Hydraulic Fracture Mechanics

Transcription

1 Hydraulic Fracture Mechanics Peter Valk6 and Michael J. Economides Texas A & M University, College Station, USA JOHN WILEY & SONS Chichester New York Brisbane Toronto Singapore

2 CONTENTS Preface Ust of Notatlon xi xiii 1 Hydraulically lnduced Fractures in the Petroleum and Related lndustries 1.1 Fractures in Well Stimulation 1.2 Fluid Flow Through Porous Media ~. 2.~ The Near-well Zone 1.3 Flow from a Fractured Well 1.4 Hydraulic Fracture Design 1.5 Treatment Execution ~.5. ~ Fracturing Fluids ~.5.2 Proppants 1.6 Data Acquisition and Evaluation for Hydraulic Fracturing Well Log Measurements ~.6. 2 Core Measurements ~. 6.3 Well Testing 1.7 Mechanics in Hydraulic Fracturing References ~ ~ ~3 14 ~4 15 ~ Linear Elasticity, Fracture Shapes and lnduced Stresses 2.1 Force and Deformation 2. ~.1 Stress Stra in 2.2 Materia! Properties 2. 2.~ Linear Elastic Materia! Materia! Behavior Beyond Perfect Elasticity 2.3 Piane Elasticity 2.3. ~ Piane Stress Stresses Relative to an Oblique Une (Force Balance l) Equilibrium Relations (Force Balance Il) Piane Strain Boundary Conditions ~

3 vi Contents 2.4 Pressurized Crack Solution of the Une Crack Problem Constant Pressure Polynomial Pressure Distribution "Zipper" Cracks "Zipper" Crack with Polynomial Pressure Distribution Stress Concentration and Stress lntensity Factor Stress lntensity Factor, Symmetric Loading Stress lntensity Factor, non-symmetric Loading Fracture Shape in the Presence of Far-field Stress. The Concept of Net Pressure Circular Crack Volume and Strain Energy Computational Methods 49 References 50 3 Stresses in Formations Basic Concepts Stresses at Depth Near-wellbore Stresses Stress Concentrations far an Arbitrarily Oriented Well Vertical Well Breakdown Pressure Breakdown Pressure for an Arbitrarily Oriented Well Limiting Case: Horizontal Well Arbitrarily Oriented Horizontal Well 70,,, 3.8 Permeability and Stress 71 Ili! l,, Stress-sensitive Permeability Measurement of Stresses Small lnterval Fracture lnjection Tests Aooustic Measurements Determination of the Closure Pressure Core Stress Measurements Critique and Applicability of Techniques 79 References 80 4 Fracture Geometry The Perkins and Kern and Khristianovich and Zheltov Geometries The Consequences of the Piane Strain Assumption Fracture lnitiation vs. Propagation Direction Fractures in Horizontal Wells Fracture Profiles in Multi-layered Formations 92 References 95 5 Rheology and Laminar Flow Basic Concepts Materia! Behavior and Constitutive Equations Force Balance 103

4 Contents vii 5.2 Slot Flow Derivation of the Basic Aelations Equivalent Newtonian Viscosity Flow in Circular Tube Basic Relations Flow Curve Equivalent Newtonian Viscosity for Tube Flow Flow in Other Cross Sections Flow in Annulus Flow in Elliptic Cross Section Limiting Ellipsoid Cross Section 124 Aeferences Non-laminar Flow and Solids Transport Non-laminar Flow Newtonian Fluid Generai Fluid Drag Reduction Turbulent Flow in Other Geometries Solids Transport Settling ot an Individuai Sphere Ettect of Shear Rate lnduced by Flow Effect of Slurry Concentration Wall Effects Agglomeration Ettects 145 References Advanced Topics of Rheology and Fluid Mechanics Foam Rheology Quality Based Correlations Volume Equalized Constitutive Equations Volume Equalized Power Law Turbulent Flow ot Foam Accounting for Mechanical Energy Basic Concepts lncompressible Flow Foam Flow Rheometry Pipe Viscometry Slip Correction 157 References Materia! Balance The Conservation of Mass and lts Relation to Fracture Dimensions Fluid Leakoff and Spurt Loss as Materia! Properties Carter Equation l Formai Materia! Balance. The Opening Time Distribution Factor 171

5 viii Contents 8.3 The Constant Width Approximation (Carter Equation Il) The Power Law Approximation to Surface Growth The Consequences of the Power Law Assumption The Combination of the Power Law Assumption with lnterpolation Numerica! Materia! Balance Differential Materia! Balance Leakoff as Flow in the Porous Medium Filter-cake Pressure Drop Pressure Drop In the Reservoir Leakoff Rate from Combining the Resistances 187 (Ehlig-Economides et al. [6]) References 187 l 9 Coupling of Elasticity, Flow and Materia! Balance Width Equations of the Early 2D Models Perkins-Kern Width Equation Geertsma-de Klerk Width Equation Radiai Width Equation Algebraic (2D) Models as Used in Design PKN-C KGD-C PKN-N and KGD-N PKN-o: and KGD-a Radiai Model Non-Newtonian Behavior Numerica! Materia! Balance (NMB) with Width Growth Differential 2D Models Nordgren Equation Differential Horizontal Piane Strain Model Models With Detailed Leakoff Description Pressure Decline Analysls Nolte's Pressure Declina Analysis (Power Law Assumption) The No-spurt-loss Assumption (Shlyapobersky method) Materia! Balance and Propagation Pressure Estimates of the Spurt Loss Resolving Contradictions Pressure Declìne Analysis With Detailed Leakoff Description (Mayerhofer et al. Technique) 230 References Fracture Propagation Fracture Mechanics Griffith's Analysls of Crack Stability Mott's Theory for the Rate of Crack Growth Classica! Crack Propagation Criterion for Hydraulic Fracturing Fracture Toughness Criterion The lnjection Rate Dependance Paradox 243

6 10.3 Retarded Fracture Propagation Fluid Lag Ti p Dilatancy Apparent Fracture Toughness Process Zone Concept The Reopening Paradox 10.4 Continuum Damage Mechanics in Hydraulic Fracturing Ti p Propagation Velocity from CDM CDM-NK Mode! CDM-PKN Design Mode! 10.5 Pressure Decline Analysis an d Ti p Retardation Resolving Contradictions with Continuum Damage Mechanics References Contents ix Fracture Height Growth (30 and P-30 Geometries) 11.1 Equilibrium Fracture Height Reverse Application of the Net-pressure Concept Different Systems of Notation Basic Equations The Effect of Hydrostatic Pressure 11.2 Three-dimensional Models Surface Integrai Method The Stress lntensity Factor Paradox 11.3 Pseudo-three-dimensional Models References Appendix: Comparison Study of Hydraulic Fracturing Models: lnput Data and Results 287 References 294 lndex 295