elgoajiroblanco@hotmail.com © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com MICHAEL J ECONOMIDES is Professor of Chemical Engineering at the University of Houston Until the summer of 1998, he was the Samuel R Noble Professor of Petroleum Engineering at Texas A & M University and served as Chief Scientist of the Global Petroleum Research Institute (GPRI).Prior to joining the faculty at Texas A & M University, Professor Economides was the Director of the Institute of Drilling and Production at the Institute of Drilling and Production at the Leoben Mining Institute in Austria (1989-1993) From 1984 to 1989, he worked in a variety of senior positions with the Schlumberger companies, including Europe Region Reservoir Engineering and Stimulation Manager and Senior Staff Engineer, North America Publications include authoring or coauthoring of textbooks and more that 150 journal papers and articles ranges of industrial consulting, including major retainers by national oil companies at the country level and by Fortune 500 companies He is the founder and a major shareholder in OTEK (Australia), a petroleum service and consulting firm with offices in five Australian cities In addition to his technical interests, he has written extensively in wide circulation media on a broad range of topics associated with energy and geopolitical issues KENNETH G NOLTE has held various senior technical and marketing positions with Schlumberger since 1986 From 1984 to 1986, he was with Nolte-Smith, Inc (now NSI Technologies, Inc.) Prior to 1984, Dr Nolte was a research associate with Amoco Production Company, where he worked for 16 years in the areas of offshore/arctic technology and hydraulic fracturing Dr Nolte holds a BS degree from the University of Illinois and received and MS and PhD from Brown University He has authored numerous journal publications and has authored numerous journal publications and has various patents relating to material behavior, drilling, offshore technology and fracturing Dr Nolte was 1986-1987 SPE Distinguished Lecturer and received the Lester C Uren Award in 1992 © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com © Schlumberger 2000 Schlumberger Dowell 300 Schlumberger Drive Sugar Land, Texas 77478 Technical editor: Elsa Kapitan-White Graphic design and production: Martha Dutton Production manager: Robert Thrasher Published by John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex PO19 lUD, England National International 01243 779777 (+44) 1243 779777 e-mail (for orders and customer service enquires): es-books@wiley.co.uk Visit our Home Page on http://www.wiley.co.uk or Visit our Home Page on http://www.wiley.com All right reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a license issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London UK WIP)HE; without the permission in writing of the publisher Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA Wiley-VCH Verlag GmbH, Pappelalee 3, D-69469 Weinheim, Germany Jacaranda Wiley Ldt, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, Clementi Loop #02-01, Jin Xing Distripark, Singapore 0512 John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W 1L1, Canada British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 471 49192 Produced from camera-ready copy supplied by Schlumberger Dowell Printed and bound in Great Britain by Bookcraft (Bath) Ltd This book is printed on acid-free paper responsibly manufactured from sustainable forestry, in which at least two trees are planted for each one used for paper production © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com ◆ Reservoir Stimulation Contents Preface: Hydraulic Fracturing, A Technology for All Time Ahmed S Abou-Sayed P-1 Chapter Reservoir Stimulation in Petroleum Production Michael J Economides and Curtis Boney 1-1 Introduction 1-1.1 Petroleum production 1-1.2 Units 1-2 Inflow performance 1-2.1 IPR for steady state 1-2.2 IPR for pseudosteady state 1-2.3 IPR for transient (or infinite-acting) flow 1-2.4 Horizontal well production 1-2.5 Permeability anisotropy 1-3 Alterations in the near-wellbore zone 1-3.1 Skin analysis 1-3.2 Components of the skin effect 1-3.3 Skin effect caused by partial completion and slant 1-3.4 Perforation skin effect 1-3.5 Hydraulic fracturing in production engineering 1-4 Tubing performance and NODAL* analysis 1-5 Decision process for well stimulation 1-5.1 Stimulation economics 1-5.2 Physical limits to stimulation treatments 1-6 Reservoir engineering considerations for optimal production enhancement strategies 1-6.1 Geometry of the well drainage volume 1-6.2 Well drainage volume characterizations and production optimization strategies 1-7 Stimulation execution 1-7.1 Matrix stimulation 1-7.2 Hydraulic fracturing 1-1 1-1 1-3 1-3 1-4 1-5 1-5 1-6 1-10 1-11 1-11 1-12 1-12 1-13 1-16 1-18 1-20 1-21 1-22 1-22 1-23 1-24 1-28 1-28 1-18 Chapter Formation Characterization: Well and Reservoir Testing Christine A Ehlig-Economides and Michael J Economides 2-1 Evolution of a technology 2-1.1 Horner semilogarithmic analysis 2-1.2 Log-log plot 2-2 Pressure derivative in well test diagnosis 2-3 Parameter estimation from pressure transient data 2-3.1 Radial flow 2-3.2 Linear flow 2-1 2-1 2-2 2-3 2-7 2-7 2-9 Reservoir Stimulation © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" v elgoajiroblanco@hotmail.com 2-4 2-5 2-6 2-7 2-8 2-3.3 Spherical flow 2-3.4 Dual porosity 2-3.5 Wellbore storage and pseudosteady state Test interpretation methodology Analysis with measurement of layer rate Layered reservoir testing 2-6.1 Selective inflow performance analysis 2-6.2 Analysis of multilayer transient test data Testing multilateral and multibranch wells Permeability determination from a fracture injection test 2-8.1 Pressure decline analysis with the Carter leakoff model 2-8.2 Filter-cake plus reservoir pressure drop leakoff model (according to Mayerhofer et al., 1993) 2-10 2-11 2-11 2-12 2-14 2-15 2-15 2-16 2-16 2-17 2-17 2-21 Chapter Formation Characterization: Rock Mechanics M C Thiercelin and J.-C Roegiers 3-1 Introduction Sidebar 3A Mechanics of hydraulic fracturing 3-2 Basic concepts 3-2.1 Stresses 3-2.2 Strains Sidebar 3B Mohr circle 3-3 Rock behavior 3-3.1 Linear elasticity Sidebar 3C Elastic constants 3-3.2 Influence of pore pressure 3-3.3 Fracture mechanics 3-3.4 Nonelastic deformation 3-3.5 Failure 3-4 Rock mechanical property measurement 3-4.1 Importance of rock properties in stimulation 3-4.2 Laboratory testing 3-4.3 Stress-strain curve 3-4.4 Elastic parameters 3-4.5 Rock strength, yield criterion and failure envelope 3-4.6 Fracture toughness Sidebar 3D Fracture toughness testing 3-5 State of stress in the earth 3-5.1 Rock at rest 3-5.2 Tectonic strains 3-5.3 Rock at failure 3-5.4 Influence of pore pressure 3-5.5 Influence of temperature 3-5.6 Principal stress direction 3-5.7 Stress around the wellbore 3-5.8 Stress change from hydraulic fracturing 3-6 In-situ stress management 3-6.1 Importance of stress measurement in stimulation 3-6.2 Micro-hydraulic fracturing techniques vi 3-1 3-2 3-4 3-4 3-5 3-5 3-6 3-6 3-8 3-8 3-9 3-11 3-11 3-12 3-12 3-13 3-14 3-15 3-19 3-19 3-20 3-21 3-22 3-23 3-23 3-25 3-26 3-26 3-26 3-27 3-28 3-28 3-28 Contents © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com 3-6.3 3-6.4 Fracture calibration techniques 3-34 Laboratory techniques 3-34 Chapter Formation Characterization: Well Logs Jean Desroches and Tom Bratton 4-1 Introduction 4-2 Depth 4-3 Temperature 4-4 Properties related to the diffusion of fluids 4-4.1 Porosity 4-4.2 Lithology and saturation 4-4.3 Permeability Sidebar 4A Permeability-porosity correlations 4-4.4 Pore pressure 4-4.5 Skin effect and damage radius 4-4.6 Composition of fluids 4-5 Properties related to the deformation and fracturing of rock 4-5.1 Mechanical properties 4-5.2 Stresses 4-6 Zoning 4-1 4-2 4-2 4-3 4-3 4-5 4-6 4-8 4-10 4-11 4-12 4-13 4-13 4-15 4-24 Chapter Basics of Hydraulic Fracturing M B Smith and J W Shlyapobersky 5-1 Introduction 5-1.1 What is fracturing? 5-1.2 Why fracture? 5-1.3 Design considerations and primary variables Sidebar 5A Design goals and variables 5-1.4 Variable interaction 5-2 In-situ stress 5-3 Reservoir engineering 5-3.1 Design goals Sidebar 5B Highway analogy for dimensionless fracture conductivity 5-3.2 Complicating factors 5-3.3 Reservoir effects on fluid loss 5-4 Rock and fluid mechanics 5-4.1 Material balance 5-4.2 Fracture height 5-4.3 Fracture width 5-4.4 Fluid mechanics and fluid flow 5-4.5 Fracture mechanics and fracture tip effects 5-4.6 Fluid loss 5-4.7 Variable sensitivities and interactions 5-5 Treatment pump scheduling 5-5.1 Fluid and proppant selection 5-5.2 Pad volume 5-5.3 Proppant transport 5-5.4 Proppant admittance 5-5.5 Fracture models 5-1 5-1 5-4 5-6 5-7 5-9 5-9 5-10 5-11 5-11 5-12 5-13 5-13 5-13 5-14 5-15 5-15 5-16 5-17 5-18 5-20 5-21 5-21 5-23 5-24 5-25 Reservoir Stimulation vii © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com 5-6 Economics and operational considerations 5-26 5-6.1 Economics 5-26 5-6.2 Operations 5-27 Appendix: Evolution of hydraulic fracturing design and evaluation K G Nolte A5-1 Chapter Mechanics of Hydraulic Fracturing Mark G Mack and Norman R Warpinski 6-1 Introduction 6-2 History of early hydraulic fracture modeling 6-2.1 Basic fracture modeling 6-2.2 Hydraulic fracture modeling Sidebar 6A Approximation to the Carter equation for leakoff Sidebar 6B Approximations to Nordgren’s equations Sidebar 6C Radial fracture geometry models 6-3 Three-dimensional and pseudo-three-dimensional models Sidebar 6D Field determination of fracture geometry 6-3.1 Planar three-dimensional models Sidebar 6E Lateral coupling in pseudo-three-dimensional models Sidebar 6F Momentum conservation equation for hydraulic fracturing Sidebar 6G Momentum balance and constitutive equation for non-Newtonian fluids 6-3.2 Cell-based pseudo-three-dimensional models Sidebar 6H Stretching coordinate system and stability analysis 6-3.3 Lumped pseudo-three-dimensional models 6-4 Leakoff 6-4.1 Filter cake 6-4.2 Filtrate zone 6-4.3 Reservoir zone 6-4.4 Combined mechanisms 6-4.5 General model of leakoff 6-4.6 Other effects 6-5 Proppant placement 6-5.1 Effect of proppant on fracturing fluid rheology 6-5.2 Convection 6-5.3 Proppant transport 6-6 Heat transfer models 6-6.1 Historical heat transfer models 6-6.2 Improved heat transfer models 6-7 Fracture tip effects Sidebar 6I Efficient heat transfer algorithm Sidebar 6J Verification of efficient thermal calculations 6-7.1 Linear elastic fracture mechanics Sidebar 6K Crack tip stresses and the Rice equation 6-7.2 Extensions to LEFM 6-7.3 Field calibration 6-8 Tortuosity and other near-well effects 6-8.1 Fracture geometry around a wellbore 6-8.2 Perforation and deviation effects viii 6-1 6-2 6-2 6-3 6-6 6-6 6-8 6-8 6-10 6-11 6-12 6-13 6-14 6-16 6-20 6-23 6-25 6-25 6-26 6-26 6-26 6-27 6-27 6-28 6-28 6-28 6-29 6-29 6-30 6-30 6-30 6-31 6-32 6-32 6-33 6-34 6-35 6-36 6-36 6-36 Contents © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com 6-9 6-10 6-11 6-12 6-8.3 Perforation friction 6-8.4 Tortuosity 6-8.5 Phasing misalignment Acid fracturing 6-9.1 Historical acid fracturing models 6-9.2 Reaction stoichiometry 6-9.3 Acid fracture conductivity 6-9.4 Energy balance during acid fracturing 6-9.5 Reaction kinetics 6-9.6 Mass transfer 6-9.7 Acid reaction model 6-9.8 Acid fracturing: fracture geometry model Multilayer fracturing Pump schedule generation Sidebar 6L Approximate proppant schedules Pressure history matching Sidebar 6M Theory and method of pressure inversion 6-37 6-37 6-38 6-40 6-40 6-40 6-41 6-42 6-42 6-42 6-43 6-43 6-44 6-46 6-47 6-48 6-48 Chapter Fracturing Fluid Chemistry and Proppants Janet Gulbis and Richard M Hodge 7-1 Introduction 7-2 Water-base fluids 7-3 Oil-base fluids 7-4 Acid-based fluids 7-4.1 Materials and techniques for acid fluid-loss control 7-4.2 Materials and techniques for acid reaction-rate control 7-5 Multiphase fluids 7-5.1 Foams 7-5.2 Emulsions 7-6 Additives 7-6.1 Crosslinkers Sidebar 7A Ensuring optimum crosslinker performance 7-6.2 Breakers Sidebar 7B Breaker selection 7-6.3 Fluid-loss additives 7-6.4 Bactericides 7-6.5 Stabilizers 7-6.6 Surfactants 7-6.7 Clay stabilizers 7-7 Proppants 7-7.1 Physical properties of proppants 7-7.2 Classes of proppants Sidebar 7C Minimizing the effects of resin-coated proppants 7-8 Execution 7-8.1 Mixing 7-8.2 Quality assurance Acknowledgments 7-1 7-1 7-6 7-7 7-7 7-8 7-8 7-9 7-9 7-10 7-10 7-13 7-14 7-16 7-16 7-18 7-18 7-19 7-19 7-19 7-19 7-21 7-22 7-22 7-22 7-23 7-23 Reservoir Stimulation © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" ix elgoajiroblanco@hotmail.com Chapter Performance of Fracturing Materials Vernon G Constien, George W Hawkins, R K Prud’homme and Reinaldo Navarrete 8-1 Introduction 8-2 Fracturing fluid characterization 8-3 Characterization basics 8-4 Translation of field conditions to a laboratory environment 8-5 Molecular characterization of gelling agents 8-5.1 Correlations of molecular weight and viscosity 8-5.2 Concentration and chain overlap 8-5.3 Molecular weight distribution 8-5.4 Characterization of insoluble components 8-5.5 Reaction sites and kinetics of crosslinking 8-6 Rheology 8-6.1 Basic flow relations 8-6.2 Power law model 8-6.3 Models that more fully describe fluid behavior 8-6.4 Determination of fracturing fluid rheology 8-6.5 Rheology of foam and emulsion fluids 8-6.6 Effect of viscometer geometry on fluid viscosity 8-6.7 Characterization of fluid microstructure using dynamic oscillatory measurements 8-6.8 Relaxation time and slip 8-6.9 Slurry rheology 8-7 Proppant effects 8-7.1 Characterization of proppant transport properties 8-7.2 Particle migration and concentration 8-8 Fluid loss 8-8.1 Fluid loss under static conditions 8-8.2 Fluid loss under dynamic conditions 8-8.3 Shear rate in the fracture and its influence on fluid loss 8-8.4 Influence of permeability and core length 8-8.5 Differential pressure effects 8-1 8-1 8-2 8-2 8-2 8-2 8-3 8-4 8-5 8-5 8-6 8-7 8-7 8-8 8-10 8-12 8-15 8-16 8-17 8-17 8-19 8-19 8-21 8-22 8-23 8-24 8-25 8-26 8-26 Chapter Fracture Evaluation Using Pressure Diagnostics Sunil N Gulrajani and K G Nolte 9-1 Introduction 9-2 Background 9-3 Fundamental principles of hydraulic fracturing 9-3.1 Fluid flow in the fracture 9-3.2 Material balance or conservation of mass 9-3.3 Rock elastic deformation Sidebar 9A What is closure pressure? Sidebar 9B Pressure response of toughness-dominated fractures 9-4 Pressure during pumping 9-4.1 Time variation for limiting fluid efficiencies 9-4.2 Inference of fracture geometry from pressure 9-4.3 Diagnosis of periods of controlled fracture height growth 9-4.4 Examples of injection pressure analysis Sidebar 9C Pressure derivative analysis for diagnosing pumping pressure 9-4.5 Diagnostics for nonideal fracture propagation x 9-1 9-2 9-3 9-3 9-4 9-4 9-6 9-9 9-10 9-12 9-12 9-14 9-15 9-16 9-18 Contents © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com C discharge coefficient Cpfl fluid heat capacity C rock type and grain size constant Cpi C shape factor accounting for wellbore curvature product concentration at the solid/liquid interface, mol/L C* critical overlap concentration, lbm/1000 gal, dL/g CR general reservoir-controlled leakoff coefficient, ft/min1/2, m/s1/2 CA reservoir geometric shape factor CRD dimensionless reservoir conductivity CA acid concentration Ct combined fluid-loss coefficient, ft/min1/2, m/s1/2 Cc compressibility control leakoff coefficient, ft/min1/2, m/s1/2 Cv viscosity-controlled leakoff coefficient, ft/min1/2, m/s1/2 Cce effective reservoir-controlled fluid-leakoff coefficient, ft/min1/2, m/s1/2 Cve effective filtrate-controlled leakoff coefficient, ft/min1/2, m/s1/2 CCO total interface concentration of carbonate species Cw wall or filter-cake fluid-loss coefficient, ft/min1/2, m/s1/2 Ccv combined leakoff coefficient, ft/min1/2, m/s1/2 CW acid concentration at solid/liquid interface Cd dimensionless discharge coefficient Cwall surface acid concentration CD dimensionless wellbore storage coefficient Cwcv total leakoff coefficient, ft/min1/2, m/s1/2 Cdiv concentration of diverter particles, lbm/gal, kg/m3 Cdiv´ net concentration of diverter particles after dissolution effects, lbm/gal, kg/m3 Cdrag drag coefficient Cdyn dynamic constrained modulus, psi, bar, Pa Ceqm equilibrium concentration CfD dimensionless fracture conductivity C0 acid concentration at capillary inlet C1 external phase constant CF concentration factor d diameter, in., m d distance, ft, in., m dprop proppant particle diameter, in., m dsol solid particle diameter, in., m dtbg tubing diameter, in., m CfDapp apparent dimensionless fracture conductivity D diameter, in., m CfD,opt optimal dimensionless fracture conductivity D distance, ft, in., m CfDtrue actual dimensionless fracture conductivity D turbulence coefficient D diffusion constant, cm2/s DA molecular diffusion coefficient, cm2/s De effective acid diffusion coefficient, cm2/s Deff effective acid diffusion coefficient, ft2/s, m2/s DH hydraulic diameter, in., m Dp pipe inner diameter, in., m Cgel gel mass concentration in fluid CH shape factor CHF concentration of hydrofluoric acid Ci reactant concentration at the solid/liquid interface, mol/L CL leakoff or fluid-loss coefficient, ft/min1/2, m/s1/2 Co rock cohesion, psi, bar, Pa Dp perforation diameter, in., m Co initial reactant concentration Dpe Cp product concentration effective diffusion coefficient for reaction products, cm2/s N-2 Nomenclature © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com Dw wire diameter, in., m FL linear flow function Da Damköhler number Fmeas fracture development function for measured data Fsim fracture development function for simulated data F0 surface flux at time zero g acceleration of gravity, ft/s2, m/s2 Damt mass-transfer-limited Damköhler number Darxn reaction-rate-limited Damköhler number De Deborah number DR damage ratio E Young’s modulus, psi, bar, Pa E´ plane strain modulus, psi, bar, Pa Eav average Young’s modulus, psi, bar, Pa Edyn dynamic Young’s modulus, psi, bar, Pa Ef reaction rate constant En effect of all previous time steps Es secant Young’s modulus, psi, bar, Pa Et tangent Young’s modulus, psi, bar, Pa Et Young’s modulus of a plastic material, psi, bar, Pa f friction factor f elastic influence function faL apparent length fraction fd drag coefficient ffL ratio of fracture to loss volume during injection fLS volume fraction lost to spurt fp pad volume fraction fpad pad volume fraction fs slurry volume fraction fv volume fraction fw water fractional flow fκ spurt fraction F g(∆tD) dimensionless fluid-loss volume function gf fracture gradient, psi/ft, Pa/m g0 fluid-loss volume function G shear modulus, psi, bar, Pa G(∆tD) dimensionless fluid-loss time function G* corrected value of G-function at closure G´ elastic storage modulus, psi, bar, Pa G´´ viscous loss modulus, psi, bar, Pa Gc fluid-loss time function G(∆tD) at fracture closure Gdyn dynamic shear modulus, psi, bar, Pa Ge strain energy release rate GRAPI gamma ray value h height, ft, m h formation or reservoir thickness, ft, m h hardening parameter h heat transfer coefficient hcp height at the center of perforations, ft, m hD dimensionless height hf fracture height, ft, m hfD ratio of the leakoff area to the characteristic length fill fraction hfo initial fracture height, ft, m F formation factor hL permeable or fluid-loss height, ft, m F force, lbf, N hpay height of the pay zone, ft, m F fracture development function hs penetration into bounding layer, ft, m Fb ratio of wellbore width or net pressure hwD dimensionless completion thickness Fc critical load H depth, ft, m Fi surface flux at the ith time step i injection rate Reservoir Stimulation N-3 © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com i interest rate kpf perforation friction proportionality constant I diffusion flux kr relative permeability, md I influence function kr reservoir permeability, md Iani index of horizontal-to-vertical permeability anisotropy kr surface reaction rate constant, cm/s J acid flux J pseudosteady-state productivity index JF fractured well productivity index JH pseudosteady-state productivity index of horizontal well Jideal ideal productivity index Jo initial productivity index Jreal real productivity index JV pseudosteady-state productivity index of vertical well k permeability, md kc damaged zone permeability, md kcake filter or diverter cake permeability, md kd dissociation constant kf fracture permeability, md kfD dimensionless fracture permeability kfil relative permeability of the formation to filtrate, md kfl thermal conductivity of a fluid kfoam foam permeability, md kfs fracture permeability in the near-well skin effect zone, md kg gas effective permeability, md kh thermal conductivity, BTU/hr ft°F, J/s m°C kH horizontal permeability, md kHs damaged horizontal permeability, md ki initial permeability, md knear wellbore near-wellbore friction proportionality constant ko oil effective permeability, md ko initial permeability, md N-4 kr,app apparent reservoir permeability, md krg gas relative permeability, md kro oil relative permeability, md krw water relative permeability, md ks damaged permeability, md ksph spherical permeability, md kV vertical permeability, md kVs damaged vertical permeability, md kw water effective permeability, md kx maximum permeability directed parallel to the principal permeability axis, md kx permeability in the x direction, md ky minimum permeability directed perpendicular to the principal permeability axis, md ky permeability in the y direction, md kz vertical permeability, md k0 reaction rate constant at the reference temperature K bulk modulus, psi, bar, Pa K mass-transfer coefficient, cm/s K equilibrium constant K coefficient of earth stress, dimensionless K empirical kinetic constant K Mark-Houwink coefficient, dL/g K power law fluid rheology consistency coefficient, lbf-sn/ft2 K´ power law effective consistency coefficient, lbf-sn´/ft2 Ka equilibrium constant of acid Kads equilibrium constant of the exothermic adsorption of molecules at surface reactive sites Kc conditional equilibrium constant Nomenclature © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com Kc G-plot slope correction factor for pressuredependent leakoff Kd dissociation constant Kdyn L horizontal well length, ft, m Lapp apparent of equivalent fracture penetration, ft, m dynamic bulk modulus Le length to tip, ft, m Keq effective equilibrium constant Lp productive length, ft, m Kfoam consistency coefficient for the foam phase Lt length of the fracture tip region, ft, m Kg mass-transfer coefficient Lv length of filtrate-invaded zone, ft, m KI stress intensity factor, psi/in.1/2, bar/m1/2 m KIc critical stress intensity factor, psi/in.1/2, bar/m1/2 slope on semilogarithmic straight line, psi/cycle (oil), psi2/cycle or psi2/cp/cycle (gas) m Archie constant m linear regression constant m reaction rate order m power law turbulence factor mbf bilinear flow slope, psi/hr1/4, bar/hr1/4 mc early-time slope used to compute the wellbore storage coefficient C mcf linear flow slope in an elongated reservoir stress intensity factor at top fracture tip, psi/in.1/2, bar/m1/2 mepr early pseudoradial flow slope mG slope of the G-plot, psi, bar, Pa Ko coefficient of earth pressure at rest, dimensionless mGc slope of the G-plot at fracture closure, psi, bar, Pa Kpipe consistency index for pipe flow, lbf-sn/ft2 mhl linear flow slope of a horizontal well Kr reaction rate mlf linear flow slope, psi/hr1/2, bar/hr1/2 Kr temperature-dependent reaction rate constant mM Mayerhofer et al (1993) method slope Ks bulk modulus of the solid constituents, psi, bar, Pa mn straight-line slope mN Nolte method slope KIceff effective fracture toughness, psi/in.1/2, bar/m1/2 KIclag critical stress intensity factor in the fluid lag region, psi/in.1/2, bar/m1/2 KIc-apparent apparent fracture toughness, psi/in.1/2, bar/m1/2 KIl KIu stress intensity factor at bottom fracture tip, psi/in.1/2, bar/m1/2 Ks solubility constant mp match pressure, psi, bar, Pa Kslot consistency index for slot flow, lbf-sn/ft2 mpp spherical flow slope Kv consistency index for a concentric cylinder viscometer, lbf-sn/ft2 mrf radial flow slope, psi, bar, Pa m3/4 slope of the G-plot at the 3⁄4 point, psi, bar, Pa l length, ft, in., m Macid moles of acid per unit rock face area l diverter cake thickness, ft, m Mgel specific density of gel mass in the fluid ld length of damaged zone, ft, m Mv viscosity average molecular weight, g/mol lp perforation channel length, ft, m MW molecular weight, g/mol L length, ft, m n number or order L fracture half-length, ft, m n index of time step Reservoir Stimulation N-5 © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com n power law fluid rheology behavior index, dimensionless pD,up dimensionless pressure for a uniformpressure, fixed-length fracture n´ power law effective index, dimensionless pe ne index of time step after shut-in constant outer reservoir pressure, psi, bar, Pa nf power law exponent of the filtrate pf fissure or fracture pressure, psi, bar, Pa nperf number of perforations pf fracturing fluid pressure, psi, bar, Pa NDH dimensionless cumulative recovery from a longitudinally fractured horizontal well pf far-field pore pressure, psi, bar, Pa pfc formation capacity pfo fissure opening pressure, psi, bar, Pa ph hydrostatic pressure, psi, bar, Pa phead hydrostatic head of wellbore fluid, psi, bar, Pa NDV dimensionless cumulative recovery from a fractured vertical well Np oil cumulative production, bbl, m3 NRe Reynold’s number NSc Schmidt number pi initial reservoir pressure, psi, bar, Pa NSh Sherwood number pif Nϕ coefficient of passive stress breakdown pressure for fracture initiation, psi, bar, Pa p pressure, psi, bar, Pa pinj injection pressure, psi, bar, Pa p reservoir pressure, psi, bar, Pa pISI instantaneous shut-in pressure, psi, bar, Pa pb bubblepoint pressure, psi, bar, Pa pit wellhead injection pressure, psi, bar, Pa pbh borehole pressure, psi, bar, Pa piw bottomhole injection pressure, psi, bar, Pa pc constant confining pressure, psi, bar, Pa pm microannulus pressure, psi, bar, Pa pc closure pressure, psi, bar, Pa pm mud pressure in the wellbore, psi, bar, Pa pc capillary pressure, psi, bar, Pa pmeas measured pressure, psi, bar, Pa pcf casing friction pressure, psi, bar, Pa pnet net pressure, psi, bar, Pa pcorr pressure corrected for non-plane-strain contribution, psi, bar, Pa pnet,crit critical net pressure, psi, bar, Pa pnet,fo pressure at center of the perforations, psi, bar, Pa net pressure required for fissure opening, psi, bar, Pa pnet,max maximum net pressure, psi, bar, Pa pD dimensionless pressure pnet,si net pressure at shut-in, psi, bar, Pa pD´ dimensionless pressure derivative pnet,so net pressure at screenout, psi, bar, Pa pDLs dimensionless pressure difference for pressuredependent leakoff po wellbore pressure, psi, bar, Pa pp far-field reservoir pressure, psi, bar, Pa ppf perforation friction, psi, bar, Pa pcp pdr downstream restriction pressure, psi, bar, Pa pdsc pressure downstream of the surface choke, psi, bar, Pa pdsv pressure downstream of the safety valve, psi, bar, Pa N-6 ppipe friction pipe friction pressure, psi, bar, Pa ppwD dimensionless pseudopressure pr reservoir pressure, psi, bar, Pa Nomenclature © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com ps pressure at the outer bound of the damaged zone, psi, bar, Pa psc standard condition pressure, psi, bar, Pa psep separator pressure, psi, bar, Pa psim simulated pressure, psi, bar, Pa psurf surface pressure, psi, bar, Pa ptf tubing flowing pressure, psi, bar, Pa ptip net pressure at fracture tip for extension, psi, bar, Pa qD dimensionless flow rate qDND dimensionless non-Darcy flow rate qe flow at tip qf volume rate of storage in a fracture qg gas production rate, Mscf/D, m3/d qi injection rate, B/D, bbl/min, m3/d, m3/s qi,max maximum injection rate, B/D, bbl/min, m3/d, m3/s qL rate of fluid loss, B/D, bbl/min, m3/d, m3/s qlast last flow rate, B/D, bbl/min, m3/d, m3/s qL,C rate of fluid loss for reservoir-controlled leakoff, B/D, bbl/min, m3/d, m3/s qn leakoff rate from one wing through two faces, B/D, m3/s pur upstream restriction pressure, psi, bar, Pa pusv pressure upstream of the safety valve, psi, bar, Pa pvapor fluid vapor pressure, psi, bar, Pa pw wellbore pressure, psi, bar, Pa pwD dimensionless wellbore pressure qo initial flow rate, B/D, m3/s pwf bottomhole flowing pressure, psi, bar, Pa qo oil production rate, B/D, m3/s pwf,ideal ideal bottomhole flowing pressure, psi, bar, Pa pwf,real real bottomhole flowing pressure, psi, bar, Pa pwfs wellbore sandface flowing pressure, psi, bar, Pa qo,max maximum oil production rate at two-phase flow, B/D, m3/s qperf flow entering a perforation qRT total reservoir production rate, B/D, m3/d pwh wellhead flowing pressure, psi, bar, Pa qs damaged flow rate pws bottomhole shut-in pressure, psi, bar, Pa qT total injection rate p1 hr pressure on extension of semilogarithmic straight line at t = hr, psi, bar, Pa qtot constant total injection rate qw water production rate, B/D, m3/d Qp cumulative production, B/D, m3/d r radial distance, ft, m r reaction rate rA rate of reactant consumption, mol/cm2/s rac stimulation radius, ft, m p3/4 pressure at the G-plot ⁄4 point, psi, bar, Pa Pe Peclet number PI productivity index, B/D/psi (oil), Mscf/D/psi (gas), m3/d/bar PIs damaged productivity index, B/D/psi (oil), Mscf/D/psi (gas), m3/d/bar PVBT number of pore volumes to breakthrough rb acid bank radius, ft, m PVinj number of pore volumes injected rbob bob radius q flow rate, B/D (oil), Mscf/D (gas), m3/d rcup cup radius qapp apparent flow rate, B/D, bbl/min, m3/d, m3/s rD rate of surface reaction, mol/cm2/s qave average flow rate, B/D, bbl/min, m3/d, m3/s re reservoir radius, ft, m qc flow rate in capillary pores reH qc injection rate into core face radius of horizontal drainage ellipse formed around a horizontal well, ft, m Reservoir Stimulation N-7 © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com rf radial extent of foam bank, ft, m rfeldspar reaction rate of feldspar, mol feldspar/cm2/s Rso solution-gas/oil ratio Rsw solution-gas/water ratio rHF radial penetration of hydrofluoric acid, ft, m Rt true formation resistivity, ohm-m rhyd hydraulic radius Rw resistivity of formation water, ohm-m ri specific reaction rate of mineral i Rxo resistivity of the flushed zone, ohm-m ri inner radius R0 filter-cake resistance, s⋅m–1 ro outer radius R0,app apparent filter-cake resistance, s⋅m–1 rp ratio of permeable area to fracture area s skin effect, dimensionless rp probe radius s reactive surface area rpD dimensionless ratio of permeable area to fracture area sc skin effect from partial penetration, dimensionless rperf perforation radius, in., m sc + θ skin effect due to partial penetration and slant, dimensionless rquartz reaction rate of quartz, mol quartz/cm2/s rs radius of damaged or displaced section, ft, m scake temporary skin effect due to diverter cake, dimensionless rv fraction of well drainage volume occupied by the crest at water breakthrough sd skin effect due to damage, dimensionless rw wellbore radius, ft, m (sd)o perforation skin effect in openhole, dimensionless rw´ effective or apparent wellbore radius, ft, m (sd)p rwD´ dimensionless effective wellbore radius rwh wormhole radius perforation skin effect due to perforations terminating outside the damaged zone, dimensionless R pore radius se vertical eccentricity skin effect, dimensionless R reaction rate sf R universal gas constant, psi ft3/mol °R, bar m3/mol K skin effect while flowing at pseudoradial conditions, dimensionless sf fracture stiffness, ft/psi/m R fracture radius, ft, m sf skin effect due to a fracture, dimensionless RA rate of appearance of reactant sfoam foam skin effect, dimensionless RAt rate of reactant consumption at the tip sfs fracture face damage skin effect, dimensionless RB rate of mineral dissolution sH plane-flow skin effect, dimensionless Rcake diverter cake resistance, m–1 so initial skin effect, dimensionless Reff effective radius, ft, m so constant skin effect, dimensionless Rf radial fracture radius, ft, m sp skin effect due to perforations, dimensionless Ri overall reaction rate of mineral i st total skin effect, dimensionless Rmf resistivity of the mud filtrate, ohm-m sV vertical skin effect, dimensionless Ro resistivity of 100% water-saturated formation, ohm-m swb wellbore skin effect, dimensionless sx vertical effects skin effect, dimensionless pressure-difference curve, dimensionless sθ skin effect due to slant, dimensionless Rp N-8 Nomenclature © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com S surface area SF surface area per unit volume of solids for fastreacting minerals tDrw′ dimensionless time referenced to the effective wellbore radius rw′ tDxf dimensionless fracture time Sg gas-phase fluid saturation, fraction te time at end of pumping or injection, min, s Si reactive surface area of mineral i tend time of the end of dual-porosity behavior So oil-phase fluid saturation, fraction Soi initial oil saturation, fraction texp time of fracture opening and initial fluid exposure Sor residual oil saturation, fraction ti injection time Sp spurt-loss coefficient, gal/ft2, m3/m2 ti time at the end of the ith step SRE rock embedment strength tknee knee time for the crossing of pressure derivatives SS surface area per unit volume of solids for slow-reacting minerals tmin minimum time tn time at the end of step n, s Sw water saturation, fraction ton time of onset of dual-porosity behavior Swc connate water saturation, fraction producing time, hr Swi irreducible water saturation, fraction pumping or injection time, hr t time, day, hr, min, s tP compressional wave arrival time t* reduced time tpss time to pseudosteady state, hr ta pseudotime tr time at reference point r taD dimensionless pseudotime tS tBT time of water breakthrough, hr time of tip screenout or injection without screenout tc closure time, tS shear wave arrival time tcD dimensionless closure time tso time at screenout tD dimensionless time tsp spurt time tDA dimensionless time referenced to reservoir drainage area t0 reference time tDblf dimensionless time at start of formation linear flow regime tαD dimensionless ratio of time to pumping time T temperature, °F, °C, K T absolute temperature, °R, K T dimensionless time for after-closure analysis T constant for tectonic effects on stress T wire tension, lbf/ft dimensionless time at which formation linear flow behavior ends Td dead weight, lbf TD dimensionless temperature time of departure from flow regime trend Tf flowing temperature, °F, °C tD,knee dimensionless knee time for the pressure derivatives Tfl fluid temperature, °F, °C Ti fluid temperature at the fracture mouth, °F, °C tDebf dimensionless time at end of bilinear flow regime tDefl dimensionless time at which fracture linear flow behavior ends tDelf tdep Reservoir Stimulation N-9 © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com Tinlet temperature at inlet, °F, °C V volume, ft3, m3 Tknee dimensionless knee time for after-closure analysis Vac acid volume injected, ft3, m3 Vf fracture volume, ft3, m3 Vf fluid volume To offset temperature, °F, °C To tensile strength, psi, bar, Pa VF volume of fast-reacting minerals Tp dimensionless time at the end of pumping for after-closure analysis Vfp fracture volume at the end of pumping, ft3, m3 Tr reservoir temperature, °F, °C Vfso fracture volume at screenout, ft3, m3 Tref reference temperature, °F, °C VHC volume of hydrocarbons, ft3, m3 Ts wire strength, lbf/ft VHCl volume of hydrochloric acid, ft3, m3 Tsc standard condition temperature, °F, °C, K VHF volume of hydrofluoric acid, ft3, m3 TR tool response Vi volume of fluid injected, bbl, gal, m3 u velocity, ft/s, cm/s Viso fluid volume injected at screenout, bbl, gal, ft3, m3 u acid flux VL leaked-off fluid volume, bbl, gal, m3 ug volumetric flux of gas VL,C uL leakoff velocity, ft/s, cm/s volume of fluid lost due to filtration, bbl, gal, m3 uP P-wave velocity, ft/s, m/s VLp leaked-off volume during pumping, bbl, m3 ur rising velocity of spherical particles in a fluid, ft/s, cm/s VLp,C volume of fluid lost due to filtration at the end of pumping, bbl, gal, m3 uS S-wave velocity, ft/s, m/s VLs usol solid particle velocity, ft/s, cm/s volume of fluid lost during shut-in, bbl, gal, m3 ut terminal settling velocity, ft/s, m/s VL,S fluid-loss component due to spurt, bbl, gal, m3 utip fracture tip velocity, ft/s, cm/s VLs,C volume of fluid lost due to filtration during shut-in, bbl, gal, m3 uw volumetric flux of water terminal proppant settling velocity, ft/s, m/s VM molar volume u∞ v flow velocity, ft/s, cm/s v^ specific volume of foam vA,x Fick’s law for the velocity of species A vc damaged zone velocity, ft/s, cm/s vfall settling rate, ft/s, cm/s v^ l specific volume of the base liquid vL fluid-loss velocity, ft/min vL specific fluid-loss volume vm matrix velocity, ft/s, cm/s vx average fluid velocity along the fracture length, ft/s, cm/s N-10 Vmineral volumetric fraction of a mineral Vp pore volume, ft3, m3 Vprop bulk proppant volume injected Vrp relative proppant volume, lbm/md-ft3 VS volume of slow-reacting minerals VS volume of fluid lost to spurt, gal/100 ft2 Vwall volume of fluid leaked off at the fracture wall w width, ft, m w fracture width, ft, in., m we average fracture width at end of pumping, ft, m Nomenclature © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com wetch acid-etched width, ft, m zw standoff from oil-water contact, ft, m wf fracture width, ft, m zw wL leakoff width, ft, m elevation of midpoint of perforations from the bottom of the reservoir, ft, m wlost proppant volume lost to the fracture width wmax maximum width, ft, m wmax,p maximum fracture width at the end of pumping, ft, m wmax,si maximum fracture width immediately after shut-in, ft, m zwD dimensionless completion elevation Z gas deviation factor, dimensionless Symbols α Forscheimer equation coefficient wn average fracture width at time step n, ft, m α Biot poroelastic constant wo wellbore width, ft, m α proportionality constant wp propped width, ft, m α specific diverter cake resistance, m/kg wp-eff effective propped width, ft, m α exponent of fracture area growth, dimensionless wso width at screenout, ft, m α order of reaction ww width at the wellbore, ft, m αbf bilinear flow constant Welas elastic energy stored in a solid αc wellbore storage constant Wext potential energy of exterior forces αcf elongated reservoir constant Wi weighting factors αf sealing fault constant Wkin kinetic energy αg geometry coefficient Ws energy dissipated during propagation of a crack αhl linear flow to a horizontal well constant αlf linear flow constant dimensionless pressure constant x linear distance, ft, m αp xe well drainage dimension, ft, m αpp partial penetration constant xe,opt optimal well spacing, ft, m αt dimensionless time constant xf productive fracture half-length, ft, m αT linear thermal expansion coefficient xfa apparent fracture half-length, ft, m α0 lower bound of area exponent, dimensionless xfeldspar feldspar volume fraction of a sandstone α1 upper bound of area exponent, dimensionless xs half-length of the skin effect zone, ft, m αθ perforation-phase-dependent variable X volumetric dissolving power β XC dissolving power of acid dissolving power coefficient related to acid strength XHCl bulk rock fraction dissolved by hydrochloric acid β stiffness β ratio of average to wellbore net pressure or width y vertical linear distance, ft, m z linear distance, ft, m β Forscheimer equation coefficient zTVD true vertical depth, ft, m β non-Darcy flow rate coefficient zw elevation from reservoir bottom, ft, m βp net pressure or width ratio during injection Reservoir Stimulation N-11 © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com βs net pressure or width ratio during shut-in γ shear strain γ channel factor γ G-plot slope correction factor for pressuredependent leakoff γ specific gravity γ shear rate, s–1 γab interaction energy per unit surface area between liquids a and b γb shear rate at the bob, s–1 γF fracture surface energy γfluid specific gravity of fluid γg specific gravity of gas γI interfacial tension γprop specific gravity of proppant γv volume-averaged shear rate, s–1 Γ foam quality, fraction Γ(d) Euler gamma function Γ(x) gamma function δ dip of the formation (angle with the horizontal), degree ∆ rate of deformation tensor ∆C ∆pfriction friction pressure ratio with and without solids ∆ph hydrostatic pressure drop, psi, bar, Pa ∆pint difference in the pressure intercept, psi, bar, Pa ∆pmisalign perforation misalignment friction, psi, bar, Pa ∆pnear wellbore near-wellbore pressure loss, psi, bar, Pa ∆pnf pressure difference due to near-face leakoff effects, psi, bar, Pa ∆ppf friction through the perforation, psi, bar, Pa ∆ppiz pressure drop across a polymer-invaded zone, psi, bar, Pa ∆pr pressure drop in the reservoir, psi, bar, Pa ∆pR total pressure increase in the reservoir beyond the filtrate invasion region, psi, bar, Pa ∆pRC pressure increase in the reservoir beyond the filtrate region due to Carter-based leakoff, psi, bar, Pa ∆pRS pressure increase in the reservoir beyond the filtrate region due to spurt, psi, bar, Pa change in concentration ∆psafe pressure safety margin, psi, bar, Pa ∆E activation energy, kcal/mol ∆pT ∆H heat of reaction total pressure difference between the fracture pressure and initial reservoir pressure, psi, bar, Pa ∆MD change in measured depth, ft, m ∆ptort ∆p pressure difference or gradient, psi, bar, Pa tortuosity component of pressure, psi, bar, Pa ∆pc pressure drop between filtrate/reservoir interface and far-field reservoir, psi, bar, Pa ∆ptotal difference between fracture pressure and far-field reservoir pressure, psi, bar, Pa ∆pcake pressure drop across the filter cake, psi, bar, Pa ∆pv pressure drop across the filtrate-invaded zone, psi, bar, Pa ∆pct total pressure drop, psi, bar, Pa ∆pw difference in wellbore pressure, psi, bar, Pa ∆pµ viscous pressure contribution, psi, bar, Pa ∆t time difference, hr ∆t sonic transit time in the formation ∆pdeparture pressure departure, psi, bar, Pa ∆pentry fracture entry pressure, psi, bar, Pa ∆pface pressure drop across fracture face dominated by filter cake, psi, bar, Pa N-12 Nomenclature © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com ∆t shut-in time, hr, min, s η efficiency ∆tae effective pseudotime η diffusivity constant ∆tc closure time ηC efficiency including spurt loss ∆tC compressional wave traveltime ηf fracture hydraulic diffusivity ∆tcD dimensionless closure time ηfD dimensionless fracture hydraulic diffusivity ∆tD dimensionless time difference ηp efficiency at end of pumping ∆tD dimensionless shut-in time ηso efficiency at screenout ∆tDso dimensionless time after a screenout θ angle, degree ∆te effective time θ fluid-loss exponent ∆tma sonic transit time in the rock matrix θ dimensionless time ∆tS shear wave traveltime κ opening-time distribution factor ∆tso time following screenout κ ∆tsup superposition time function ratio of fracture-opening stress to minimum stress κ spurt effect factor κ thermal diffusivity κ overall dissolution rate constant, cm/s ∆Tsurf temperature change at the surface, °F, °C ∆TBT change in true bed thickness, ft, m ∆TVD change in true vertical depth, ft, m ∆V change in volume κso spurt correction at screenout ∆ρ density difference, lbm/ft3, g/cm3 λ interporosity constant λ experimental coefficient for the tortuosity reorientation of a fracture path ∆φmax difference in maximum porosity ∆σ stress difference, psi, bar, Pa ∆$n λ apparent time multiplier incremental revenue ε λ characteristic relaxation time introduced error ε λrt total mobility ratio of closure time to the time interval ε λt multiphase mobility longitudinal strain εa Λ length scale corresponding to pore size axial strain εe µ viscosity, cp elastic strain εf µa apparent viscosity, cp acid front position divided by the linear flow core length µbase viscosity of base fracturing fluid, cp εh minimum tectonic strain µeff effective viscosity, cp εH maximum tectonic strain µfil viscosity of fracturing fluid filtrate, cp εp plastic strain µfluid fluid viscosity, cp εr radial strain µg gas viscosity, cp εS specific volume expansion ratio µgi εV gas viscosity at initial reservoir pressure and temperature, cp volumetric strain η µi intrinsic viscosity, cp poroelastic stress coefficient Reservoir Stimulation N-13 © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com µinh inherent viscosity, cp ρr ratio of slurry density to fluid density µo oil viscosity, cp ρrock rock density, lbm/ft3, g/cm3 µr viscosity ratio ρs slurry density, lbm/ft3, g/cm3 µr viscosity of reservoir fluid, cp ρsol solid particle density, lbm/ft3, g/cm3 µr relative viscosity, cp σ stress, psi, bar, Pa µslurry slurry viscosity, cp σ′ effective stress, psi, bar, Pa µsp specific viscosity, cp σa axial stress, psi, bar, Pa µw water viscosity, cp σBL stress acting on a bounding layer, psi, bar, Pa µ0 zero-shear viscosity, cp σc uniaxial compressive strength, psi, bar, Pa µ100 viscosity at 100°F [40°C], cp σc closure pressure or stress, psi, bar, Pa µ∞ high-shear-limiting viscosity, cp σc′ effective confining stress, psi, bar, Pa ν Poisson’s ratio σf normal stress across a fissure, psi, bar, Pa ν stoichiometric ratio of reactants to products σfailure′ effective ultimate strength, psi, bar, Pa νCaCO3 stoichiometric coefficient of CaCO3 σh minimum horizontal stress, psi, bar, Pa νdyn dynamic Poisson’s ratio σh′ νHCl stoichiometric coefficient of hydrochloric acid minimum effective horizontal stress, psi, bar, Pa νu undrained Poisson’s ratio σH maximum horizontal stress, psi, bar, Pa ξ fluid-loss term σH,max maximum horizontal stress, psi, bar, Pa ξ dimensionless area or position σh,min minimum horizontal stress, psi, bar, Pa ξf forward rate constant σm mean stress, psi, bar, Pa ξr reverse rate constant σmin ρ minimum stress or minimum principal stress, psi, bar, Pa density, lbm/ft3, g/cm3 ρa σn normal stress component, psi, bar, Pa mass of diverter cake per unit area of available sandface, g/cm3 σn′ effective normal stress, psi, bar, Pa ρA density of reactant, g/cm3 σo equal-stress constant, psi, bar, Pa ρb bulk density, lbm/ft3, g/cm3 σpay stress of the pay zone, psi, bar, Pa ρbrine brine density, lbm/ft3, g/cm3 σPZ stress acting on the pay zone, psi, bar, Pa ρC density of acid solution σr radial stress, psi, bar, Pa σref constant state of stress, psi, bar, Pa ρCaCO3 density of calcium carbonate, lbm/ft3, g/cm3 ρdiv density of diverter particles, kg/m3 σrθ shear stress, psi, bar, Pa ρf fluid density, lbm/ft3, g/cm3 σv vertical stress, psi, bar, Pa ρl liquid density, lbm/ft3, g/cm3 σv′ effective vertical stress, psi, bar, Pa ρma density of matrix components, lbm/ft3, g/cm3 σ1 maximum principal stress, psi, bar, Pa ρo oil density, lbm/ft3, g/cm3 σ1′ ρp maximum principal effective stress, psi, bar, Pa proppant or particle density, lbm/ft3, g/cm3 N-14 Nomenclature © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO" elgoajiroblanco@hotmail.com σ2 σ2′ intermediate principal stress, psi, bar, Pa φcake diverter cake porosity, fraction intermediate principal effective stress, psi, bar, Pa φD porosity from density, fraction effective porosity, fraction σ3 φeff minimum principal stress, psi, bar, Pa φf fracture porosity, fraction σ3′ minimum principal effective stress, psi, bar, Pa φi initial porosity, fraction σθ tangential stress, psi, bar, Pa φN porosity from neutron, fraction σθθ circumferential stress, psi, bar, Pa proppant pack porosity, fraction τ φp shear stress, psi, bar, Pa φS porosity from sonic, fraction τ time of fracture opening τ φtotal total porosity, fraction dimensionless slurry time ϕ angle, degree τc characteristic time for fracture propagation, dimensionless Φ channel flow function τo foam yield stress, lbf/ft2, bar, Pa ψ change of angle, degree τoct octahedral shear stress, psi, bar, Pa Ψ dimensionless hydrofluoric acid concentration τw wall shear stress, lbf/ft2, bar, Pa Ψ dimensionless rock dissolution rate τyp yield point, lbf/ft2, bar, Pa ω angular velocity, rad/s, rpm φ porosity, fraction ω storativity ratio φ angle of internal friction, degree Ω stoichiometric coefficient Reservoir Stimulation N-15 © 2010 COPYRIGHT MERCADO NEGRO, LAS PLAYITAS MARACAIBO-EDO ZULIA, VENEZUELA PARA COMPRAR AL DETAL O AL MAYOR, ESTE Y OTROS PRODUCTOS, FAVOR PREGUNTAR POR EL GÖAJIRO BLANCO, EN EL MERCADO LAS PLAYITAS ADVERTENCIA: "EL DERECHO DE AUTOR NO ES UNA FORMA DE PROPIEDAD SINO UN DERECHO CULTURAL EXIGE TU DERECHO"