe rate-normalized pressure and its derivative.

Our presentation of both the Nolte and the Mayerhofer techniques considers onlythe basics of the methods. The interested reader may find several extensions, varia­tions and considerations in the original papers [12-17,22,25,26].

Coupling of elasticity, flow and material balance232

In Chapter 9 we presented a vision of a fracture resulting from the combination of afundamental law (material balance) and constitutive equations which describe elas­ticity and fluid rheology. The models were intended to be representations of certainphysical processes having major influence on the final geometry of the created frac­ture. However, in the case of hydraulic fracturing, these models present a rather sterileview. Stating explicitly, assuming implicitly, or sometimes even unconsciously, thefracture propagation velocity with which the tip propagates away from the well isoften considered to be a trivial consequence of the other processes described in amodel. Of course, more realistic fracture propagation modeling requires the under­standing of the processes involved at the tip.

The fracture propagation velocity is a crucial issue in predicting not only theinstantaneous geometry but also the behavior of the propagating fracture which, intum, has a compounding effect on the resulting geometry.

What should the purpose of propagation modeling be? If the purpose is to repro­duce a specific observed behavior, then two problems may arise. First, the observedbehavior (such as the fracture treatment pressure) may constitute only a small windowin a far more complicated global process. A model that can match such a windowmay fail considerably in accounting for unobserved behavior before or after thespecific observation. Second, more than one model may reproduce the same data.This can be done by emphasizing or even arbitrarily weighing different factors whichmay prove wrong outside the window of observation or even .be incompatible withother physical phenomena.

The process of selecting a model or parameters that can both reproduce observedbehavior and conform to other information about the system, while respecting funda­mental laws and/or common sense, is the essential element of data interpretation.Modelers should never become overconfident just because they are able to repro­duce one set of observations or a particular presentation style for acquired data(especially if they achieve this by adjusting several parameters). The significantcontribution from a model is the identification of characteristic behavior trends in itsforward simulations that are distinct from the behavior of other models and whetherthese trends have been observed in acquired data. The applied value of the model isfurther enhanced with analyses demonstrating the sensitivity of the model behaviorto parameter values that are important for practical decision-making.

FRACTURE PROPAGATION10

...__ ._._- ...-- ..__ .... -_ ... -_ .... __ .__ ....._- ..,,--_ ..- .... ----------.------_._-_ .._-- .---

The scientific-engineering discipline of fracture mechanics has evolved from thenecessity to avoid the dangerous collapse of engineering structures such as bridgesor dams. With loading tests of carefully selected specimens, the limits of safe loadingcan be determined experimentally. After experimental data are collected, one canattempt to formulate rules to avoid failure. In formulating a failure criterion, conceptsof the theory of elasticity are used (e.g. deformation, stress, stress intensity factor),but the criterion itself does not follow from the theory of elasticity alone. As a matterof fact, the process of failure or fracture lies outside the area of elasticity.

For a long time the theory of fracture mechanics has been biased by its ultimateaim, i.e, to avoid fracture. Von Karman [2] realized that there are two schools ofthought in fracture mechanics: one aimed to provide simply definable conditions toremain on the safe side and the other more concerned about the physical image of theevolution and propagation of the fracture once the loading limit has been exceeded.While the overwhelming majority of works in fracture mechanics corresponds to thefirst group, the second group is of major interest to hydraulic fracturing where theaim is to create a fracture and not to avoid it.

The history of fracture mechanics is described in excellent books by Timo­shenko [3] and Gramberg (4). The first and up to now the most influential ideasgo back to Coulomb [5] and Mohr [6], who assumed that the collapse of the mate­rial takes place when the state of stress of a material point reaches the boundaryof the safe domain. They assumed that the boundary can be described in simplealgebraic terms and considered the location of the boundary as a material property.

Some selected points on this boundary are referred to as the strength of thematerial. The tensile strength is the limiting value of stress the material can bear in auniaxial test where the loading is tensile. The compressive strength is another value(usually an order of magnitude higher) which has to be applied to cause failure if inthe uniaxial test the stress is compressive.

While the Coulomb-Mohr theory provides a framework to describe the strengthof materials with respect to the geometry of loading conditions, it was Griffith [7,8]who first gave a theory to explain some aspects of the strength behavior. He wasconcerned with "brittle" materials, first of all glass. Jaeger and Cook [9], lookingback to the history of fracture mechanics, gave a very objective appreciation ofGriffith's contribution as follows: "His concept of failure, based on the existenceof minute internal and surface flaws, is questionable to rock. However, the body of

10.1 Fracture Mechanics

theories. At issue is the velocity with which the fracture tips (lateral and vertical)travel, the causes of possible retardation (with respect to the one obtained fromsome trivial assumptions) and the resulting observable net fracturing pressure vs.time behavior.

Clearly, it is this debate that could benefit from some common language anduniformity in the representation of results.

In the following, we give a brief overview of fracture mechanics concepts relevantto tip propagation and describe their usage in hydraulic fracture modeling.

It would be useful if the developers of hydraulic fracture models were to adopt acommon method of presentation with which to distinguish the various models and theeffects of various phenomena. Petroleum engineers have adopted the log-log plot ofpressure change and its derivative with respect to the log of elapsed time as a standardfor presentation of pressure transient data acquired at a constant flow rate: Variationson this plot for data acquired with a constant wellhead pressure or for variablepressure and rate conditions have also been developed. The derivative presentationhas proved to be highly effective in its sensitivity to model and parameter variationswhile, at the same time, these presentations reflect simple straight-line derivativetrends with which behavior of practical interest can be readily identified. With theexception of the qualitative Nolte-Smith [1] analysis for the simple 2D models, noserious attempt has been made among hydraulic fracturing workers in this direction.

There are three basic requirements in constructing a fracture propagation model:

L Fundamental laws such as material and energy balances must be obeyed in ademonstrable way.

2. A complete mathematical formulation of the governing and boundary equations,without resorting to arbitrary "weighing factors", should be derived. Any simpli­fying assumptions must be defined clearly and their effects quantified. Theformulation of the model has to be separated from its computer algorithm.

3. A fracture tip propagation criterion and its interaction with the provided energymust be explicitly stated.

A potentially very powerful tool is the formulation of analytical solutions withexplicit simplifying assumptions in the governing equations. These solutions shouldserve as limiting forms of the same governing formulations without the simplifyingassumptions. The analytical forms may suggest presentations of the numerical simu­lations (such as the log-log plot of pressure change and its derivative) that may helpin the intuitive understanding of the implications of the non-simplified cases. A suit­able presentation also provides a means to demonstrate the sensitivity of the modelbehavior to the parameters eliminated from the analytical solution.

A common methodology of representation, emanating from the analytical solution(e.g., net pressure during fracturing on log-log coordinates) should be adopted also fornumerical modeling, not only to provide a recognizable pattern but also to indicate theextent and manner of deviation from the simplifying assumptions. Many numericalsimulations of the same observed behavior which may look credible in a Cartesianrepresentation, for example, may fail miserably when subjected to the rigors andscrutiny expected from an analytical solution. More to the point, the use of analyticalsolutions should be preferred for cases in which aspects that would require use ofthe numerical model are insignificant.

Fracture propagation is an area that has spun considerable debate and controversy.While in Chapters 1 to 9 of this book we introduced several new developments, andwhile several issues still need investigation, the means of coupling material balance,rock elasticity and fluid rheology is reasonably well understood and accepted. Themechanism of fracture propagation, on the other hand, is reflected in several different

237Fracture mechanicsFracture propagation236

.---~-.----.,.

(10.10)

holds during the whole process.

It is not difficult to discover the stress intensity factor, K/ = pocl/2, on the left­hand side (see Eq. 2.53). Thus, the Griffith inequality can be interpreted in terms ofstress intensity. There exists a critical value of the stress intensity factor, often calledfracture toughness, KIC, with the following property: If the stress intensity factor atthe tip is less than the fracture toughness, the fracture is stable. Otherwise, the fractureis unstable, and starts to propagate. A propagation process through equilibrium stateswould imply that the equality

(10.9)

(10.8)

(2aE')1/2

poc1/2:::; --n

i.e,

The inequality can be rearranged such that the variables characterizing a given stateof the material are on the left and variables, characterizing the material behavior areon the right-hand side. Then, the inequality becomes

2 2aE'poc:::; --,

n

(10.7)

Comparing Eqs. 10.3 and 10.4 Griffith [7,8] realized that'half of the work done byth~ inner pressure, i.e, ~ Wp/2, is not converted into strain energy. What happensWIth the other half of the work? Part of it is used to create new surface and ifstill any surplus energy is available, it is dissipated as heat. Whatever the' energyconsumption is during the creation of the new surface, in a given material it shoulddepend only on the area of the created new surface. Griffith postulated that thereis a material property, the specific surface energy, a, that characterizes the energyconsumption while a unit area of new surface is created. The specific surface energyis measured in J/m2•

Once we accept that a is a property of the material, an energy balance can bewritten as

~ W p = ~Wo + 2ct86.c + generated heat. (10.5)

The crack will not propagate if the left-hand side is not enough to cover the two firstitems on the right-hand side. This is the famous Griffith stability criterion, statingthat a crack is stable if

~Wp:::; ~Wo+2aMc. (10.6)

After substitution of Eqs. 10.3 and 10.4 into inequality 10.6, the inequality becomes

2JrP60c~c JrP50c~cE' :::; E' + 2a8~c,

(10.4)

239

is easily derived from Eq. 2.68. The result is

~w - ~V _ 21(pooc~C 21(P68c~cp - Po - Po E' E'

Fracture mechanics

.------.~-- ,,---,--,,----

Figure 10.1 Propagation of a line crack

f.+.----- c

Strain energy stored

The work done by the inner pressure while moving the fracture faces apart andahead is obtained as pressure multiplied by volume change, where the volume change

(10.3)

(10.2)W + ~W = Jrp6°(c +~da 0 2£"

from which the increase of strain energy is obtained (neglecting the second-orderterm) as

(10.1)W = JrP58?o 2£"

If the crack propagates, a next state (also depicted in Figure 10.1) will be character­ized by the strain energy:

Let us consider a slice of thickness 8 from an infinite material containing a three­dimensional extension of a "line crack" of half-length c, as depicted in Figure 10.1.The situation is described well by a plane strain condition. We assume that the crackis opened by a constant pressure, po, both with respect to location and time. Thestrain energy stored in one half of the medium (i.e. corresponding to one wing ofthe two-wing crack) is calculated as (see Eq. 2.64)

10.1.1 Griffith's Analysis of CrackStability

theory which has grown up around his concept is very useful. .. and there seems tobe no reason why it should not be applied to the behavior of much larger cracks ... "

Fracture propagation238

(10.20)

i.e.

(10.19)

(10.18)where we have substituted 11c= (dcfdt)Lll. Combining Eqs. 10.3, 10.4, 10.14 and10.18 gives the energy balance in the form

Again, for a stable crack the product of the (net) pressure and the square root of thecharacteristic dimension (now radius) has to be below a certain critical value.If this inverse proportionality can be observed in practice, there is no reason

to preclude LEFM. However, for more sophisticated problems, including effects oflarge-scale yielding, time dependence, non-trivial loading conditions and effects ofdamage, the inverse square root proportionality Simply does not hold and it is difficultto use classical fracture mechanics concepts (Lamaitre [10]). The train of thoughtapplicable under these "extreme" conditions is termed local approach. Very oftenother names-process zone theory (Boone et al. [11]), continuum damage mechanics(Kachanov [12]) and crack layer' theory (Chudnovsky [13])-are used to indicate devi­ations from LEFM.

[d2c (dC)2]2kopp~c cd(2 + dt LlC

£'2

(10.13)2 1/2- poR ..s K/c·it

where k is a numerical factor with unit order of magnitude. The change in the kineticenergy during a time interval Llt iswhich can be rewritten using Eq. 10.11 as

(10.17)

at least near the fracture surface, i.e. in an area approximately c2 large. Thus,Eq. 10.15 can be rewritten as

( dC)2 P6c2WK = kop dt £'2'(10.12)

(10.16)du Podc ::::::£1'

where p is the density. Mott did not carry out a rigorous calculation of the integral.Instead, he used an order of magnitude estimate. From Eq. 2.36,

(10.15)I (dC) 2 JOOJC (dU)2WK = 'i_op dt dc dxdy,cc 0

Mott gave a gross estimate of the kinetic energy for a crack increasing its half lengthand width. If u(x, y) is the displacement at any point corresponding to a half lengthc, the velocity at any point is the first derivative of the displacement with respectto c multiplied by the tip velocity, dc/dr, Once the velocity is known, the kineticenergy can be calculated according to

(10.14)LlW p = LlW0 + 2ao!:.c + 11WK.generation term:

Unfortunately, Eq. 10.11 is not very useful to determine the fracture toughness,because a is not an easily observable or derivable quantity. On the other hand, it isnot difficult to accept that the stress intensity factor (even if it is connected with anotherwise infinite stress state and even if it has a rather unusual physical dimension)is a state variable and that the material responds to it by rupture if the value is abovea certain limit.

The fracture toughness of a given material can be determined from experimentson specimens of specific shape, observing the critical loading conditions when thefracture starts to propagate. It is not necessary to establish a given stress intensityfactor by pressurizing a crack; the same effect can be achieved by applying a tensilestress, 0"0, because then K/ = O"OCI/2 at the tip., Griffith's concept has given birth to the scientific and engineering discipline calledlinear elastic fracture mechanics, LEFM. The discipline provides an efficient frame­work to measure, distill, store and reveal information on the rupture properties ofmaterials. It is particularly useful in predicting critical loading conditions for struc­tures under design and in use.

The success of LEFM lies in the fact that, for certain materials and for a certainscale domain, the load-carrying capability of the material containing a crack isinversely proportional to the square root of the crack length. The longer the crack,the easier it is to propagate.

An analogous derivation for a radial crack results in the condition of stability

When a crack propagates, the velocity of certain material points has to be altered.Griffith's criterion is not sufficient to describe the situation because of its staticnature. Matt [14] presented a simple theory of the crack propagation velocity takingaccount of the kinetic energy change. He augmented the energy balance, i.e. Eq. 10.5,introducing the change of kinetic energy, LlWk. but neglecting the irreversible heat

(10.11)( 2a£/) 1/2K/c = --

n

10.1.2 Mott's Theory for the Rate of Crack GrowthFrom inequality 10.9 the fracture toughness is obtained as

241Fracture mechanicsFracture propagation240

-- .._-. _-- •.._ ....__ ...__ ....__ ..---.~- .._--- ...-..--- .•....------------

8.89 X 106 psi2000 psi- ftI/250/2 =25 bpm

6.13 x 1010 Pa7.6 x 106 Pa. mI/2

0.0662 m3Js

Table 10.1 Input data for Example 10.1

IjZ itpoR = -K/c·2

"equation" 10.13On the other hand, since the fracture propagates according to LEFM, from

8R3 PnO .--=It.3£'

volume:Since there is no leakoff, the volume of one wing (see Eq. 2.30) equals the injected

Solution

Based on the original ideas of Griffith, fracture propagation is often modeled byapplying the following cycle of steps: (1) calculate the stress intensity factors fora given geometry and loading condition, (2) apply the stability criterion involvingfracture toughness to determine whether the fracture will propagate, and (3) if thefracture is unstable, propagate it a certain distance corresponding to the selectedtime step.

The first step is accomplished by the perhaps most successful scientific-engineeringmethod of this century, the finite element method (Zienkiewicz [16]) or by one of itsvariations. The second step is the application of inequality 10.9, or one of its manygeneralizations (Erdogan and Sih [17]; Sib [18]). The third step is often arbitrary,and sometimes hidden in the computer program, and can be as simple as "add oneelement per time step". The tip propagation velocity is obtained implicitly, throughthe rate at which the loading conditions build back up to instability. The numer­ical models often use interactive computer graphics (Boone et al. [19]) and adaptiveremeshing (Gerstle and Xie [20]).

In hydraulic fracturing, the geometry and loading conditions can be assumed tobe rather simple, e.g. an ellipse may be assumed (Bouteca [21]) because detailedinformation on the peculiarities of the shape is not available. The stress intensityfactor at the tip, KJ, is calculated from the actual pressure distribution inside thefracture. The propagation criterion is usually the simplest one: "crack advance occurs

10.2.1 Fracture Toughness Criterion

Calculate the net pressure at t = 200 min in a radially propagating fracture for the planestrain modulus, fracture toughness and injection rate given in Table 10.1. Assume thatthere is no leak-off, the viscosity of the fluid is so low that the pressure drop in thefracture is negligible and that the fracture propagates according to the LEFM criterion,i.e. at every time instant the radius R satisfies inequality 10.13 "sharply", with theinequality sign becoming an equal sign. Finally, repeat the calculation of net pressureassuming an injection rate that is ten times smaller.

10.2 Classical Crack PropagationCriterion forHydraulic Fracturing

Example 10.1 Radial Fracture Propagating According to LEFM

To illustrate why the classical crack propagation criterion has proved less thansuccessful in hydraulic fracturing, we consider a somewhat simplified but illustrativeexample.

10.2.2 The Injection Rate DependenceParadoxAccording to the theory, once the crack starts to propagate, it rapidly accelerates,and at later times, when the length is much larger than the initial length, the prop­agation rate is almost constant. Experiments show that for brittle solids, a limitingvelocity is often about half the shear wave speed (Billington and Tate [15]). Clearly,Mott's theory provides only an upper estimate on the propagation rate since theenergy dissipation due to irreversible processes is neglected.

(10.22)dc = r;E (1 _ Co) .dt V 2kP C

Mott assumed that the crack propagates from an equilibrium state, so the initial valueof c can be expressed from inequality 10.8, used as an equality:

in such a way that the stress intensity factor, K[, is kept nearly equal to the criticalstress intensity factor, K[c, during crack extension at each node (Clifton and Abou­Sayed [22]) or, in other words, "growth occurs when K[ reaches K[c" (Thiercelinet at [23]). The rebuilding of the critical loading conditions is accomplished throughfluid flow, which is described in different levels of complexity from one-dimensionalsteady-state flow to two-dimensional unsteady flow involving a possible non-wettedzone at the tip.

Unfortunately, the direct application of classical fracture mechanics concepts inhydraulic fracturing has not been as successful as in many other fields, e.g. in theanalysis of concrete structures.

(10.21)2E'a

co=--·:rcp~

Also, he assumed that dc/dt and d2c/dt2 are zero at the start of propagation. The solu­tion of Eq. 10.20, assuming the above initial conditions, was found by Mott [14] as

243Classical crack propagation criterionFracture propagation242

,,-_._ _ ..__ _- _-_ .. --,,-_ _-_ ---

Plastic rock behavior near the tip may cause rock dilatancy, constrain the openingand lead to increased resistance of fluid flow [32-34]. Shlyapobersky and Chud­novsky [31] find that " ... tip dilatancy can only affect the net pressure if the charac­teristics of non-linear deformation and failure of rock mass are subject to a strongscale effect, which may be difficult to predict based on laboratory data."

The idea of a region with a "narrower" flow channel near the tip (i.e. a stifferregion with higher elastic modulus) has been investigated using a fully coupled model

10.3.2 TipDilatancy

In order to overcome the paradoxical predictions of LEFM, the concept of fluid lag(non-wetted zone, non-penetrated zone) has been reintroduced (Jeffrey [29]). Theconcept was first introduced in the literature by Khristianovitch and Zheltov [30].The additional degree of freedom allows one to "tune" the pressure without givingup Eq. 10.10.

Is there a fluid lag in reality? Of course there is. It occurs because fracture prop­agation is a discrete process with "jumps". The exact location of the fluid frontdepends on several factors, e.g. when the last jump occurred and the microscopicshape of the flow channel. Additional factors such as interfacial tension and thewetting angle might be of primary importance when formulating a moving boundarycondition for the fluid. Without a clearly stated boundary condition the COnceptdoesnot enlighten the mechanism controlling tip propagation.

We agree with Shlyapobersky and Chudnovsky [31) who state that "... often thesemodels, however, simulate too high net pressures at early injection time and too lownet pressures for a large, poorly contained fracture at later time ... The inability ofthese numerically very accurate hydraulic fracturing models to simulate high netpressures calls for modifying these models by including additional effects that maybe left unaccounted for. .. "

10.3.1 Fluid Lag

If we repeat the calculation with the much smaller injection rate, i = 0.0066 m3/s(2i = 5 bpm), then the net pressure will be higher, i.e. 1.1 MPa (159 psi), and thisresult is difficult to accept. 0In laboratory experiments the decreasing pressure vs. time curve can be observed

clearly (Heuze et al. [24]; De Pater et al. [25]), and this is often considered as proofof the LEFM theory. However, the result that the net pressure is inversely propor­tional to the !power of the injection rate has never been confirmed in laboratoryexperiments and obviously contradicts the everyday field experience of fracturingworkers.

The LEFM theory gives a false picture of the fracturing process in large scale andunder compressive far field stress states. What is the reason for this discrepancy?In large scale hydraulic fracturing, Eq, 10.5 still holds (with the modification toinclude the kinetic energy change, as suggested by Matt). The dissipated energy inthe form of heat is, however, no longer a negligible term. It becomes more and moreimportant with size. The existence of a large fracture generates a zone of microcracksaround and ahead of the fracture. This zone can be called damaged zone, cohesivezone, process zone, crack layer or crack band. At large scales more and more energyis used up by this process and by possible other irreversible changes generated inthe surrounding formation. Thus, with increasing scale the predictions of LEFM,neglecting the dissipation term, become more and more unrealistic and contradictcommon sense.

When viscous forces become important, an additional dissipation of energy occursdue to fluid flow. This part of the energy dissipation, i.e. the frictional pressure drop,is incorporated into almost all existing models as was shown in Chapter 9.It is not difficult to see that classical fracture mechanics, coupled with mate­

rial balance, elasticity and fluid flow description, gives an exaggerated estimate ofthe tip propagation rate, especially at larger injection times. The two-dimensionalmodels of Chapter 9, namely PKN, KGD, radial and especially the Nordgren-Kempformulation estimate the tip velocity more realistically. In these models, the fracturepropagation rate is determined as the limiting velocity of the fluid approaching thetip, calculated as if the fluid were flowing in a static ("frozen") counterpart of theinstantaneous geometry. In other words, no additional dissipation of energy near thetip region is considered. Technically, this is achieved by explicitly (or unconsciously)setting the net pressure to zero at the tip. An analogy might be a knife cutting butter,where the velocity of the knife is controlled by the friction between the surface ofthe knife and the butter but not by the processes at the edge. In the following werefer to the "knife in the butter" analogy as unretarded propagation rate.

There is continuously growing evidence that existing fracture propagation modelsneglect some important aspects of the near tip region. Often, the observed fieldtreating pressure is much higher than predicted by these models (Medlin andFitch [26]; Palmer and Veatch [27]). The net pressure is rather insensitive to limitedrate variations and especially to fluid viscosity changes (Cleary [28]).

Several efforts have been made to build more realistic models. The difficulty liesin the fact that fracture propagation is a result of several mechanisms, i.e. a modelcontains several elements. If one of these elements is modified, the performance canbe predicted only using a fully coupled implementation of the new modeL Often theresults are disappointing because the introduced modification does not generate thedesired change in overall behavior. In the following we give a brief description ofsome current trends in modeling propagation of hydraulically induced fractures.

[(7.6 X 106)6 ] 1/5 ( 1 ) 1/5=2.09 x --6.1 x 1010x 0.0662 12000

= 0.69 MPa (100 psi).

= (T(6) 1/5 (KYc) 1/5 (~) 1/5p".o 24 E'i t

10.3 Retarded Fracture PropagationEliminating the unknown R from the above two equations we obtain

245Retarded fracture propagationFracture propagation244

- ._-_.__ -- --- ._---- ._---_ ...----- - __-_ ..._--

The engineering (or macro-) scale in hydraulic fracturing is of the order of onehundred meters. With respect to this scale the damage due to microcracks can beconsidered continuous. The material damage at a given point is characterized bythe dimensionless variable D. It varies between zero (undamaged state) and unity(total damage). The evolution of damage is described by a constitutive equation,the Kachanov law [12} which relates the damage growth, dDjdt, to the net section

10.4.1 Tip Propagation Velocity from CDM

10.3.4 .Process Zone Concept

In a review paper by Shlyapobersky and Chudnovsky [31}, a promising train ofthought is outlined concerning the tip effects present in hydraulic fracturing. Two

Perhaps the conceptually most difficult issue in modeling fracture propagation liesin understanding the problem of different scales [10]. In processes such as hydraulicfracturing, at least two different scales have to be considered simultaneously, namelythe scale of the fracture (tens or hundreds of meters) and the scale of the inhomo­geneities or microcracks (one meter or less). Continuum damage mechanics (CDM)is a convenient vehicle to handle the communication between the scale levels.Table 10.2 "Toughness-dominated relations" from Hagel and Meyer [38J

Model XI (X w(x p. (X

PKN [ £'V ] [K/~t] [KIC]K/chj2 hIt[~r3 [Ktcvr3 [K4 h f3GDK ._..K_LKIChf £12hf E'V

Sneddon (Radial) [£'VfS [K7c V] 1/5 [K~c rsKIC £'4 E'V

10.4 Continuum Damage Mechanics inHydraulic Fracturing

Before the main hydraulic fracturing treatment, an injection test with the same fluidand the same injection rate but smaller injected volume is often executed to revealthe Ieakoff characteristics of the formation. Thus, during the main treatment - orat least during the initial phase of the main treatment - there is no tip propagation.Rather an existing fracture is reopened. Very often the treating pressure is essentiallythe same for the reopening as it was for creation of the original fracture. Since inreopening the tip does not propagate, no tip retardation effect can be manifested.

A possible resolution of the reopening paradox was provided by Shlyapoberskyand Chudnovsky [31). They suggest that when a hydraulically induced fracture hasclosed and is "reopened", the same main hydraulic channel is unlikely to be formed.A new main channel is formed during each injection, and a new process zone (damagezone) is created.

10.3.5 The Reopening Paradox

This approach has emerged from the observation that observed fracture toughness isnot a material property (Shlyapobersky et al. [36]). The increase of apparent fracturetoughness with high confining stress can be predicted by theory (Yew and Liu [37]).Apparent fracture toughness is widely used in the industry as a calibration parameterin order to reproduce elevated net pressures. The idea is to reproduce the high netpressure in the injection period either exactly the same way as our Example 10.1shows or to postulate that the net propagation pressure is a sum of the unretardedpropagation pressure plus the propagation pressure increase due to fracture toughness.As usual in the petroleum engineering literature, these models are presented in theform of proportionalities. A present collection of such proportionalities given byHagel and Meyer [38] is reproduced in our Table 10.2. (The connection between thelast entry of the table and our Example 10.1 is obvious.)

Field tests showed that apparent fracture toughness is an order of magnitudelarger than the laboratory value. Moreover, it is rate dependent and increases withthe fracture length [31). Variations of fracture toughness are usually treated withinthe R-curve approach (Bazant et al. [39]) in fracture mechanics, but extrapolation ofan R-curve to hydraulic fracturing scales seems to be impossible. Since no physicallysound scale law is available to predict the variation of the fracture toughness withfracture dimension, the "calibration" experiment should be as close to the actualtreatment as possible.

extreme cases are considered. In the "cooperative" fracture case, the intensity of thedamage formed as a response to the stress concentration at the tip of a propagatingcrack is much greater than the intensity of pre-existing defect population. This caseis well described by the crack layer theory [13]. The other extreme mode is whencrack propagation is controlled by the field of pre-existing defects and does not causenoticeable changes in the field. This case is described by statistical fracture mechanics(Chudnovsky and Gorelik [40]). Both mechanisms are present in hydraulic fracturingand are subject to scale effects. The translation of these creative concepts into workingmodels of hydraulic fracturing might be a major improvement in the future.

10.3.3 Apparent Fracture Toughness

(Gardner [35J). It seems that such modification cannot reproduce the increasing char­acter of the net pressure curves at late injection times.

There is growing evidence that any attempt to vary numerical values of the param­eters involved in the classical models (e.g. elastic modulus), especially if done onlyin the near tip region, is insufficient to achieve a breakthrough in understanding thepropagation process.

247Continuum damage mechanicsFracture propagation246

(10.31)and a similar relation for I is simply

I = clID•

(10.30)C_ crH,minC- _2 D,

Cz'4

(10.29)(

1/2 ) 2-2 XfD 2UfD = CDiD WD.x=Xj·

ID+xfD

where the relation between the Kachanov parameter, C, and its dimensionless coun­terpart, CD. is given as

uo.zs)Uj = .si: (_xyz )2 W;=XjrrcrH,min I +x f

can be used as a boundary condition replacing the "net pressure equals zero" boundarycondition in the two-dimensional differential model given by Eqs. 9.67 to 9.69.

Remarkably, the propagation velocity explicitly depends on the minimum principalstress and is inversely proportional to the fracture length, at least for long fractures.

(In Chapter 11 we will discuss the "stress intensity paradox" connected with thepropagation of elongated elliptical three-dimensional cracks. From those results itwill be clear that the scale parameter, i, has another easily acceptable interpretationas well. For a homogeneous and perfectly elastic formation it can be substituted bythe height of a well contained elliptic fracture. In that case the stress intensity factorat the tip calculated from an analytical solution will be indeed proportional to theinverse of the square root of the fracture length.)

It is convenient to consider the model in dimensionless form (see Eqs. 9.75 to9.79). The dimensionless form of Eq. 10.28 is given by

A simple estimation of the nominal stress intensity factor is Pn.[xjZ, where Pn,f isthe net pressure at the fracture tip. If this approximation is substituted into Eq. to.27,the resulting condition

10.4.2 CDM-NK Model

(10.27)-2 ( )2CI Kl_nUj---- ---

- rrcrH.rnin I+xf

where the product of the Kachanov parameter and the square of the scale parameter,-?

Cl", will playa central role.

pronounced with larger ratio xf /1, and (2) the presence of damage does not affectthe stress field far from the damaged area.

Assuming linear damage growth (K = 1), substitution of the specific stress distri­bution into Eq. 10.25 and applying the rupture criterion yields the following form ofthe tip propagation velocity [43]. '

249Continuum damage mechanics

~~-~~------------ ..--.- --..

where KI.n is the nominal stress intensity factor calculated from horizontal planestrain approximation and I is a scale variable measured in meters. It represents thescale of damage and can be considered as the "average distance of microcracks".

The factor l/ (I +xf) in Eq. 10.26 can be interpreted as the reciprocal number ofthe microcracks crossed by an elongated fracture. If the fracture is short with respectto I, the stress-ahead function is the one known from elasticity theory (see Chapter 2).If the fracture is long, the stress is much less than it would be if calculated fromthe nominal stress intensity factor because the presence of microcracks decreasesthe stress concentration capability of the macrocrack. Equation 10.26 expresses acoupling between damage and stress for massive hydraulic fracturing in a highlysimplified form. It should be considered as a working postulate reflecting the twomain known features of the stress-ahead function: (1) the effect of damage is more

(10.26)

Equation 10.25 was used by Wnuk and Kriz [42) in a local manner to describemacrocrack propagation in a damaged material. It is reasonable to assume that inthe intact state of the rock D = O. As the macrocrack approaches a given materialpoint, the actual stress increases because the macrocrack acts as a stress concentrator.In the mean time, the local damage is increasing according to Kachanov's law. Thematerial point joins the macrocrack when the damage reaches its critical value, D = 1.Assume for the moment that the stress distribution moving ahead of the propagatingcrack is known. This determines the velocity of the tip because a tip velocity thatis too fast would not leave enough time for the damage to reach its critical valuewhile a tip velocity that is too slow would cause the damage to evolve beyond itscritical value without the rupture of the materiaL In Valko and Economides [43,44)the stress ahead of the moving fracture was assumed to be of the form

(10.25)dD (cr)k--C --dt - I-D

where a is the nominal stress calculated without considering damage. SubstitutingEq. 10.24 into Eq. 10.23, the damage accumulation equation is given by

(10.24)

where the Kachanov parameter, C, and the exponent, k, are material properties,at least with respect to the considered scale. According to Rabotnov's interpreta­tion [41], if a material point is partly damaged, only (1 - D) fraction of an elementarysection is able to carry the load, and hence the net section stress is defined as

(10.23)dD-=C~dt n'

stress, cr,,, in a power law form:

Fracture propagation248

· .._-_ .._-- .. -- --- _-- '--'- ------

toDimensionless maximum width (or net pressure) vs. time, depending on the combinedparameter (After [43J)

Figure 10.3

10S10110-1 10710-310-1

Figut'e 10.5 Relative net pressure increase with respect to unretarded propagation. Dependenceon the dimensionless combined parameter and dimensionless time.

other words, the CDM model automatically reduces to its ancestor model if thecombined parameter is large enough.

If CDI; < 1, the fracture propagation is retarded by the rupture process andhence the length will be less and the wellbore width and efficiency (ratio ofthe fracture volume to the fluid volume injected) will be greater than the corre­sponding Nordgreri-Kemp value. As a consequence, the net treating pressurewill be a multiple of the Nordgren- Kemp value. The factor by which the netpressure is higher than the unretarded value depends on the dimensionless timeas well. Figure 10.5 shows the relative pressure increase with respect to theunretarded value.

(b)

C j20.0001

D D

.>:V 0.001

cf2DD

V 0.01 ~L 0,' --..__1

V

I 11111

100

0.010.001 0.1

Figut'e'10.2 Solution of the CDM-NK model, Dimensionless length vs. time, depending on thecombined parameter (After [43])

16

a: 14~~ 12§ 10.:~ 8

& 6

4

2

o ~~--~--~--~~~----~~~--~--~~0.0001

~c------------------------------------18

Figure 10.4 Fluid efficiency vs. dimensionless time, depending on the combined parameter(After [43])

to

10110-3

1()3

1Q2

10'

X'D100

10-'

10-2

1Q-3

1Q-410-3 10-1 101 1()3 10S 107

to

10S

-2If ColD > 1, the fracture propagation is not restricted by the rupture processand the solution path does not differ from the original Nordgren-Kemp, In

(a)

10-2

C /2DD

.'-0.0001

~0.001

0.1 t 0.01

I 111111111! 11111I

10-'

The constants Ch C2, and C4 are the Nordgren-Kemp constants introduced inSection 95.1.

The CDM version of the Nordgren-Kemp model (CDM-NK) can be solved bya finite differences shooting algorithm [44]. The effect of the two CDM parameterscan be summarized as follows:

1. For a large fracture (small damage scale), when TD«xfD the dimensionlesssolution path is affected only by the combined parameter CDT~.Figures 10.2 to 10.5show computational results obtained for this case.

10°

Fracture propagation250 251Continuum damage mechanics

--~- ,-_.- .._-,. ----,------------ ----,----------------------------

(10.35)7r (7r 7r)YCDM = 4' + "5 - 4' ko,

and

Table 10_3 Formation and fluid characteristics forExample 10.2

£' 8.5 x 1010 Pa 1.2 x 107 psihi 40 m 131ftP. 55 MPa 7980 psiCL 8.5 x 10-5 m/sl/2 0.0022 ft/min1/2Sp 0.015 m 0_59 inJL 0.001 Pa· s lcp

0.016 m3/s 24_2/2 = 12.1 bpmt, 6000 s 100 min

(1034)

where ww.O,PKN is the maximum wellbore width calculated from the PKN widthequation (Eq. 9AO). The factor ICDM is introduced to account for the larger widthat the wellbore and the factor YCDM is somewhat different from the one used in thePKN model (i.e. in Eq. 9A1) because the distribution of the pressure, and hence, thevariation of the width, is different from the PKN case. Both these factors depend onthe combined parameter, CD1b. The Seiler correlation is given by

(10.33)and

Determine the created fracture length, maximum wellbore width, average width andfluid efficiency for a waterfrac treatment from the basic data given in Table 10.3. Firstapply the PKN model without tip effect and find the propagation pressure at the endof pumping. Then repeat the calculations using the additional information that thepropagation pressure at the end of pumping is 63_2 MPa (9180 psi).

(10.32)Ww,O= ICDMWw.O.PKN

Example 10.2 CDM-PKN Simulation of a Water FracturingTreatment

While the CDM theory resulted in the introduction of two new parameters, namelythe Kachanov parameter, C, and the average distance between microcracks, T, a~arge-sca~~treatment is affected only by the combination of these two parameters,I.e. by C I . From now on we restrict our consideration to such large treatments anduse only the combined parameter to characterize the retardation effect.

Careful examination of the results of the CDM-NK model shows that a small valueof the combined parameter, CDI~, has two effects on the net pressure: It elevatesthe level of the net pressure curve and it causes a more uniform distribution of netpressure inside the fracture (Economides and Valko [45]).It is possible to account for these two effects directly in a simplified design

version. Seiler [46] fitted the results of the CDM-NK model and obtained a simplemodification of the PKN model. The CDM version of the PKN model (CDM-PKN)is defined by the two equations

The logic behind the formulas is simple. The factor IcDM reduces to unity if kois unity, i.e. the tip propagation is unretarded. For ko =0, the factor YCDM reducesto the PKN value, i.e. 7r/5. Thus, the CDM-PKN model reduces to the PKN modelexactly if ko is set to unity. Once tip retardation is significant, i.e, the value of kobecomes less than unity, the factor ICDM starts to grow. However, for very largedimensionless limes (long fractures) the ICDM factor reduces to unity because allthe curves on Figure 10.2 to 10.4 join the unretarded curve at very late times. Thevariable k3 in the correlation is designed to ensure this "late-time" convergence.

In the typical design mode, the desired fracture length x f is specified. The PKNwidth, Ww.O.PKN, may then be calculated from Eq. 9.14 and, considering CDM,adjusted according to Eq. 10.32. The average width can be obtained from Eq. 10.33.The subsequent steps follow a conventional design procedure presented e.g. byMeng [47].

The CDM-PKN model is especially useful for the analysis and design of massivehydraulic fracturing with low viscosity fluid (e.g. water) where the observed highnet pressures cannot be related to viscous dissipation.

10.4.3 CDM-PKN Design ModeJ

(10.36)

ko = min(CDI~, 1)

kl = 0.62 + 0.38 x k5k - k-O.372 - °

(E'h4 CB) 1/3

k3 = 4.75 x ko x --J-I=.IJ J1-

where2. When TD cannot be neglected with respect to XfD, the corresponding CDT;pressure-time solution path is reached asymptotically for late times, while in earlytimes the curve has a sharp decreasing character [44].

As seen from Figure 10.5, the combined parameter CDT~ can be used to describethe. "abnormally high pressures" during the whole fracturing treatment. For largescale treatments it is reasonable to assume that the fracture is large with respect tothe damage scale, and hence, the only relevant parameter is the combined param-

-2eter, CDlD'

If the combined parameter is relatively high, tip propagation rate is controlled bythe viscosity of the fluid (similar classical fracturing models.) When the combinedparameter is low, propagation is retarded by the damage evolution, and hence thefluid viscosity has very limited effect on the propagation rate.

253Continuum damage mechanicsFracture propagation252

-_ .._-._-_._---------------------_._-- ..- .....__ .-

Equations 9,40, 9,46, 10.32 and 10.33 can be reduced to one equation, eliminating allvariables but xf' Thus,

Determine the created fracture length, maximum wellbore width, average width, fluidefficiency and propagation pressure for a waterfrac treatment if the basic data is the sameas in Table 10.3 of Example 10.2 except with a double injection rate, i =: 0.032 m3jsper one wing (48,4 bpm per two wings). Use the combined CDM parameter obtainedin Example 10.2, i.e. cr = 5.5 x 10-7 m1/(Pa. s) [0.042 ftz/(psi· s)).

)

1/4_ -O.00039xf (0.001 x 0.016 x xf- [1+ 7 x e ] x 8.5 X 10iO

Example 10.3 CDM-PKN Simulation with Increased Injection Rate

where Xt is still unknown. From Eq, 10.35, y = 7'(/4.From Eqs. 9,40, 10.32 and 10.33,

From Eq. 10.34,

Substituting the closure pressure into the minimum principal stress and using the expres­sions for coefficients Cj, Cz and C4 given by Eq. 9.76, the following value is obtained:

-2 ( ihf ) 2/3 -2Cl = 0.0992 x jJ.CLE'z x Pc x CDID

(0.016 x 40 )2/3 7

= 0.0992 x 0.001 x 8.5 x 10-5 X (8.5 X 1010)2 x 5.5 x 10 x 0.001

= 5.5 x 10-7 m2j(pa. s) [0.042 ftZj(psi· s)].

The dimensioned combined parameter clz =: 5.5 x 10-7 mZj(pa· s) [0.042 ftz/(psi· s))is considered to be a property of the formation. 0The real power of the CDM-PKN theory is the ability to extrapolate to other

conditions. In the following example we calculate the propagation pressure at theend of another 100 minute treatment with injection rate twice the original value.

leo =: min(CDlb. 1) = 0.001

kl = 0.62 +0.38 x kJ = 0.62

k2 = koO.37 = 12.9

k3 = 4.75 x ko x (E'~}Cf) = 4.75 x 0.00115u:

[010 04 8 10-5)8] 1/3x 8.5 x 1 x 4 x ( .5 x = 0.000390.0165 x 0.001

where gO~ =: 1.415. The solution of the equation and the other results stemming fromit are presented in Table 10.5.

Compared to the results in Table 10,4, the length is smaller and the width and netpressure are considerably larger than the PKN values. The retarded propagation allowsless surface for leakoff and spurt loss, so the fluid efficiency increases compared to theunretarded case.

It is useful to calculate the combined parameter, ct', characteristic for the formation,from its dimensionless value, 0.001. Combining Eq. 10.30 and 10.31 we obtain

60000.016 x 40

xf = -'----{ ~-----:-:-:--:-1/4} .7'( -0.00039I/ (0.016 x 0.001 x Xf)4"[1 +7 x e ] 8.5 x 1010

+ 2 x 8.5 x 1O-5J6000 x go 0) +2 x 0.015

Calculations similar to Example 9,4 give the PKN-a results which are shown inTable 10,4.

Clearly, the PKN propagation pressure is very low compared to the value given asknown in this example. This happens often when the fluid is water because most of thehydraulic fracturing models consider only viscous dissipation of energy, and water hasa very low viscosity. The likely explanation of the high propagation pressure observedin practice is that the propagation of the tip is retarded. Within the CDM-PKN model,this would mean a smaller than unity value of the combined parameter CDlb. A trialand error procedure can be used to reveal the value of the dimensionless combinedparameter, which gives the specified propagation pressure, 63.2 MPa (9180 psi). Hereonly the last iteration is shown. Assume that the current estimate of the CDlb parameteris 0.001. According to Eq. 10.36 the constants leo, kb k2 and k3 are obtained as

Solution

T/ = 11.1%pw.pr = 63.2 MPa(9170 psi)

XI = 43.9 m(144 ft)w.·.o =: 0.0078 m(O.31 in)W = 0.0061 m(O.24 in)

T/ =: 1.3%P"'pr = 56.1 MPa(8130 psi)

XI =: 48.7 m(160 ft)w •..o = 0.0010 m(O.04O in)w =: 8.48 x 10-3 m(O.00064 in

Table 10.5 CDM-PKN simulation for Example 10.2Table 10.4 PKN-a simulation without tip effect for Example 10.2

255Continuum damage mechanicsFracture propagation254

----~-----,·· ..... • ·_·_· .,w • •• • .• " ",_, ., •

...

is a clear indication of tip retardation.Note that the viscosity of the fluid is in the denominator of Eq. 10.41, so for thin

fluids, the basic inequality 10.42 will often signal tip retardation.Once the phenomenon of tip retardation is established, the use of a model capable

to reproduce the phenomenon is a necessity. One of the models providing the descrip­tion of tip retardation is the CDM-PKN model which uses the combined parameter,c12•This parameter modifies Eq. 10.41 and keeps the length below the limit posedby Eg. 10.40.

(10.42)Xf,ur > xf.max

If the unretarded fracture length estimate is longer than the maximum possible valuegiven by Eq. 10.40, then the original PKN width equation (i.e. Eq. 10.41) did nothold during propagation and the tip was retarded. The inequalitySome of these questions were answered by the previous examples. We saw how

the tip retardation manifests itself as high propagation pressure once we know theparameters of the fluid loss process. We saw how to determine the combined param­eter, and how to apply it in design (even if some of the treatment characteristics aredifferent from the calibration test.)

What is still not clear is how to obtain the leakoff, spurt loss and tip retardationcharacteristics simultaneously. The remaining part of this chapter is devoted to thisprincipal issue.

The essential idea is that the presence of tip retardation does not change the lawsof linear elasticity and leakoff which are in effect during the decline period. Forthe PKN geometry, i.e. for the vertical plane strain assumption, Eq. 9.105 is stillin effect:

(10.41)4 ( hite )Xf.ur = 0.140(Pw.pr - Pc) £'3J1.V; .

Xf.max = 7Th}Cb- Pc)'

On the other hand, the propagation pressure provides an estimate of the fracturelength if the propagation is unretarded. Rearranging Eq. 9.89 we obtain the unre­tarded fracture length, xf,ur, as

(10.40)2E'Vi

While the continuum damage mechanics model may not be the ultimate solution tothe challenging problem of modeling hydraulic fracture propagation, it provides aframework to predict fracture performance under variable conditions while avoidingsome unreasonable features (such as independence or decrease of treating pressurewith increasing injection rate) often inherent in other theories. However, some ques­tions (relevant to any practical theory of hydraulic fracturing) have to be answered.

1. Can the tip retardation phenomenon be revealed from field data?2. Can the combined parameter be determined from an injection test?3. If the combined parameter is known, can we predict the main characteristics of

the fracture created in the main treatment? Can we do this even if some of theparameters (e.g. injection rate) are different in the calibration test than in themain treatment?

(10.39)s _ ~ 7Th/(pc - b)P - lxfhf + 4E'

Equations 10.38 and 10.39 are not affected by the fact that a shorter length is createdduring the injection period.It is easy to see that the second term on the right-hand side of Eq. 10.39 is

negative. Since the spurt loss cannot be negative (a flow into the fracture duringfracture propagation is absolutely not possible), the fracture half length, xf' shouldbe less than a maximum value, xf,max, where

10.5 Pressure Decline Analysis and Tip Retardation

An additional relation between the fracture half-length and spurt loss coefficientexists in terms of the intercept, b, of the straight line as follows:

When the injection rate is 0.032 m3/s, the dimensionless parameter, CDI~, will bedifferent from its previous value, 0.001. It is easily seen that for an injection ratetwice the previous value the dimensionless combined parameter will be less by afactor of 2-2/3. Thus Col; = 0.001 x 2-2/3 =0.00063. With this new dimensionlesscombinedparameter,the solutionof the CDM-PKNmodelgives the results summarizedin Table 10.6.

Compared to the results in Table 10.5, the increasedinjection rate brings about anincreasedpropagationpressure and increased efficiency.0

Solution(10.38)

From Eq. 10.37, it is seen that the leakoff coefficient can still be obtained from theslope, m, of the straight line on the g-plot as

T!hfCL= --(-m).

4...;t;E'

(10.37)

2E'V; 4E' ( 4£' )Pw=Pc+-h, --h Sp- hCL~ xg(LltD,a)

T! fX f T! f T! f

= b + m x g(LltD, a).I)= 17.2%P....p, = 68.7 MPa (9980 psi)

Xj = 8.1 m(268 ft)ww.o = 0.013 m (0.51 in)W = 0.010 m (0040 in)

Table 10.6 CDM-PKNsimulationwith increased injectionrate,Example10.3

257Pressure decline analysis and tip retardationFracture propagation256

'-"._-_ ... _._ .._-_ .._- .._-" -._---~- .•.-- __ -_---------- ..-.--.-_- ..

A parameter set is accepted if both and ~X f and Spare positive real numbers.(Otherwise the assumed tip retardation is still not enough to keep the propagationlength below its physically possible maximum.) It is suggested to start the procedure

_ (b + 1.415 x m - Pc)1r:hjwISI = 2£' .

2. xj and ~X j are related through

1. WISI satisfies5. Calculate the spurt loss coefficient from

_ { wpr }4 (Elte)- 3.27 x [1+(k1k2-1)exp(-k3Xj,pr)] x [rr(0.25-0.05ko)] fJ-Vj'

4. Solve for ~x j the following quadratic equation (select the positive root)

_ _ _ 2rrhj(pc - b)~xj ~Xj ViWISrxf.pr - WprXj.pr + wISJ~xf + ,+ = O.

4E (xf.pr+~j)hj

1. formation data: E', hj, and Pc;2. treatment data: Vj, i, and u;3. observed data: m and b of the g-plot straight line and the propagation pressure,

Ppr'The following conditions have to be met:

Xj.pr

3. Solve the following nonlinear equation for Xj.pr

E'

_ (b+ 1.415 x m - pc)rrhjWISI = ------___;;'---~

2E'_ 2rrhf(0.25 - 0.05ko)(Ppr - Pc)wpr ==

4. the half-length at the last instant of propagation, x j,pro5. the average width at the last instant of propagation, Wpro and6. the average width following the short after-growth process, WlS!.

It is assumed that the fracture propagates according to the CDM-PKN model(Eqs. 10.32 to 10.36) combined with the a-type material balance (Eq. 9.46). Inaddition, it is assumed that the fracture propagates after the pumps stop, with theincremental length given as the after-growth parameter. It is assumed that the after­growth happens in a short time; this assumption has to be checked after the modelis fitted. The last assumption means that the pressure after shut-in is modeled byEq. 10.37, where Xf is the sum of the half length at the last instant of propagationplus the after-growth.Input data required for the estimation procedure are the following:

ko = CDl1

kl = 0.62 + 0,38 x 14k2 = kOO.37

(E'h4 C8~) 1/3

k3 = 4.75 x ko x fs L eVifJ-

In addition, we will use the following variables:

-21. the combined parameter, Cl ,2. the spurt loss parameter, Sp» and3. the after-growth, ax],

1. Assume a dimensionless combined parameter, CDlb, less than unity.2. Calculate

The algorithm to determine the parameters is numerically simple:

WISIXj + 2Sp~xj = WprXf.pr'

4. Eqs. 9.40 and 10.32 give the wellbore maximum width and Eq. 10.33 the averagewidth at the last instant of propagation.

5. Net wellbore pressure calculated from Eq. 9.43 (with the wellbore width obtainedfrom Eq. 10.32) should equal the observed propagation pressure minus theclosure pressure.

In the following we suggest a procedure to obtain a set for parameters of the CDM­PKN model which reproduce the main characteristics of an injection test withoutinner contradiction. The procedure is counterpart to the pressure decline analysissuggested in Section 9.6.2.

We start from the straight-line fit of the g-plot. Once m and b are determined,the leakoff coefficient, CL, is obtained from Eq. 10.38. The next step is to calculateXj,max from Eq. 10.40 and Xj,ur from Eq. 10.41. If the basic inequality 10.42 doesnot hold, there is no reason to use a tip retardation model and the procedure ofSection 9.6.2 should be followed. If the basic inequality holds, the fracture propa­gation is interpreted in terms of the CDM-PKN model. The following parametersare unknown:

3. The fracture volume at the end of injection equals the volume after the after­growth process plus the spurt loss during the after-growth process:

10.5.1 Resolving Contradictions with ContinuumDamage Mechanics

259Pressure decline analysis and tip retardationFracture propagation258

~------------------_---- _ ..

CL = 1.7 x 10-5 rn/Slf2(4.3 x 10-" ft/minl/2)Sp = 0.0075 m(0.30 in)C/2 = 1.6 x 10-5 m2/(Pa· 5) [1.2 ft2/(psi. 5)]AtQ= 11 min

Table 10.8 Accepted parameter set forExample lOA

The results of the parameter estimation procedure are summarized in Table 10.8.

c72 = 0.0992 x ( Vih f ) 2!3 -2f.JCLE'2te X Pc X Colo

(125 x 51.8 ) 2/3= 0.0992 x x 39.3 X 106 x 0.07

0.1 x 1.7 x 10-5 X (6.1 X 1Ol0)2 x 2220

= 1.62 x 10-5 m2J(Pa· s)[1.2 ft2/(psi· s)].

that the total after-growth time is 6.ta.o = 10 min, where we use an extra subscript,"0", to denote that this value is observed. Using the CDM-PKN model, we make asimulation run for every parameter set (Col;, CL, Sp) shown in Table 10.7. At the endof pumping we calculate the fracture propagation rate from the length increment duringthe last minute of injection, and denote it by Uf. Assuming that this propagation rate willlinearly decrease to zero, the calculated after-growth time is given by Ata = 2Axt/uf.We accept the data set for which the calculated and observed after-growth time differsthe least, i.e. we minimize the deviation of 6.ca from Ala.a.

In the given example, the procedure leads to the data set corresponding to CDl~=0.07, Sp = 0.0075 m (0.30 in) and 6.ta = 11 min. The suggested criterion to select theappropriate parameter set is rather selective. If the previous parameter set (Cot; = 0.08)is selected, the after-growth time is too large (Ata = 14 min) and if the next parameterset (CDI~ = 0.06) is selected, the after-growth time is too low (Ata = 8.5 min).

The combined parameter in dimensioned form is obtained from

Pressure decline analysis and tip retardation 261

Table 10.7 Consistent data sets determined from the pressure decline datafor Example 10.4

-,CDr; Sp (m) x{.pr (m) 6.xf (m) ~~Sl (m) ~pr (m)

0.12 0.000356 211 116 0.0044 0.007200.11 0.00126 187 76.1 0.0044 0.007220.10 0.00231 163 51.3 0.0044 0.007230.09 0.00359 140 34,4 0.0044 0.007250.08 0.00526 118 22.7 0.0044 0.007260.D7 0.00750 97.1 14.4 0.0044 0.007280.06 0.0107 77.5 8.72 0.0044 0.007290.05 0.0155 59.3 4.88 0.0044 0.007310.04 0.0234 42.7 2.44 0.0044 0.007320.D3 0.0384 27.9 1.01 0.0044 0.007340.02 0.0738 15.3 0.298 0.0044 0.00735om 0.215 5.5 0.0377 0.0044 0.00737

Since xf.ur > xf.max, the tip propagation is retarded and the CDM-PKN model shouldbe used.

Following the procedure delineated above, we start the calculation with a dimension­less combined parameter equal to unity and decrease it step by step. The first positive

-2real root of the quadratic equation for 6.xf appears when Colo = 0.12. The results areshown in Table 10.7.

The obtained parameter sets describe the pressure behavior equally well during theinjection period. In the time period after the pumps stop the parameter sets still result inan equally good match of the pressure decline curve when only that part is consideredwhich gives a straight line on the g-plot.

Additional considerations are needed to select one of the above parameter sets. Onepossible procedure is based on the early behavior of the pressure decline curve. Thetechnique is the following. The approximate time of after-growth is estimated from theg-plot. From Figure 9.2 we see that for the SFE3 injection test approximately 10 minutescan be considered as a potential time interval for after-growth because the observedpressure points are located markedly off the straight line in the first 10 min. Assume

51.84 x 2220= 0.140 x (5.56 x 106)4 0 3 = 742 m(2440 ft).. (6.13 X 101 ) x 0.1 x 125

4 ( hite )xf·ur = 0.140(Pw.pr - Pc) E'3f.JVi

_ 2E'Vi _ 2 x 6.13 X WiD x 125 _ 362 (1190 f)xfmax- , - 6- m t I.. TChj(b - Pc) tt X 5.182 x (44.32 - 39.3) x 10

The unretarded propagation length from Eq, 10.41 is

The slope and intercept of the straight line are m = -1.208 MPa and b = 44.32 MPa.The leakoff coefficient is the same as in Example 9.7, i.e, CL = 1.70 X 10-5 m/s1/2(4.32 x 10-4 ft/mini(2).

The maximum possible length is determined from Eq. 10.40.

Solution

Consider the formation, fluid, injection and pressure decline data analyzed inExamples 9.7 and 9.8. Repeat the pressure decline analysis without artificially changingthe viscosity, i.e, use the originally accepted estimate, f.J = 0.1 Pa· s (100 cp). Accountfor possible tip retardation.

Example 10.4 Pressure Decline Analysis with the CDM-PKN Model

with a near unity dimensionless combined parameter, CD1b' If the parameter isdecreased step by step, the first feasible parameter set appears when the spurt losscoefficient is zero. Further reduction of the combined parameter will yield additionalfeasible parameter sets. This is not surprising because we posed five constraints onsix variables. The following example illustrates the procedure and addresses the issueof uniqueness.

260 Fracture propagation

--_._-_.:;;--"~---.--------.

1. Nolte, K.G. and Smith, M.B.: Interpretation of Fracturing Pressures, J. Petrol. Technol.,1767-1775,1981.

2. Von Kinnan, Th.: Festigkeitsversuche unter allseitegem Druck, Z. Ver. Deutscher Inge­nieure, 55 (42),1749-1757,1911.

3. Timoshenko, S.P.: History of Strengths of Materials, McGraw Hill, New York, 1953.4. Gramberg'J.: A Non-conventional View on Rock Mechanics and Fracture Mechanics,

Balkema, Rotterdam, 1989_5. Coulomb, C.A: Sur Une Application des Regles de Maximis et Minimis a Quelques

Problemes de Statique Relatifs a I' Architecture, Acad Roy. des Sciences Memoires demath. et de physique par divers savans, 7, 343-82. 1773.

6. Mohr 0_: Welche Umstande bedingen die Elastizitatgrence und den Bruch eines Mate­rials? Z. VeT. dt. Ing., 44, 1524-1530, 1572-1577,1900.

7. Griffith A.A. The phenomena of Rupture and Flow in Solids, Phil. Trans. Royal Soc.London, Ser. A 221, 163-198, 1920.

8. Griffith AA The Theory of Rupture Proc. First International Congress on AppliedMechanics, Delft, 55-63, 1924.

9. Jaeger J.C and Cook N.G.W.: Fundamentals of Rock Mechanics, Chapman and Hall,London, 1976.

10. Lamaitre J.: Local Approach of Fracture, Engineering Fracture Mechanics, 25(5/6),523-537, 1986.

11. Boone T.J., Wawryznek, P.A and lngraffea, AR.: Simulation of the Fracture ProcessZone in Rock with Application to Hydrofracturing, Int. J. Rock Mech. Min. & Geomech.Abstr., 23(3), 255-265, 1986.

12. Kachanov L.M.: Time of Rupture Process under Creep Conditions, Izv. Akad: Nauk SSR,Otd. Tekh., 8, 1958.

References

propagation pressure at the end of pumping and the decline pressure curve are matchedcorrectly. At early times the treatment pressure is overestimated. Reasons for this mightbe numerous and are beyond the scope of this chapter. Any attempt to "improve" the fitby reproducing the small but evident changes in the trend of the pressure decline curvewould be meaningless because these capricious changes are due to systematic errors of.the particular measurement system. Also, we recall that the purpose of this exercise isto illustrate the flexibility of the tip retardation concept. The results depend heavily onthe selected values of plane strain modulus, closure pressure and height. The decisionto accept, discard or modify these values is an important engineering activity, but isalso beyond the scope of this chapter.

At this point it is in order to recall that the selected viscosity, I-' = 0.1 Pa s (100 cp),corresponds to average shear rate conditions in the fracture with average width of 8 mm(0.3 in) as seen from Table 9.15. Since the final width shown in Table 10.9 is 7.3 mm(0.29 in), there is no reason to adjust the viscosity further.

The simulation stops when the pressure decreases below the closure pressure (i.e. thewidth reaches zero.) The remaining part of the pressure decline curve (after 180 min)cannot be simulated with the Carter leakoff concept. If we wish to describe that periodas well, the detailed leakoff model of Section 9.5 should be used. It is obvious that theCDM-PKN concept can be applied together with any fluid-loss model, including theparticular one of Section 9.5. 0

263References

'.--" _._- --- --------------------------- ----

4S 120X E

!IS XQ. 44 100~ £3:: C.0. 43" 80 t::~e e:::l...

60::::I.. 42 Ul!! :c.

e 40 "'C

s <11 ~~

"540 20 0

iiipc, dosure pressure o

39 '00 50 1® 150 200

Time,min

Figure 10_6 Observed and calculated pressure curve for Example 10.4

Using the treatment data and the parameters in Table 10.8 we simulate the injectiontest as follows. During the injection period, the CDM-PKN width equation (Eqs. 10.32,involving Eqs. 10.33, 10.34, 10.35, 10.36 and 9.40) is solved together with the materialbalance Eq. 9.46. The results, shown in Table 10.9, are obtained for the end of theinjection period.

Taking into account that the after-growth process lasts about 11min and the propaga­tion rate decreases linearly during this period, the fracture lengths shown in Table 10.10are easily determined from the formula of constant deceleration.

Now we have. all the information needed to use the first line of Eq. 10.37 withthe partly varying lengths given in Table 10.10 and the pressure decline can be easilysimulated. The simulated total pressure curve is shown in Figure 10.6. Note that the

I, min Xj

37 9S.3 m 323 ft38 101 m 330 it40 105 m 344 ft42 lOS m 354 ft44 110 m 362 ft46 111m 364 ft

from 48 111 m 365 ft

Table 10.10 Fracture length variationduring the after-growthprocess in Example 10.4

TJ = 29.7%uf = 0.039 m/s(7.S ft/min)pw.pr =44.88 MPa(6509 psi)

Xf.pT = 98.3 m(323 ft)w,.,.o = 0.00943 meO.37! in)iii=0.0073 m(0.287 in)

Table 10.9 CDM-PKN simulation result at the end of the injec­tion for Example lOA

Fracture propagation262

47.

46.

45.

44.

43.

42.

41.

40.

39.

38.

37.

36.

35.

34.

Cleary, M.P. Wright, C.A Wright, T.B.: Experimental and Modeling Evidence forMajor Changes in Hydraulic Fracturing Design and Field Procedures, Paper SPE 21494presented at the Gas Technology Syrnp., Houston, TX, Jan. 23-25, 19~1. .Van den Hoek, PJ., Van den Berg, J.T.M., Shlyapobersky, J.: Theoretical and Experi­mental Investigation of Rock Dilatancy Near Tip of Propagating Fracture, Int. J. RockMeeh. Min. & Geomech. Abstr., 30, 1261-1264, 1993.Gardner, D.C.: High Fracturing Pressures for Shales and Which Tip E~e~~s May BeResponsible, SPE paper 24852 presented at Techn. Conference and Exhibition, Wash-ington, D.C., Oct 4-7, 1992. . .Shlyapobersky, J. Wong, G.K and Walhaugh, W.W.: Overpressure-Calibrated Designof Hydraulic Fracture Simulations, Paper SPE 18194 presented at the 63rd TechnicalConference and Exhibition, Houston, TX, Oct 205, 1988.Yew, C.H. and Liu G.: The Fracture Tip and K1C of a Hydraulically Induced Fracture,Paper SPE 22875 presented at 66th Ann. Techn. Conf. and Exh. of SPE, Dallas, Oct.6-9,1991. .Hagel, M.W. and Meyer, B.R.: Utilizing Mini-Frac Data to Improve Design andProduction, J. Canadian Petro Technol., 33, 26-35, 1994. .Bazant, Z.P., Gettu, R. and Kazemi, M.T.: Identification of Nonlinear Fracture Prop­erties From Size Effect Tests and Structural Analysis Based on Geometry DependentR-curves, Int. J. Rock Mech. Min. & Geomech. Abstr., 28(1), 43-51, 1991.Chudnovsky, A. and Gorelik, M.: Statistical Fracture Mechanics, - Basic C~nce?ts a~dNumerical Realization, Probabilities and Materials: Tasks, Models and Applications, inBreysee D. (ed.) Kluwer, Boston, 1994.Rabotnov, Y.N.: Creep Rupture in Proc. XII Intern. Congress Appl. Mech., Stanford,Springer, Berlin, 1969. .' .Wnuk, M.P. and Kriz, R.D.: CDM Model of Damage Accumulation In LaminatedComposites, International Journal of Fracture, 28, 121-138, 1~85. .Valko, P. and Economides, M.J.: Continuum Damage Mechanics Model of HydraulicFracturing, J. Petrol. Technol., 198-205, 1993. .Valko, P. and Economides, M.J.: Propagation of hydraulically induced fractures - acontinuum damage mechanics approach, Int. J. Rock Mech. Min. & Geomeeh. Abstr.,31(3) 221-229, 1994. . .Economides, M.J. and Valko, P.: Interpretation and modeling of hydraulic fractu~gphenomena with continuum damage mechanics - An application to engineering design,In. Siriwardane H.J. and Zaman, M.M. (ed.) Computer Methods and Advances InGeome-chanics, 1579-1583, Balkema, Rotterdam, 1994. .Seiler, R.: Development of a Fracture Design Procedure Based on Continuum Damagemechanics, Diploma Thesis, Mining University Leoben, 1993.. .,Meng, H-Z.: The Optimization of Propped Fracture Treatments, In Reservo.lr Stimu­lation, Economides, M.J. and Nolte, KG. (ed.) Prentice Hall, Englewood Cliffs, N.J.,1989.

33.

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D. Chudnovsky A: Crack Layer Theory, NASA CR-17463 , Case Western University.Cleveland,OH, 1984.

14. Mott N.F. Engineering, 165, 16-18, 1948_15. Billington, E. W. and Tate, A: The Physics of Deformation and Flow, McGraw Hill,

New York, 1981.16. Zienkiewicz, O.C.: The Finite Element Method in Structural and Continuum Mechanics,

McGraw Hill, London, 1976.17. Erdogan, F. and Sih, G.C: On the Crack Extension in Plate under Plane Loading and

Tranverse Shear, J. of Basic Eng., ASME, 85, 519-527, 1963.18. Sih, G.C: Strain-Energy-Density Factor Applied to Mixed-Mode Crack Problems, Int.

J. Fracture Mechanics, 10, 305-321, 1974.19. Boone T. J., Ingraffea AR. and Rogiers J.-C: Visualization of Hydraulically Driven

Fracture Propagation in Poroelastic Media, J. Petrol. Technol., (June), 574-580, 1969.20. Gerstle, W. H. and Xie, M.: FEM Modeling of Fictitious Crack Propagation in Concrete,

1. Engrg. Mech., ASCE 118(2) 416-434,1992.21. Bouteca, M.L Hydraulic Fracturing Model Based on a Three-dimensional Closed Form:

Tests and Analysis of Fracture Geometry and Containment, SPE Production Engineering(Nov.), 445-454, Trans. AIME, 285, 1988.

22. Clifton Rl and Abou-Sayed AS.: On the Computation of the Three DimensionalGeometry of Hydraulic Fractures, Paper SPE 7943, Symp. on Low-Permeability Res.Denver, 1979.

23. Thiercelin, M., Jeffrey, R.G. and Ben Naceur, K: Influence of Fracture Toughness onthe Geometry of Hydraulic Fractures, SPE Production Engineering, (Nov.), 435-442,1988.

24. Heuze, F.E., Shaffer, R.J., Ingraffea, A.R. and Nilson, R.H.: Propagation of Fluid­Driven Fractures in Jointed Rock. Part 1 - Development and Validation of Methodsof Analysis, Int. J. Rock Mech. Min. & Geomech. Abstr., 27, 243-257, 1990.

25. De Pater, C.J., Weijers, L. Van den Hoek, P-I. Barr, D.T.: EXperimental Study of Non­linear Effects in Hydraulic Fracture Propagation, Paper SPE 25893 presented at the SPEJoint Rocky Mountain Regional Meeting and Low-Permeability Symposium, Denver,CO, April 12-14, 1993.

26. Medlin, W.L. Fitch, lL.: Abnormal Treating Pressures in Massive Hydraulic FracturingTreatments, J. Petrol.Technol., 633-642, 1988.

27. Palmer, tD. and Veatch Jr., RW.: Abnormally High Fracturing Pressures in Step-RateTests, SPEPE, (Aug.), 315-323; Trans. AlME, 289.

28. Cleary, M.P.: Rate and Structure Sensitivity in Hydraulic Fracturing of Fluid-SaturatedPorous Formations, Proc. 20th u.s. Rock Mechanics Symosium, Austin, TX, 124-142,1979.

29. Jeffrey, R.G.: The Combined Effect of Fluid Lag and Fracture Toughness on HydraulicFracture Propagation, Paper SPE 18957 presented at the 1988 SPE Joint Rocky MountainRegional!Low Permeability Reservoirs Symposium, Denver, CO, March 6-8.

30. Khristianovitch, S.A. and Zheltov, Y.P.: Formation of Vertical Fractures by Means ofHighly Viscous Fluids, Proc. World Petroleum Congress, Rome, 2, 579, 1955.

31. Shlyapobersky, J. and Chudnovsky, A: Review of Recent Developments in FractureMechanics with Petroleum Engineering Applications, Paper presented at EUROCK'94,Delft, August 29-31, 1994.

32. Desroches, J. Lenoach, B. Papanastasiou, P. and Thiercelin, M.: On Modelling of NearTip Processses in Hydraulic Fractures, Int. J. Rock Mech. Min. & Geomeeh. Abstr., 30,1127 -1134, 1993.

264 Fracturepropagation

._--_ .._._-_ ........,._-_.__ ..-._--_ .._--------_ ..---_ .. _..----,.

Geological formations consist of more or less distinct layers. As usual, only oneor few of the layers contain hydrocarbons. The well is perforated to access thehydrocarbon-containing target layer(s). A hydraulic fracture is intended to remainwithin the perforated interval, i.e. to be contained. However, simple observations oflithological logs frequently suggest no clear lithological barriers for fracture heightcontainment, although reservoir height delineation logs often suggest a much clearerlimit for the productive interval. In certain cases, lithological homogeneity maysuggest penetration by a growing fracture height into adjoining porous or even non­porous media. Indeed, there is evidence that "the fracture may grow up, out anddown the perforated interval" (Rahim and Holditch (1D.

In the models considered in Chapter 9, strong assumptions are posed on theresulting geometry of the fracture. These assumptions are designed to leaveonly one degree of freedom besides the width, and hence the description two­dimensional (2D).

If the height is considered to be known, the additional degree of freedom (besidesthe width) is the length. The different variations of the PKN and KGD models aresuch constant-height models. The constant-height models are the most typical butnot the only two-dimensional models.

The radially propagating fracture model is not of constant-height type, but still onlyone degree of freedom is left (the radius); thus, it also belongs to the two-dimensionalgroup of models.

In contrast to the two-dimensional models, three-dimensional (3D) models describethe change of the geometry allowing for complex shape variation which can bedetermined using only three space coordinates. Three-dimensional hydraulic frac­turing models envision the formation to consist of horizontal layers of differentthickness, mechanical properties and minimum horizontal stress. This detailed infor­mation is combined with the characteristics of fluid flow inside the fracture (including

FRACTURE HEIGHTGROWTH (3D AND P-3DGEOMETRIES)

11

.._- ---- .._- .__ .._ __ ---_ .•...---------------_ __ ..

To understand the concept of equilibrium height, some abstraction is necessary. Aline crack is considered in the vertical plane perpendicular to the lateral direction ofpropagation. At a given lateral location, x, the pressure along the vertical direction isconsidered to be either constant (if the hydrostatic pressure component is neglected)or varying linearly with depth (a result of increasing hydrostatic pressure). Simonsonet al. [4] considered the simplest case neglecting hydrostatic pressure and assumingthat the material properties of the upper and lower layers are identical. They assumedthat the minimum horizontal principal stress has the same direction in all the threelayers. In addition, the value of O"H.min in the upper layer (0"2) and in the lower layer(0"3) were assumed equal to each other and higher than in the target layer (0"1). Alsothe critical stress intensity factor in the upper layer (K[C.2) and in the lower layer(K[C.3) were considered equal. If all of these assumptions hold, the geometry of theequilibrium crack is symmetric.

To calculate the stress intensity factor at the tip, Simonson et al. [41 used theconcept of superposition, defining a different net pressure in the target and in thetwo other layers. This was done not because the pressure varies inside the fracture(it does not) but because the boundary condition at infinity changes with the verticallocation (aH.min is different in the target and in the other layers). Note that in thetheory of elasticity these minimum principal stresses are used as boundary conditionsat infinity. The use of superposition with respect to the change in the boundarycondition at infinity is not a standard technique of mathematical physics, and froma strictly mathematical point of view it might be questionable. It may be difficultto visualize that a discrete jump in a condition at infinity will cause a jump in thestress at the surface of the crack. Nevertheless, if we accept that the net pressure,calculated as the difference p - O"H.min, can be used to calculate the stress intensity

11.1.1 Reverse Application of the Net-pressure Concept

Griffith's theory of equilibrium allows us to calculate the length of a line crackif the inner pressure is known as a function of location. Applying this theory toheight-determination involves some technical difficulties, e.g. complexity of notationinvolved, possible non-existence or multiplicity of solutions to the resulting non­linear system of equations. Also, some basic assumptions considered trivial in thetechnique need particular caution. In all of our derivations we will consider only thethree-layer case. Some remarks concerning more layers will be given at the end ofthe discussion.

The approach of Simonson et al. [41 has been used widely in the petroleumindustry. The penetration of fracture into the upper and lower layers surroundingthe target layer is determined by obtaining the equilibrium height for a given netpressure. The equilibrium height satisfies the condition that the computed stressintensity factor at the vertical tip equals the critical stress intensity factor. The latterquantity, often called fracture toughness, is considered to be a material property ofthe rock.

11.1 Equilibrium Fracture Height

269Equilibrium fracture height

_._---_ .._.•....•_---

.: ;

the resulting pressure distribution) to determine the direction and velocity withwhich the fracture boundaries move. The width at a given lateral and vertical loca­tion is calculated from a detailed description of the stress and displacement field.The numerical methods of linear elasticity theory are used extensively. Fluid flowis considered two-dimensional, assuming quasi-steady-state flow between (locally)parallel plates.

Often the name pseudo-three-dimensional (P-3D) model is also used in hydraulicfracturing (Settari and Clearly [2]). Most P-3D models do not consider vertical fluidflow but, instead, predict height from net pressure as a function of lateral locationand with the local use of fracture mechanics criteria. Analytical solutions for simpli­fied cases are used (Bouteca [3]) in place of detailed numerical methods of linearelasticity. In some cases the difference between three- and pseudo-three-dimensionalmodels is not well defined.

Both in three- or pseudo-three-dimensional models the aim of departure fromtwo-dimensional theory is to describe the process of hydraulic fracturing with betteraccuracy. Since the geometry is known in more detail, it is natural to use this infor­mation to refine the description of all other elements. In particular, in 3D modelsthe fluid flow submodel may take into account the movement of the fluid not onlyin the lateral but also in the vertical direction (2D flow). In the future, it may evenbe possible to track the motion of fluid particles in a direction perpendicular to thefracture surface by introducing a three-dimensional description of fluid flow. Oncethis Pandora's box is open, the simple concept of quasi-steady-state flow might bedropped in favor of unsteady flow equations which include inertia effects. An addi­tional step is to consider the fluid front moving somewhat behind the boundaries ofthe fracture, i.e. to incorporate the description of the fluid film formed near to thetip. With a detailed geometry, it is also possible to consider stress intensity factors orsimilar characteristics far more complicated than the simple K[, i.e. opening modesfar more complex than the simple "mode I opening" . These considerations are ofprimary importance in research where the aim is to discover the governing lawsand/or their combined effect.

In applied engineering, the aims are somewhat different. Here the primary concernis not to neglect one of the main factors that affect the resulting design, whilemicroscopically accurate description of the known subprocesses is de-emphasized.

The admirable numerical arsenal in linear elasticity computations has evolved incivil engineering. Prior to the finite element method, a huge amount of availableinformation on the geometry of the structures was simply discarded for lack of amethod to apply it. The possibility of using this information was a revelation andbecame the engine of development. In hydraulic fracturing, unfortunately enough, thesituation is different. Here we generate the geometry through mathematical modelsconsisting of partly less accurate submodels. Therefore, the accuracy of a modeldepends less on representing the geometry in detail and more on accounting for allthe significant phenomena (such as energy dissipation in the rock.)

In this chapter we give an overview of existing methods to handle the issueof height growth. In doing this, we will point out some conceptual and technicaldifficulties inherent to existing theories.

Fracture height growth268

.----_.-- ... ,----, '---_.,-,--_., .._------ ..__ ."._-----,---

p=koo+k1yY. + Ydkoo = Pcp +pg--­Y. - Yd

Zhpkl =-pg--­

yu-yd

Pcp

P= Pcp+

(t::.hd - 6h. . )

pg 2 - YR

PcpPcp

P = Pcp+

pg (b2 - h; - yw)

CoordinateCenter of crackCenter ofperforationUpper heightgrowthLower heightgrowth

Top ofperforation

Bottom ofperforation

Total height

Perforatedheight

Pressure atcenter ofperforation

Pressure vs,vertical location

After Ref. [5) After Refs. [6-10] Dimensionless(Figure 11.1) (Figure 11.2) (Figure 11.2)

Yw YR Y0 0 0

b» _ hp t::.hd - tlhu Y. + Yd- 2 2 2

e- bz Sh; 1- Y.

e - (hp - b1) tlhd 1+ Yd

b2hp - tlhu + t::.hd

y.2

-b3 =b1 - hp-hp - t:,.hu+ t::.hd

Yd22c hp + Sh; + t:,.hd 2

hp hp Y.- Yd

System ofNotation

Table 11.1 Relation between different notations for the three-layer equilibrium problem

Figure 11.1 is the notation taken largely from Warpinski and Smith [4]. One minordifference is that here we use the letter c for the half-length of the vertical line crack(to be consistent with the other chapters of this book) while those authors use the lettera. Here we use Yw for the vertical coordinate to make sure it is not confused withthe dimensionless coordinate to be introduced later. The thickness (i.e. the height)of the perforated (or target) interval is denoted by hp• Those authors introduce the"geometry factors" b: and b3. In fact the "geometry factor" h3 is not an independentvariable, but rather the difference hp - bz- We have two degrees of freedom, b2and c.

The same problem with a somewhat different system of notation is depicted onFigure 11.2. This is the system used by Ahmed [6] and Economides [10]. Here thekey variables to determine are the upper and lower height growth, Sh; and /)"hd.Also, a dimensionless coordinate system is introduced in the figure. The dimension­less variable y is zero at the center of the crack and unity at the top. The relationsbetween the different notations are given in Table 11.1. In Table 11.2, the dimen­sionless variables are expressed in terms of the key variables for both systems ofnotations.

271Equilibrium fracture height

----- -_ ..-.--_ .... ----------------------1

Figure 11.2 Equilibrium height, system of notation after [6,7]

Yuo

-.---­L>hu...--~ t_

Yd-1

Y

Figure 11.1 Equilibrium height, system of notation after [5]

1c

c___ • __Center off crack

1. 02 and 03 may be different but still higher than OJ,

2. the critical stress intensity factor may be different in the upper and lower layers,and

3. the density of the fluid is accounted for.

The analysis of Warpinski and Smith [5] has been generalized to account for someor all of the above variations by many authors [5-9], but often the results are difficultto compare because of the different systems of notations. Because of the technicalimportance of the results, we try to cover the basic issues using two basic systemsof notations accepted in practice.

.~..

Here we will consider the more complex situation where

11.1.2 Different Systems of Notation

factor even in this case (where 0H,min is not constant any more), we arrive at a usefultechnique of characterizing height containment.

Fracture height growth270

(11.9)(11.3).__ h_.p__ x e pn(y)J_1_+_YdyJr(Yu-Yd) L, 1-y

n10[+1, 4J, kI] = 4 (+24J - kt), and

Jr .fl[+l,ko,kI] = -(+2ko+kl),

4n

fl[ -1, 4J, kI] = -( -2ko - kJ),. 4

The following limits are needed:

There are only two unknown variables (c and b2) if the pressure, p, is given. Anyroot-finding numerical method can be used to obtain the solution.

However, here we are interested in the more realistic case where the density ofthe fluid is not zero. In the following we derive the system of equations for the moregeneral case.

The stress intensity factor for asymmetric loading is given by Eq. 2.55, whichhas to be applied in the vertical direction. Eq. 2.55 is rewritten in terms of thedimensionless variable, Y, as

(11.8)

{l=Y [ (24J - k1)y kly2]/b[y, ko, kI] = V 1+Y x ko - kl + 2 +2

(2ko - k1) (Yv'(I- y)/(l + y»)- arctan .2 -1+ Y

andand

(11.7)

Selecting suitable integration constants, the integration results in

(l+Y [ (2ko+k1)y kly2]ft[Y, ko, kI] = V 1-Y x -ko - kl + 2 +2

(2ko + k1) (Yv'(l + y)/(1 - y»)+ arctan ,2 l+y

..Jii(K{C.2 +K{C,3) ( ). (b2)r;: = (T2 - (Tl arCSIn -

2v~ c

+ (0"3 - (Tl) arcsin (hp ~ b2) - «(T2+ 0"3 - 2p)i (11.1)

(11.6)J ~ -Yf b[Y, 4J, kI] = (4J + klY) --dy.l+y

and

(11.5)The two unknowns can be determined from two equations. The two equations arederived from the conditions of equilibrium. The stress intensity factor at the top(K/,top) calculated from the net pressure distribution should equal the fracture tough­ness of the upper layer (K{C,2), and the stress intensity factor at the bottom (KI.boltom)calculated from the same net pressure distribution should equal the correspondingmaterial property of the lower layer, (K{C.3)'

For the case where the density is considered zero, Warpinski and Smith [5] givethe following very elegant and concise system of equations:

11.1.3 Basic Equations

Relation to kl

Relation to koo

Relation to Y.where the net pressure varies with the dimensionless vertical coordinate y, for tworeasons: The minimum horizontal principal stress is different in the different layersand the hydrostatic pressure increases with depth.

The last entry in Table 11.1 gives the pressure as a function of the vertical coor­dinate. The net pressure will be almost the same function, except the constant termko will be the difference between koo and the minimum principal stress in the givenlayer. In general, the net pressure will be a piecewise linear function (with jumpsat the layer boundaries). Hence, it is advantageous to determine two indeterminateintegrals (primitive functions) for the general case where the net pressure is a linearfunction of the vertical location. The following two functions are defined:

b2-hYt=--Pc

2b2 -hkoo = Pcp +pg--2-P

ckl = -pg-

2

Relation to Yd

K/,lx)ttom =hp + 6hu + l!ihd

YR"" 2 Yzs».

Yu =1 - .,-----,--:--=--:­hp + l!ihu + 6hd

zs»,Yd = -1+ ---=-­

hp + l!ihu + 6hd6hd -l!ihu

koo = Pcp +pg 22hp

k1=-pg--Yu - Yd

Yw "" cy

b2Yu"" -c

Relation to y(11.4)hp 11. ~-y--'--- x Pn(y) --dy,

Jr(Yu-Yd) -1 1+ Y

andTable 11.2 Dimensionless variables in terms of the key variables of the differentsystems of notation

273Equilibrium fracture heightFracture height growth272

._._--- .__ ._-_ ..__ ...----------------------

Once the input data hp, 0"1, 0"2, 0"3, K1C.2, K1C,3, and Pcp are given, Eqs, 11.12 and11.13 contain only two unknowns, namely Yu and Yd. If another set of key variablesis preferred, Table 11.2 can be used to obtain a system containing only the pair Sh;and t:,.hd, or the pair b2 and c. Using a suitable root finding method, the solution canbe easily found.

The above relationships were obtained and used with the algebraic manipulationsystem Mathematica (Wolfram [11]); a program was published in Valko and Econo­mides [12]. In the following we consider a series of examples extracted from pages70 and 71 of Warpinski and Smith [5] and compare our results with the ones giventhere. Once the solution corresponding to the simpler data set (with zero density) isfound satisfactorily, we can proceed to examine the more complex problem involvinghydrostatic pressure effects.

(11.13)and,~

.~.

where all the functions and variables are defined as before.The system of equations to solve is simply

hpK',bottom(Yu, }a) = ( ) X {!b[Yd, koo - 0"3, kl] - !b[-I, koo - 0"3.kl]

7r Yu - Yd

+ h[yu. koo - 0"1, k1] - !b[Yd. koo - 0"1>kl] + !b[+I, koo - 0"2, kl]

- !b[y", koo - 0"2. kl]}, (11.11)

Table 11.3 Input data for Example 11.1 (After Simonson et al. [4])

No (Tl 0) K1C.2 K1C,) Ppc

1 3500 psi 3500 psi 0 0 3350 psi24.13 MPa 24.13 MPa 23.10 MPa

2 3500 psi 3500 psi 1000 psi inl/2 1000 psi- in1/2 3360 psi24.13 MPa 24.13 MPa LOI MPa· ml/2 1.01 MPa· ml/2 23.17 MPa

3 3500 psi 4000 psi 1000 psi in1/2 1000 psi- inl/2 3360 psi24.13 MPa 27.58 MPa 1.01 MPa· ml!2 1.01 MPa· ml/2 23.17 MPa

4 3500 psi 4000 psi 4000 psi- inl/2 1000 psi- inl/2 3360 psi24.13 MPa 27.58 MPa 4.04 MPa· ml/2 1.01 MPa· m 1/2 23.17 MPa

p=o 01 ::: 3000 psi (20.68 MPa) hp = 50 ft (15.24 rn)

Table 11.4 Solution of Eqs. 11.12 and 11.13 for the data sets of Table 11.2

No f)..h. f)..hd Eq. 11.1 Eq. 11.1 Eq. 11.2 Eq.IL2left right left right

(MPa) (MPa) (MPa)· m (MPa) -m

30.07 ft 30.07 ft 0 0 0 09.164 m 9.164 m

2 26.08 ft 26.08 ft 0.4936 0.4936 0 07.950 m 7.950 rn

3 22.75 ft 4.241 ft 0.5686 0.5686 0 06.934 m 1.293 m

4 9.239 ft 4.051 ft 1.568 1.568 -2.921 -2.9212.816 m 1.235 m

(11.12)

where the variables koo and kl are given in Table 11.1 in terms of the two keyvariables, Yu and Yd. The function It is defined by Eq. 11.7, and the two limitvalues needed are given by Eq. 11.9. For the stress intensity factor at the bottom asimilar expression is obtained,

( )X {I,[Yd, koo - 0"3. k1] - I I[ -1. koo - 0"3. k1]

it Yu - Yd+ I,[yu, koo - 0"1. kd - I,[Yd, koo - O"J, kl]+ 11[+1, koo - 0"2,kl]

The system of Eqs. 1L 12 and 11.13 are solved using the program of Valk6 and Econo­mides (12]. The results are presented in Table 11.4. Though not justified by the likelyaccuracy of the input data, we use four significant digits to avoid further misunder­standing. After the solution is obtained, we substitute it into Eqs. 11.1 and 11.2 tocheck consistency.

The solution for the first case agrees with Warpinski and Smith [5] very well, withan upper height growth of 30 ft calculated for the specified pressure. (In Warpinskiand Smith [5] the pressure was calculated from the height growth, but the calculationsshould result in the same consistent data set either way.) Moreover, Eqs. 11.1 and 11.2

(11.10)

Solution

Calculate the upper height growth and lower height growth for the selected datasets given in Table 11.3. (Note that in Warpinski and Smith [5J, the inverse problemis considered to determine the pressure providing exactly 30 ft [9.16 m] of heightmigration.)

Example 11.1 Equilibrium Height Neglecting the Density of the FluidFrom Figure 11.2 it is obvious that the integral in Eq. 11.3 consists of three partscorresponding to the intervals (1) from -1 to Yd, (2) from Yd to Yu, and (3) from Yuto 1. The slope, k1, of the net pressure is identical in these intervals and it is givenin Table 11.1. The constant term, ko varies from interval to interval. For instance, inthe interval from -1 to Yd, it is given by koo - 0"3, where koo is the pressure at themiddl~ o~ the crack (note that it is not the pressure at the middle of the perforation).Substituting the values of ko into Eq. 11.3 provides the expression for the stressintensity factor at the top:

275Equilibrium fracture heightFracture height growth274

._------ ._---------------_._--_ .._-_ ..-

Height map for Example 11.2. The first solution pair of the height migrat!onis shown with sclidIine. The dashed line corresponds to the second solution

Figure 11.3

Treating pressure

I21

hp 50 ft 15.24 m0"1 3000 psi 20.68 MPaO"z 3500 psi 24.13 MPa0"3 4000 psi 27.58 MPaK1C•2 1000 psi inl/2 1.01 MPa· ml/2K1C•3 1000 psi- inl/2 1.01 MPa· mliZp 62.4 lbm/ft" 1000 kg/rrr'

-300

-200

-100The system of Eqs. 11.12 and 11.13 is solved using the program of Valko and Econo­mides [12]. In order to reveal possible multiplicity, different starting points are usedin the solution algorithm. The solution of the system is shown for net pressure 3 MPato illustrate the case of a unique solution. A similar case with 400 psi net pressure

oSolution

100

200

Calculate the upper and lower height growth from the data given in Table 11.5. Investi­gate the range of pressure which results in a unique solution-pair (i.e. upper and lowerheight migration), two possible solution-pairs, and no solution.

Tip location1m]

300L_

V-:...........v I-V -r-....

\/I

.//

Tip location[ft]

1000

800

600

400

200

o-200

-400

.s00

-800

-1000

-12003000 3100 3200 3300 3400 3500 3600 3700 3800 psi

Table 11.5 Input data for Example 11.2

Example 11.2 Equilibrium Height Taking into Account the Density ofthe Fluid

From now on we do not neglect the hydrostatic pressure gradient in the fluid. Becauseof the hydrostatic term, the system of non-linear equations, i.e. Eqs. 11.12 and 11.13,becomes more complicated and the problem of multiple solutions cannot be excluded.The issue is illustrated by a numerical example which is essentially case No. 3of Example 11.1, except that the density of the fracturing fluid is not taken tobe zero.

is provided in field units. Two possible height pair solutions are shown for 5 MPa toillustrate multiplicity, and a similar case with 700 psi net pressure is provided in fieldunits. The results are given in Table 11.6.

The "height map" of the system, i.e. the location of the upper and lower tips vs.the treating pressure, is shown in Figure 11.3. The location of the second solution isindicated in the figure by dashed lines. At lower net pressures the second solution is veryfar from the known solution and might explain why it has been neglected. However,as the treating pressure increases, the two pairs of solutions converge. In this case,there is no justification in neglecting the second solution-pair. There is only one pairof solutions below the net pressure of 3660 psi (25.2 MPa). From 3670 to 3780 psi

11.1.4 TheEffect of HydrostatiCPressure

Firstsolution Second solutionPcp - 0"1

llh. llhd llhu llhd

3 MPa 13.1 m 2.32 m400 psi 29.9 ft 5.9 ft5 MPa 208 m 34.8 m 197 m 238 m700 psi 602 ft 84.3 ft 521 ft 889 ft

Table 11.6 Solution of Eqs, 11.12 and 11.13 for the data setof Table 11.4

are satisfied by our solution perfectly as seen from Table 11.3; here we present bothsides of Eqs. 11.1 and 11.2 as calculated at the upper and lower height growth shownin the same row of the table.

The second case raises some doubts because while Warpinski and Smith [5] calcu­lates both the upper and lower height growth to be 30 ft, our value is only 26 ft.However, because our solution satisfies exactly Eqs. 11.1 and 11.2, we conclude thatthe discrepancy is not conceptual.

The third case shows a larger discrepancy between the numerical results of ourcalculations and those of Simonson et al. [4] where 30 it is calculated for the upperheight growth and 3 ft for the lower. The reason for this discrepancy seems to be thatthe fracture toughness was neglected by them. In this example, neglecting the fracturetoughness has a large impact on the calculated height.

The fourth case is not considered in Warpinski and Smith [5]. It is selected here toillustrate non-identical critical stress intensity factors. 0

277Equilibrium fracture heightFracture height growth276

(lLl8)Kw = T.6.p,The equilibrium height theory in Section 11.1 was presented for a vertical line crack.A natural generalization is to consider a fully three-dimensional fracture and assume

In the original paper by Bui [14], the problem of a plane crack under pressureacting on its faces but without any far field stress is considered. Introduction ofa~(x, y, 0) (the value of the normal compressive stress on the crack surface at timet = 0) into the definition of net pressure by clifton et al. [15] is an attempt to avoidthe use of the concept of varying far field stresses.

Both the crack opening and the pressure are represented as linear combinationsof suitable selected trial functions [13]. To construct the trial functions, the cracksurface is covered by a quadrilateral mesh as shown in Figure 11.4. The i-th trialfunction has the value of unity at the i-th node, varies linearly over adjacent triangles,and vanishes along the opposite side of these triangles. Essentially the same trialfunctions are used for the crack opening and the pressure, except for the specialelements introduced in the near-crack-tip zone for the crack opening. (These specialtrial functions vary as the square root of distance from the tip.) Substituting therepresentation of the pressure and crack opening results in a large (and dense) set oflinear equations:

11.2 Three-dimensional Models

(11.17)R = [(x - x? + (x - x?]1/2.

(11.16)- a - 0-"1;;: -i+-jox' oy"

and the distance R between point (x', y') at which the integrand is evaluated andpoint (x, y) at which the pressure is evaluated is

i.e. E. = E'/(8:rr) with the notation used in this book; V is the gradient operator

(11.15)GEe = ,

4:rr(1 - v)

where A is the fracture surface, Ee is the effective elastic modulus

(11.14)

Clifton [13] refers to Bui [14] when stating that the change in normal stress on thecrack plane is related to the crack opening, w(x, y), by an integral of the form

ts pt»; y) =: p(x, y) - (J"~(x, y, 0) = s,J1V'w· V'(l/R)dA',

11.2.1 Surface Integral Method

In the previous example (and in every other three-layer equilibrium problem), thesecond solution appears because the system is coupled. Downward height migrationprimarily decreases the stress intensity factor, because the net pressure in the lowerlayer is primarily negative and adds a negative term to the integrals in Eqs. 11.3and 11.4. However, hydrostatic pressure has a secondary effect of increasing the netpressure at the bottom. The larger the downward height migration, the larger is thissecondary effect. Here a second solution may appear.

For the sake of simplicity, we consider only constant minimum stress inside agiven layer, but the developed mathematical apparatus can be readily applied to stressprofiles varying linearly in a layer with different coefficients from layer to layer.

The second solution is not stable in the sense that the downward tip "runs away"for any positive perturbation of the downward height migration. (In the case ofnon-zero stress gradients in the formation, the non-stable tip might be the upper tipdepending on the relative magnitude of the hydrostatic pressure gradient and stressgradient.)

It is well known that there is an upper net pressure limit for which an equilibriumheight exists. The important additional messages of Figure 11.3 are that (1) theequilibrium height is not unique above a certain net pressure, which is considerablyless than the "run-away pressure", and (2) there is a maximum equilibrium heightwhich can be created without a run-away.

In the petroleum engineering literature, it has been assumed that a graduallyincreasing pressure establishes a (large enough) equilibrium height. Height growth,in tum, limits the pressure rise. In other words, equilibrium height theory (withincomplete numerical formulas) has suggested that the system is self-regulating. Thefact that there is a limit to the height which can be achieved in a stable mannercreates a conceptual problem; i.e. the self-regulating mechanism is mostly an artifactof neglecting gravity.

For problems with more than three layers, several solution-pairs may be possiblefor a given treating pressure and more than one solution pair may be stable.

The equilibrium fracture height concept can be used to determine a realistic heightfor a constant height model. The general procedure is to assume a constant height,run a simulation, determine a representative treating pressure, obtain the equilibriumheight corresponding to the representative treating pressure and repeat the constantheight simulation with the improved estimate of fracture height. Such a procedure issuggested in Rahim and Holditch [1].

If the height is to be calculated simultaneously with the other two characteristicdimensions (length and width), then three-dimensional models are needed.

that it evolves through equilibrium states. Again, the reverse application of super­position seems to be necessary to use net pressure calculated with varying (J"H.min,i.e. with varying boundary condition at infinity. To our knowledge, all the three­dimensional models of hydraulic fracturing assume that the net pressure (calculatedas the difference between the pressure inside the fracture and the varying (J"H.min) canbe used to determine stresses and displacements as if this net pressure were actinginside the fracture and the far-field stress state were identically zero.

(25.3 to 26 MPa), there exist two solution-pairs.Above 3780 psi (26 MPa), there isno solution. The maximum equilibriumheigh! is approximately 1400 it (430 m). Noequilibriumcan exist beyond this limit.0

279Three-dimensional modelsFracture height growth278

-_ ....,,-- •...__ ..._- --_ ..._-_ .... - ' .._'._- ... ..-.

Clifton [13] presents a series of case studies. Inhis example Case A (Newtonian fluid,no leakoff, high stress barrier) the fracture propagates with essentially constant heightafter the first few seconds, The important result is that the treating pressure startsto increase after a short time and "the three-dimensional predictions of width andpressure agree quite well with the predictions of the two-dimensional PKN model",(It is interesting to note that Barree [16] arrived at a similar conclusion, finding thata three-dimensional model behaves similarly to a KGD model at early times and toa PKN model at late times.)

Simple reasoning suggests that a three-dimensional model based on the frac­ture toughness criterion would give decreasing treating pressures with time, so theincreasing treating pressure might initially seem a bit surprising, Let us assume

11.2.2 The Stress Intensity Factor Paradox

(11.22)ri' ::::in'+l)K' (2+ :,) n' ,

with K' and n' being the "usual power law coefficients". At sufficiently low valuesof shearing rates (jqjlwZ), the power law fluid is replaced by a Newtonian fluid toavoid singular coefficients in Eqs. 11.21 and 11.22_

Equations 11.20 to 11.22 are not used directly but are reformulated using a vari­ational approach. The boundary condition at the moving boundary is derived fromthe requirement that the flow into the near-crack-tip region is balanced by the sumof three phenomena: leakoff rate, spurt loss and local fracture volume expansion.

Advance of the crack occurs in such a way that the stress-intensity factor, K/, iskept nearly equal to the critical stress-intensity factor, K{c, during crack extensionat each node. In principle, the stress intensity factor can be calculated once thenodal crack openings are known for the given time step. For technical reasons, themodels may avoid calculating the stress intensity factor explicitly but make use ofan equivalent equation to fulfill the equilibrium requirement.

The three-dimensional model described above is based on the displacement discon­tinuity method, which is only one of the many available numerical methods in linearelasticity _Any of the methods belonging to thelarge arsenal of numerical mechanicscan be selected as a starting point for a three-dimensional hydraulic fracturing model.Finite elements, discrete elementsand their numerous variations are potential candi­dates [17-19]_ The fluid flow description submodel might also utilize one of thesemethods, In general, the overall behavior of the model should not depend signifi­cantly on the selection of the numerical method. The crack advance criterion and itspractical implementation. should be more influential.

(11.21)ap + I (!1!)n'-l qy = Fa 'I 2 3 P y,Y w w

where the product pF y is the body force per unit volume caused by the weight ofthe fluid. The "viscosity parameter", 'I', is given by

and

281Three-dimensional models

(ll.20)ap + ri' (lil)n'-l qx = 0,ax w2 w3

where qx is the volume flow rate in the x direction per unit length in the y directionand_qy is the volume flow rate in the y direction per unit length in the x direction;q{ IS the volume injection rate per unit fracture area (non-zero only in the near­wellbore elements); the product CL,O x (p - Plr) is the Carter leakoff coefficient(now depending linearly on the difference of the pressure inside the fracture andthe fluid pore pressure in the reservoir, pfr); and rex, y) is the opening time of thesurface element located at the position (x, y). Two pressure gradient equations areincluded:

(11.19)2CLO(p - Plr) aw-r===;:::::::::=';:-'- - - +ai./t - rex, y) at '

where w is the vector of nodal displacements, K is the stiffness matrix, T is thematrix of element areas and !!..p is the vector of nodal net pressures. The derivationof the mat~x coefficients in K and T is presented by Clifton and Abou-Sayed [15]where t~e interested reader may find further details. The main advantage of thesurface mtegral method (or in other words, the boundary integral method) is thatthe stiffness matrix has less rows (and columns) than the similar matrix of the finiteelement method. On the other hand, it is dense while in the finite element methodsit is sparse,

Another large set of equations involving the same variables is obtained from thetwo-dimensional continuity equation:

Figure 11.4 Surface boundary elements used in one of the three-dimensional fracture propagationmodels, after [13]

_--:.">-:".:x

--

lip element

Wellbore elementy

Fracture height growth280

._- ---.-- .-_. --_ ..._----------------------

The pseudo-three-dimensional (P-3D) concept of mode!ing ~y~raulic fract.uring wasintroduced by Settari and Cleary [2]. A P-3D model IS a limited extension of the

Figure 11.5 Pressurized plane crack of elliptical shape. The inner pressure p and the height isconstant, the length is varying

11.3 Pseudo-three-dimensional Models1----- x, --- .....~I

cp is the angle with the x axis (it is zero for the lateral tip stress intensity factorshown in Figure 11.6 and ](/2 for the stress intensity factor at the vertical tip) andE(k) is the complete elliptic integral of the second kind.

(11.24)

where the k parameter depends only on the aspect ratio,

(11.23)

The stress intensity factor at the lateral tip, normalized by pVhfl2 is shownas curve a of Figure 11.6. As seen from the figure, when the aspect ratio (~f / ~f)reaches the value five, the normalized stress intensity factor becomes almost Identicalto the function (2xtIhf) -1/2, also shown in the figur~ as c~rve c. Thus, t?e ch~gein the aspect ratio Significantly decreases the stress intensity factor and. mcreasl~gtreatment pressure is needed to compensate this effect. From the asymptotic behaviorof the stress intensity factor, it follows that in a three-dimensional model of a well­contained hydraulic fracture the pressure increases with the square root of lengthat late times. (Eq. 9.11 shows a somewhat different result obtained from the PKNwidth equation where the pressure increases with the fourth root of l~ngth.) .

While the analytical solution for the stress intensity factor at the tip of an elhp­tical crack seems to explain the "stress intensity factor paradox", it is not quiteclear whether other idealized geometries, e.g. a rectangular fracture with circularpropagating edges, would also result in a decreasing stress intensity factor at thelateral tip. . .

For comparison, the normalized stress intensity factor at the vertical ttp of ~heelliptical crack is also shown in Figure 11.6 (as curve b). At lar~er aspect ratl~sit converges to unity, i.e. to the value anticipated from the vert.ICal plane st~amcondition. (This illustrates that for the equilibrium height calculations, the verticalline crack approximation is reasonable.)

Figure 11.6 Normalized stress intensity factor at the tip of the pressurized plane crack of ellipt~calshape (curve a). For comparison the normalized stress intensity factor at the verticaltip is shown as curve b. Curve c is the function (2xtIhf )-1/2

283Pseudo-three-dimensional models

temporarily that the stress intensity factor at the lateral tip can be estimated as iheproduct of the square root of half-length and some kind of average net pressure.Since for a well contained fracture (e.g. Case A of Clifton [13]) the length and thepressure are increasing simultaneously, it is difficult to see how the stress intensityfactor remains "nearly constant". The apparent contradiction between the behavior ofthe three-dimensional model and the intuitive comprehension of the stress intensityfactor can be resolved in two ways.

One possibility is that the net pressure is decreasing with time in average while thevalue at the wellbore is still increasing. According to our understanding in that casethe pressure distribution in the fracture would become steeper and steeper near thewellbore with increasing time. (However, it is difficult to see a mechanism providinglarger pressure drops near the wellbore with increasing time, knowing that also thewidth is increasing and the flow rate is nearly constant in this region.)

The other possibility is that the stress intensity factor at the lateral tip is verymuch influenced by the fact that the aspect ratio, (2xf /hf) is increasing with time.To understand the phenomenon, we substitute the fracture with an idealized three­dimensional elliptical crack with constant height, constant inner pressure and variablehalf-length. Such a crack is shown in Figure 11.5. The stress intensity factor at theelliptical boundary is given analytically by Kassir and Sih [20] (see page 82 of theirbook). In our notation,

Fracture height growth282

1~<,

0.8-N...........~0.6'-'

.3........~0 0.4~a:

0.2

0a 2 4 6 8 10

Aspect ratio, 2x,/h,

1. Rahim, Z. and Holditch, S.A: Using a Three-dimensional Concept in a Two-dimensionalModel to Predict Accurate Hydraulic Fracture Dimensions, Paper SPE 26926 presentedat the Eastern Regional Conference and Exhibition, Pittsburgh, Nov. 2-4, 1993.

2. Settari, A. and Cleary, M. P.: Development and Testing of a Pseudo-Three-dimensionalModel of Hydraulic Fracture Geometry, SPEPE, (Nov.), 449-466, 1986; Trans.AlME,283,1986.

3. Bouteca, M.J.: Hydraulic Fracturing Model Based on a Three-dimensional Closed Form:Tests and Analysis of Fracture Geometry and Containment, SPEProductionEngineering,(Nov.), 445-454, Trans.AlME, 285, 1988.

References

4. Simonson, E.R., Abou-Sayed, AS. and Clifton, RJ.: Containment of Massive HydraulicFractures, SPEJ, (Feb.), 27-32, 1978.

5. Warpinski, N.R. and Smith, M.B.: Rock Mechanics and Fracture Geometry, in RecentAdvances in Hydraulic Fracturing, J. Gidley et al. (eds.): Monograph Series, SPE,Richardson, TX (1989), SPE, Richardson, TX, 1989.

6. Ahmed, U.: Fracture-Height Predictions and Post-Treatment measurements, in ReservoirStimulation (2nd ed.), Econornides, M.J. and Nolte, K.G. (ed.) Prentice Hall, EnglewoodCliffs, N.J., 1989.

7. Newberry, B.M. Nelson, R.F. and Ahmed, U.: Prediction of Verital Hydraulic FractureMigration Using Compressional and Shear Wave Slowness, Paper SPE 13895, 1986.

8. van Ekelen, H.AM.; Hydraulic Fracture Geometry: Fracture Containment in LayeredFormations, SPEJ, (June), 341-349, 1982.

9. Fung, R.L., Vijayakumar, S. and Cormack, D.E.: Calculation of Vertical FractureContainment in layered Formations, SPEFormationEvaluation, (Dec.), 518-522, 1987.

10. Economides, MJ.: A Practical Companion to Reservoir Stimulation, Elsevier,Amsterdam, 1992.

11. Wolfram, S.: Mathematica:A System for Doing Mathematics by Computer, 2nd ed.Addison-Wesley, Reading, MA 1991.

12. Valko, P. and Economides, M.J.: Fracture Height Containment with Continuum DamageMechanics, Paper SPE 26598 presented at the Annual Technical Meeting and ExhibitionHouston, 1993.

13. Clifton, R.J.: Three-dimensional Fracture propagation Models, in Recent Advances inHydraulic Fracturing, (J. Gidley et al. (eds.j), Monograph Series, SPE, Richardson,TX,1989.

14. Bui, H.D.: An Integral Equations Method for Solving the Problem of Plane Crack ofArbitrary Shape, J.Mech. Phys. Solids, 25, 29-39, 1977.

15. Clifton R.J. and Abou-Sayed AS.: On the Computation of the Three-dimensional Geom­etry of Hydraulic Fractures, Paper SPE 7943, Symp. on Low-Permeability Reservoirs,Denver, 1979.

16. Barree, R.D.: A Practical Numerical Simulator for Three-dimensional Fracture Propaga­tion in Heterogeneous Media, Paper SPE 12273 presented at the Reservoir SimulationSymposium, San Francisco, Nov. 15-18, 1983.

17. Advani, S. H.: Finite Element Model Simulations Associated With Hydraulic Fracturing,SPEJ, (April), 209-218, 1982.

18. Thiercelin, MJ, Ben-Naceur, K and Lernanczyk, Z.R.: Simulation of Three­dimensional Propagation of a Vertical Hydraulic Fracture, Paper SPE 13861 presentedat the Low-Permeability Gas Reservoir Symposium, Denver, May 19-22, 1985.

19. Morita, N., Whitfill, D.L. and Wahl, H.A: Stress Intensity Factor and Cross SectionalShape predictions from 3D Model for Hydraulically induced Fractures, IPT, (Oct.),1329-1342, 1987.

20. Kassir, M.K. and Sih, G.C.: Three-dimensional crack problems, in Mechanics of Frac­ture, Vol. 2, Sih, G.C.(ed.), Noordhoff, Leyden, 1975.

21. Nordgren, R. P.: Propagation of a Vertical Hydraulic Fracture, SPEJ, (Aug.), 306-314,1972;Trans.AlME, 253.

22. Setrari, A: Quantitative Analysis. of Factors Influencing Vertical and Lateral FractureGrowth, SPE ProductionEngineering, (Aug.), 310-322, 1988.

23. Palmer, J.D. and Caroll, H.B. Jr.: Numerical Solution for Height of Elongated HydraulicFractures, Paper SPE 11627 presented at the Low Permeability Gas Reservoirs Sympo­sium, Denver, March 13-16, 1983.

differential two-dimensional model (in most cases that of Nordgren [21]), allowingfor variable height. The height as a function of the lateral position is calculated inevery time step. Height growth is controlled by the pressure which is often consid­ered to be a function of the lateral position only. For the determination of height,every vertical cross section is considered separately; the resultant height-growth rates(Settari [221) or the resultant height-growth values (Palmer and Caroll [23]) are thenused in the continuity equation.

The concept was used by Nolte [24] and Palmer and Luiskutty (25] and subse­quently by many other authors. In principle, the methods of Section 11.1 can beapplied at any lateral cross section to obtain the equilibrium height from the actualpressure. The equilibrium height (if it exists) is considered as an upper boundand some models attempt to limit the height growth by other factors, e.g. fluidflow [26-28]. If the rate of height growth is limited by the energy dissipationdue to creating a damaged zone, a continuum damage mechanics approach mightbe useful [12]. A detailed discussion of the P-3D or 3D computer models actuallyused in the petroleum industry is beyond the scope of this book as these computerprograms themselves are sophisticated systems with numerous additional algorithmicrules. These rules reflect the aggregate experience of modelers involved in everydayfield jobs for many years and are extremely useful in bridging the gap between theoryand practice.

In the past, a certain hierarchy of models was accepted. Two-dimensionalmodels have been considered as the routine vehicle for everyday design andanalysis of fracturing jobs. Pseudo-three-dimensional models have been used asimproved engineering tools needing more accurate input data and somewhat increasedcomputational time. Fully three-dimensional models have been thought of mainly ascomputationally demanding research tools which need even more input informationbut provide a deeper understanding of fracture propagation. With the developmentof computer hardware and software, this strict hierarchy of models is becomingobsolete and the emphasis is shifted toward understanding the coupling of varioussubprocesses. The inclusion of previously overlooked (but lately proven effects) mayimprove the prediction of fracture geometry and propagation even more significantlythan the attempt to incorporate the latest numerical methods found promising in otherfields of mechanics.

285ReferencesFracture height growth284

Warpinski et al. (1] is a summary of a comparative study of hydraulic fracturingmodels using test data from the GRI Staged Field Experiment No.3. More detailsof the SFE3 experiments are given in Refs. 2-5.

Models compared in the study include 2D, pseudo-3D, and 3D codes, run on up toeight different cases. The purpose of the study was "to provide production engineerswith a practical comparison of the available models so that rational decisions can bemade as to which model is optimal for a given application".

Table Al shows the relevant rock and reservoir information for the comparativestudy. Three different physical configurations were considered: single-layer (2D)case, three-layer (3D) case and five-layer (3D) case. Table A2 gives the character­istics of a treatment. The participants were asked to model the treatment. The tablesare reproduced from Warpinski et al [1]. We added the symbols as used in this bookwhenever they were available.

Each participant could model a total of eight cases. These were GDK (KGD),PKN, three-layer and five-layer cases, with separate runs for a constant Newtonianviscosity and a constant n' and K' power-law fluid. The PKN and GDK cases wererun with a constant height (2D) set at 51.8 m (170 ft). The three- and five-layer caseswere run with a 3D or pseudo-3D model, allowing fracture height be determined bythe model.

The results provided by the modelers are summarized in Tables A3-ASTables A6-A8 show those names which were used to identify the computer codes(model runs) by Warpinski et al. [1]. An interesting discussion of the study waspublished in the same issue of the journal (Cleary [6]).

APPENDIX: COMPARISONSTUDY OF HYDRAULICFRACTURING MODELS:INPUT DATA AND RESULTS

24. Nolte, K.G.: Principlesof FractureDesignBased on PressureAnalysis SPEPE (F b)22-30, 1988. ' ,e . ,

25. Palmer, LD. a~d Luiskutty,C.T.: A Model of the Hydraulic Fracturing Process forElongated Vertical Fractures and Comparisons of Results with Other Models, PaperSPE 13864,presentedat the Low Permeability Gas Reservoirs Symposium, DenverMay, 1985.· ,

26. Moral~s,R'!1' and Abu-Sayed,A.S.: MicrocomputerAnalysis of Hydraulic FractureBehavior with a Pseudo-Three-dimensionalModel,SPEPE, (Feb.), 198-205, 1989.

27. ~eyer, ~.R, Cooper,G.D. and Nelson, S.G.: Real-Time 3-D Hydraulic FracturingSlmul~llon;Theory and Field CaseStudies, Paper SPE 20704 presented at the AnnualTechnicalConf. and Exhibition,NewOrleans, 1990.

28.. Weng,X.: Incorporationof 2D FluidFlow into a Pseudo-3DHydraulicFracturingSimu­lator, Pap~r SPE 21849 presented at the Rocky Mountain Regional meeting and LowPermeabilityReservoirSymposium,Denver, Co, April 15-17, 1991.

286 Fracture height growth

_._--_._----_._._._----------------

• No symbol is used for this variable ill this book

61.873.69386.474.388.379897976.98685.276.57881.7

0.60.280.750.820.360.970.460.810.470.380.840.7330.4150.4560.504

0.850.420.821.040.51.240.641.030.590.531.070.9330.6220.580.641

0.7380.817

0.850.540.981.040.641.240.811.030.750.681.070.933

18801986

6l.8.1067.51624117

1397161177489

17541474

97

170170204170170170170170170170170170170170170

254246292516209841181808339521423347404620312304365633963155

202122232425262728293031323334

85.572.39383.172.285.476.68475867781.97373.8'85.982.574.47575

0.60.360.320.770.60.370.3870.434

0.6050.2890.730.620.320.740.40.610.37

0.430.980.7670.5540.4920.553

0.77

0.8490.3970.760.790.430.940.530.780.46

0.6270.704

0.8480.5020.910.790.550.940.680.780.590.740.500.770.630.540.98

15951684

6210941685

701188

97147453

137771

92581.9

1380118282

170170204170170170170170170200200170170170170170170170170

2542485525842659450722883803272440392480415713472029459522122716398638663556

2345678910111213141516171819

2000~OOO200020002000

0.50.06

5010000

None2Vi

•Assumed unit, no! given in Warpillski et al. [1]

246p, 3600s, 0.0Entire fracture heightCL 0.00025f.1. 200

K', Ibf·so.5IW (*)Injection rate, bpmFluid Volume, bbiProppant

n'

Bottomhole temperature, 'FReservoir pressure, psiSpurt lossFluid leakoff heightFluid leakoff coefficient, ftlmin 1/2

Viscosiry - case A, cpRheology - case B

6.58.55.47.94.0

0.300.210.260.200.30

Table A2 Treatment data

180 7150170 570040 735075 5800

195 8200

91709340938094559650

89909170934093809455

12345

Five-layer (3D) case

200020002000

200 cp

6.58.55.5

0.300.210.29'

180 7150170 5700310 7350

917093409650

899091709340

123

Three-layer (3D) case

(*)pn,w8.50.215700170934091701

2000

Efficiency(%)average

fracturewidth(in)

OverallAveragewidth atweUbore(in)

Maximum width(in)

Pressure(psi)

Height(ft)

Length(ft)

Single layer (2D) case

Evmin max

Fracturetoughness(psi/in 1/2)

289Appendix

Table A.3 2D-Results at end of pump

---~-------- ------------~---------------------- -----

young'smodulus(lQ6 psi)

poisson'sratio

III situstress(psi)

zonethickness

(ft)

Inter­val

Table Al Rock and reservoir dataDepth(ft)

Appendix288

290 AppendixAppendix 291

TableA.4 Three-Layer results at end of pump TabJeA,S Five-layer results at end of pump, Symbols given are notations used in this bookLength Height Pressure Maximum width Average Overall Efficiency Length Height Pressure Maximum width Average Overall Efficiency(ft) (ft) (psi) (in) width at average (%) (ft) (ft) (psi) (in) width at average (%)

wellbore fracture wellbore fracture(in) width (in) width

(in) (in)xf hj P".l\· WI1'.O (*) iii

xf hf P.,k' W"".O (0) w

200 cp200 cp

3408 318 1009 0,65 0.35 0.3 77 1 2905 394 960 0.72 0.42 0.31 80.12 3750 903 283 0.56 0.32 0.25 66 2 3709 361 852 0.63 0.38 0.25 663 1744 544 1227 0.9 0.54 0.36 80 3 1754 501 1119 0.83 0.6 OA 824 l360 442 1387 L04 0.68 0.64 96 4 1224 476 1250 1.03 0.7 0.65 975 3549 291 987 0.58 0.35 0.29 70.3 5 2962 328 669 0.5 0.36 0.28 70.56 2697 360 1109 0.72 0.41 0.34 74.3 6 2407 327 768 0.6 0.46 0.35 74.87 3598 306 992 0.57 0.31 0.25 67 7 3399 394 944 0.64 0.36 0.24 688 1938 435 1132 0.72 68 8 2011 428 1008 0.68 699 2089 357 1113 0.66 0.33 0.25 43 9 1594 438 1129 0.81 0.45 0.36 58.1«.s: «.s:

10 3259 371 1093 0.75 0.38 0.31 77.6 10 2647 430 1035.5 0.82 0.46 0.31 81.811 3289 329 1005 0.67 0,35 0.26 68 11 2765 388 935 0.71 0.42 0.25 012 902 596 1428 1.1 0.74 0.49 62 12 1042 600 1358 1.18 0.9 0.6 8713 1326 442 1433 L08 0.71 0.66 96 13 1156 476 1262 1.04 0.71 0.66 9314 2915 337 1094 0.69 0.4 0.32 72.7 14 2535 330 766 0.6 0.46 0.37 73.715 2120 413 1212 0.86 0.48 0.4 76.9 15 1980 349 891 0.75 0.57 0.42 77.816 3235 353 1083 0.65 0.33 0.26 69 16 2926 405 968 0.7 7017 2424 435 1171 0.74 0.34 0.21 47 17 3124 449 1160 0.74 6218 1125 602 1270 1.11 76"No symbol is used for this variable in this book

19 2636 391 934 0.49 6220 1870 458 1151 0.85 0.47 0.34 64

"No symbol for this variable is used in this book

--'_"-"'-- ,-- ---_.

293Appendix

SAHNSIRESMarathonMeyer-lMeyer-2Arco-StimplanTexaeo-FPAdvaniSAHNSIRESMarathonMeyer-lMeyer-2Arco-StimplanArco- TerrafracTexaco-FPTexaco-FPNOTIPAdvani

1234567891011121314151617181920

Model name asgiven in Warpinski et al. [I]

Table A.S Model names corre­sponding to entries inTable A.5

11 NSI12 RES13 Marathon14 Meyer-l15 Meyer-216 Arco-Stimplan17 Advani

SAHNSIRESMarathonMeyer-IMeyer-2Areo-StimplanTexaco-FPAdvaniSAH

12345678910

Model name asgiven in Warpinski et al. [1]

Table A.7 Model names corre­sponding to entries inTable A.4

• S. A. Holditch & Assocs.Inc.

22 Marathon23 Meyer-l (GDK)24 Meyer-l (PKN)25 Meyer-2 (GDK)26 Meyer-Z (PKN)27 Shell (GDK)28 Shell (PKN)29 Advani30 Halliburton31 Conoco (GDK)32 Conoeo (PKN)33 ENERFRAC-l34 ENERFRAC-2

SAH· (GDK)SAH (PKN)MarathonMeyer-l (GDK)Meyer-l (PKN)Meyer-2 (GDK)Meyer-2 (PKN)Shell (GDK)Shell (PKN)Texaco-FP (GDK)Texaeo-FP (PKN)Chevron (GDK)Chevron (PKN)AdvaniHalliburtonConoco (GDK)Conoeo (PKN)ENERFRAC-lENERFRAC-2SAH (GDK)SAH (PKN)

123456789101112131415161718192021

Model name asgiven in Warpinski et a/. [I]

Table A6 Model names corre­sponding to entries inTable A.3

Appendix292

failure 237far-field stress 55, 279·filter-cake 169

resistance 184, 21i, 231finite conductivity fracture 5, 232finite elements method 49, 242

damageaccumulation 248residual 11zone 247

Darcy slaw 2Deborah number 136deviated well 90differential strain curve analysis (DSCA) 77dilatancy 245dilatant fluid 99displacement 22dissipation 115drag

coefficient 139reduction 134

drilling direction 71

effective stress 55, 66, 71horizontal 73

elastic energy 47elastic region 26equilibrium height 275equivalent Newtonian viscosity 110, 119,224

constant height model 267contained hydraulic fracture 267continuum damage mechanics (CDM) 240,

247, 258convection 145Couette flow 97crack layer theory 240crack opening 279critical depth 58crosslinker 13

Carterequation I 169, 180fluid loss model 183leakoff 210, 212, 263equation II 172IImaterial balance 199IIsolution 197

CDM versionof the Nordgren-Kemp model (CDM-NK)

250of the Perkins-Kern-Nordgren model

(CDM-PKN) 252,260characteristic dimension 216characteristic length 105circular crack 45, 47, 86energy of a 48

closure pressure 13, 71, 73, 76, 79, 216, 263closure quality 71Colebrook -White equation 131combined parameter 249, 252

dimensionless 258comparative study of hydraulic fracturing

models 287compressive strength 237conductivity/porosity factor 71confining stress 246consistency index 99

bilinear flow 5Bingham plastic model 99BNS equation 133boundary element method 49boundary integral method 280breakdown pressure 65, 68brittle material 237brittle solid 242

after-growth 228, 230, 258time of 260

anelastic strain recovery (ASR) 15,77average width 166

INDEX

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1. Warpinski N.R., Moschovidis, Z.A., Parker C.D. and Abou-Sayed, 1.S.: ComparisonStudy of Hydraulic Fracturing Models: Test Case - GRI-Staged Field Experiment No.3, SPE Production & Facilities, 9 (1), 7-16, 1994.

2. Holditch, SA. et al.: The GRI Staged Field Experiment, SPE Production Engineering,(Sept.), 519, 1988.

3. Robinson, B.M. Holditch, S.A. and Peterson, R.E.: The GRI's Second Staged FieldExperiment: A Study of Hydraulic Fracturing, Paper SPE 21495 presented at the GasTechnology Symposium, Houston, Jan. 22-24, 1991.

4. Robinson, B.M. et al.: Hydraulic Fracturing Reserarch in East Texas: The GRI StagedField Experiment, Journal of Petroleum Technology, (Jan.), 78, 1992.

5. Saunders, B.F. et al.: Hydraulic Fracturing Reserarch in the Frontier Formation throughthe GRf's Fourth Staged Field Experiment, Paper SPE 24845 presented at the AnnualTechnology Conference and Exhibition, Washington, Oct. 4-7, 1992.

6. Cleary, M.P. Discussion of Comparison Study of Hydraulic Fracturing Models: TestCase-GRF-Staged Field Experiment No.3, SPE Production & Facilities, 9 (1), 17-18,1994.

References

294 Appendix

screenout 93shape factor 85, 190, 192shear

rate 97nominal Newtonian 109

stress 20, 54, 97skin effect 4, 15slip

apparent 158coefficient 157correction 157velocity 157

smooth closing 38Sneddon crack 87specific

surface energy 239volume expansion ratio 148

spurt loss 169, 171, 256coefficient 217

negative 221steady-state 3Stefan'S boundary condition 182step-rate test 76Stokes' law 139strain 21

energy 47, 238relaxation 78

stressabsolute 56absolute vertical 56anisotropy 72compressive 20

stress (continuetf)contrast 14distribution 35, 42intensity factor 42, 43, 239, 240, 269, 281

Rabinowitsch-Mooney equation 117radial width equation 196radially propagating fracture 267real-gas pseudopressure 2relative roughness 131reservoir pressure 55retardation 237retarded tip propagation 226, 254Reynolds number 104, 108

generalized 133particle 139wall 135, 152

rheologicalconstitutive equation 98curve 98, 103, 110, 116, 118

rotating cup viscometer 161run-away pressure 278

Index 297

perforated interval 267perforation phasing 61Perkins-Kern

model 83no leakoff 191

width equation 190, 192permeability 71

proppant-pack 11fracture 8reservoir 15, 231

PKNgeometry 31, 83, 88, 214model 202, 267width equation 204

plane strain 30horizontal 83, 85, 193modulus 263vertical 83,193

plane stress 27plastic behavior 99plastic region 26plug with uniform velocity 106, 113Poiseuille flow 97poisson ratio 24, 44, 55, 75pore volume 2poroelastic constant 55, 75porosity 2power law

assumption 173, 178generalized 133length growth 201model 99width growth 180

pressure decline analysis 212, 215, 260Nolte's 212

pressure transient testing 4principal stress 21, 54superposition 44process zone 246, 247propagation

criterion 242pressure 218, 220, 227rate 208, 242velocity 235

of longitudinal wave 25proppant 13

carrying capacity 138materials 8settling 138transport 12

pseudo-three-dimensional model 268pseudoplastic behavior 99pseudoradial flow 6pseudosteady-state 3

net pressure 48fracturing 76

net present value (NPV) 7, 14Nolte bounds 200non-Newtonian behavior 202non-wetted zone 243Nordgren equation 206Nordgren-Kemp model 183, 209numerical material balance 179

Oldroyd-Jastrzebski plot 160opening time 181distribution factor 171

overburdenpressure 56stress, absolute 55

matrix hydraulic diffusivity, dimensionless 186matrix stimulation 4maximum drag reduction asymptote (MDRA)

134, 136, 152maximum velocity 104, 107maximum width 34

at the wellbore 85minifrac test 217minimum horizontal stress 58, 89, 92, 269Mooney plot 160moving boundary 245

Kachanov law 247Kachanov parameter 248KGDgeometry 31, 85, 88, 214model 203, 267

Khristianovich and Zheltov model 83kinetic energy correction factor 104, 137, 153

leakoff 11,76volume 165coefficient 258history of 187

LEFM 243line crack 32, 238

energy of a 48two-wing 47vertical 271

linear elastic fracture mechanics(LEFM) 240

load-carrying capability 240longitudinal fracture 90

injected volume 166injection test 79~nstantaneous shut-in pressure 74, 219mtaleyer stress contrast 79

g-plot 230, 257Geertsma-de KIerk (GDK).width equation 193model, no leakoff 194

general fluid 117, 132geometry factor 271Griffith crack 32Griffith stability criterion 239

Hagen-Poiseuille law 103 115height map 277 'high net pressure 253Hook's law 214horizontal well 90breakdown pressure 69

hydrostatic pressure gradient 276

induced stress 59infinite conductivity fracture 5, 72

fissure total compressibility 186flowbehavior index 99curve 103, 109, 110, 115, 116, 118laminar 97turbulent 131in annulus 122in elliptic cross section 123in fracture 127in limiting elliptic cross section 124

flow-back test 76fluidefficiency 166lag 245leakoff 169loss parameters 215

foam 13quality 13, 147

formal material balance 178frack & pack 8, 184fracturecoalescence 67compliance 214conductivity 15, 185

dimensionless 5, 186half length 167height migration 92network area, dimensionless 186surface 166toughness 239, 249, 269, i81apparent 246

fracturing fluid 11friction factor 134, 153Fanning 104, 133Weisbach 104

296 Index

Tvshaped fracture 94tensile failure 61tensile strength 65, 237terminal settling velocity 140three-dimensional model 267tip propagation velocity 249tip retardation 7, 247tip screen-out 8tortuousity 89, 227transient leak-off 185transverse fracture 90treatment pressure 235

uniaxial test 25unwetted zone at the tip 165

velocity profile 132virial equation of state 153viscosity 97apparent 99apparent wall 110wall 119

volume equalizedconstitutive equation 149power law 151, 155quantity 148shear rate 149yield-power Jaw 149Bingham plastic 151

yield stress 99Young's modulus 23, 44, 55, 75

zero absolute pressure at the tip 195zero net pressure at the tip 209zipper crack 38, 40equation 38

turbulent flowin ellipsoid cross section 137

nominal 248normalized 283

in-situ 63maximum principal 43minimum principal 43net section 248nominal 248normal 20singularity 35tectonic 56tensile 20wall 117

stress-ahead function 248

298 Index

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