QUANTITATIVE METHODS IN RESERVOIR ENGINEERING
CONTENTS 1 MOTIVATING IDEAS AND GOVERNING EQUATIONS, EXAMPLES OF INCORRECT FORMULATIONS, DARCY’S EQUATIONS FOR FLOW IN POROUS MEDIA, LOGARITHMIC SOLUTIONS AND BEYOND, FUNDAMENTAL AERODYNAMIC ANALOGIES, PROBLEMS AND EXERCISES, 2 FRACTURE FLOW ANALYSIS, EXAMPLE 2-1 SINGLE STRAIGHT-LINE FRACTURE IN AN ISOTROPIC CIRCULAR RESERVOIR CONTAINING INCOMPRESSIBLE FLUID, EXAMPLE 2-2 LINE FRACTURE IN AN ANISOTROPIC RESERVOIR WITH INCOMPRESSIBLE LIQUIDS AND COMPRESSIBLE GASES, EXAMPLE 2-3 EFFECT OF NONZERO FRACTURE THICKNESS, EXAMPLE 2-4 FLOW RATE BOUNDARY CONDITIONS, EXAMPLE 2-5 UNIFORM VERTICAL VELOCITY ALONG THE FRACTURE, EXAMPLE 2-6 UNIFORM PRESSURE ALONG THE FRACTURE, EXAMPLE 2-7 MORE GENERAL FRACTURE PRESSURE DISTRIBUTIONS, EXAMPLE 2-8 VELOCITY CONDITIONS FOR GAS FLOWS, EXAMPLE 2-9 DETERMINING VELOCITY FIELDS, 3 FLOWS PAST SHALY BODIES, EXAMPLE 3-1 STRAIGHT-LINE SHALE SEGMENT IN UNIFORM FLOW, EXAMPLE 3-2 CURVED SHALE SEGMENT IN UNIFORM FLOW, EXAMPLE 3-3 MINERALIZED FAULTS, ANISOTROPY, AND GAS FLOW, 4 STREAMLINE TRACING AND COMPLEX VARIABLES, DISCUSSION 4-1 THE CLASSICAL STREAMFUNCTION, DISCUSSION 4-2 STREAMFUNCTION FOR GENERAL FLUIDS IN HETEROGENEOUS AND ANISOTROPIC FORMATIONS, DISCUSSION 4-3 SUBTLE DIFFERENCES BETWEEN PRESSURE AND STREAMFUNCTION FORMULATIONS, DISCUSSION 4-4 STREAMLINE TRACING IN THE PRESENCE OF MULTIPLE WELLS, DISCUSSION 4-5 STREAMFUNCTION EXPRESSIONS FOR DISTRIBUTED LINE SOURCES AND VORTEXES, DISCUSSION 4-6 STREAMFUNCTION SOLUTION USING COMPLEX VARIABLES TECHNIQUES, DISCUSSION 4-7 CIRCLE THEOREM: EXACT SOLUTIONS TO LAPLACE’S EQUATION, DISCUSSION 4-8 GENERALIZED STREAMLINE TRACING AND VOLUME FLOW RATE COMPUTATIONS, DISCUSSION 4-9 STREAMLINE TRACING IN 3D FLOWS, DISCUSSION 4-10 TRACER MOVEMENT IN 3D RESERVOIRS, FLUID FLOW INSTABILITIES, 5 FLOWS IN COMPLICATED GEOMETRIES, WHAT IS CONFORMAL MAPPING? USING “SIMPLE” COMPLEX VARIABLES, EXAMPLE 5-1 THE CLASSIC RADIAL FLOW SOLUTION, EXAMPLE 5-2 CIRCULAR BOREHOLE WITH TWO SYMMETRIC RADIAL FRACTURES, EXAMPLE 5-3 CIRCULAR BOREHOLE WITH TWO UNEVEN, OPPOSITE, RADIAL FRACTURES; OR A SINGLE RADIAL FRACTURE, EXAMPLE 5-4 CIRCULAR BOREHOLE WITH MULTIPLE RADIAL FRACTURES, EXAMPLE 5-5 STRAIGHT SHALE SEGMENT AT ARBITRARY ANGLE, EXAMPLE 5-6 INFINITE ARRAY OF STRAIGHT-LINE SHALES, EXAMPLE 5-7 PATTERN WELLS UNDER AQUIFER DRIVE, THREE-DIMENSIONAL FLOWS, EXAMPLE 5-8 POINT SPHERICAL FLOW, EXAMPLE 5-9 FINITE LINE SOURCE WITH PRESCRIBED PRESSURE, EXAMPLE 5-10 FINITE LINE SOURCE WITH PRESCRIBED FLOW RATE, EXAMPLE 5-11 FINITE CONDUCTIVITY PRODUCING FRACTURE HAVING LIMITED AREAL EXTENT, EXAMPLE 5-12 FINITE CONDUCTIVITY NONPRODUCING FRACTURE HAVING LIMITED AREAL EXTENT, BOREHOLE INTERACTIONS, EXAMPLE 5-13 PRODUCING FRACTURE NEAR MULTIPLE WELLS UNDER AQUIFER DRIVE, EXAMPLE 5-14 PRODUCING FRACTURES NEAR MULTIPLE WELLS IN SOLID WALL RESERVOIRS, EXAMPLE 5-15 STRAIGHT-LINE SHALE SEGMENT NEAR MULTIPLE WELLS IN UNIFORM FLOW, EXAMPLES 5-16 AND 5-17 NONPRODUCING FAULTS IN SOLID WALL AND AQUIFER-DRIVEN RESERVOIRS, EXAMPLE 5-18 HIGHLY CURVED FRACTURES AND SHALES, 6 RADIAL FLOW ANALYSIS, EXAMPLE 6-1 STEADY LIQUIDS IN HOMOGENEOUS MEDIA, EXAMPLE 6-2 SIMPLE FRONT TRACKING FOR LIQUIDS IN HOMOGENEOUS, ISOTROPIC MEDIA, EXAMPLE 6-3 STEADY-STATE GAS FLOWS IN HOMOGENEOUS, ISOTROPIC MEDIA, TRANSIENT COMPRESSIBLE FLOWS, EXAMPLE 6-4 NUMERICAL SOLUTION FOR STEADY FLOW, EXAMPLE 6-5 EXPLICIT AND IMPLICIT SCHEMES FOR TRANSIENT COMPRESSIBLE LIQUIDS, EXAMPLE 6-6 TRANSIENT COMPRESSIBLE GAS FLOWS, 7 FINITE DIFFERENCE METHODS FOR PLANAR FLOWS, FINITE DIFFERENCES: BASIC CONCEPTS, FORMULATING STEADY FLOW PROBLEMS, STEADY FLOW PROBLEMS: SEVEN CASE STUDIES, ISOTROPY AND ANISOTROPY: FLUID INVASION IN CROSS-BEDDED SANDS, 8 CURVILINEAR COORDINATES AND NUMERICAL GRID GENERATION, GENERAL COORDINATE TRANSFORMATIONS, THOMPSON’S MAPPING, SOME RECIPROCITY RELATIONS, CONFORMAL MAPPING REVISITED, SOLUTION OF MESH GENERATION EQUATIONS, 9 STEADY-STATE RESERVOIR APPLICATIONS, GOVERNING EQUATIONS, STEADY AREAL FLOW: GENERALIZED LOG R SOLUTION, STREAMLINE TRACING IN CURVILINEAR COORDINATES, CALCULATED STEADY FLOW EXAMPLES, EXAMPLE 9-1 WELL IN HOUSTON, EXAMPLE 9-2 WELL IN DALLAS, EXAMPLE 9-3 WELL IN CENTER OF TEXAS, EXAMPLE 9-4 FRACTURE ACROSS TEXAS, EXAMPLE 9-5 ISOTHERMAL AND ADIABATIC GAS FLOWS, MESH GENERATION: SEVERAL REMARKS, 10 TRANSIENT COMPRESSIBLE FLOWS: NUMERICAL WELL TEST SIMULATION, EXAMPLE 10-1 TRANSIENT PRESSURE DRAWDOWN, EXAMPLE 10-2 TRANSIENT PRESSURE BUILDUP, 11 EFFECTIVE PROPERTIES IN SINGLE AND MULTIPHASE FLOWS, EXAMPLE 11-1 CONSTANT DENSITY LIQUID IN STEADY LINEAR FLOW, EXAMPLE 11-2 LINEAL MULTIPHASE FLOW IN TWO SERIAL CORES, EXAMPLE 11-3 EFFECTIVE PROPERTIES IN STEADY CYLINDRICAL FLOWS, EXAMPLE 11-4 STEADY, SINGLE-PHASE, HETEROGENEOUS FLOWS, EXAMPLE 11-5 TIME SCALE FOR COMPRESSIBLE TRANSIENTS, 12 MODELING STOCHASTIC HETEROGENEITIES, OBSERVATIONS ON EXISTING MODELS, A MATHEMATICAL STRATEGY, EXAMPLE 12-1 CONTRACTIONAL FRACTURES, 13 REAL AND ARTIFICIAL VISCOSITY, REAL VISCOSITY AND SHOCKWAVES, ARTIFICIAL VISCOSITY AND FICTITIOUS JUMPS, 14 BOREHOLE FLOW INVASION, LOST CIRCULATION, AND TIME LAPSE LOGGING, BOREHOLE INVASION MODELING, EXAMPLE 14-1 THIN LOSSY MUDS (THAT IS, WATER), EXAMPLE 14-2 TIME-DEPENDENT PRESSURE DIFFERENTIALS, EXAMPLE 14-3 INVASION WITH MUDCAKE EFFECTS, TIME LAPSE LOGGING, LOST CIRCULATION, 15 HORIZONTAL, DEVIATED, AND MODERN MULTILATERAL WELL ANALYSIS, FUNDAMENTAL ISSUES AND PROBLEMS, GOVERNING EQUATIONS AND NUMERICAL FORMULATION, EXAMPLE CALCULATIONS, EXAMPLE 15-1 CONVERGENCE ACCELERATION, TWO DEVIATED HORIZONTAL GAS WELLS IN A CHANNEL SAND, EXAMPLE 15-2 DUAL-LATERAL HORIZONTAL COMPLETION IN A FRACTURED, DIPPING, HETEROGENEOUS, LAYERED FORMATION, EXAMPLE 15-3 STRATIGRAPHIC GRIDS, DRILLING DOME-SHAPED STRUCTURES, EXAMPLE 15-4 SIMULATING-WHILE-DRILLING HORIZONTAL GAS WELLS THROUGH A DOME-SHAPED RESERVOIR, EXAMPLE 15-5 MODELING WELLBORE STORAGE EFFECTS AND COMPRESSIBLE BOREHOLE FLOW TRANSIENTS, 16 FLUID MECHANICS OF INVASION, QUALITATIVE IDEAS ON FORMATION INVASION, BACKGROUND LITERATURE, DARCY RESERVOIR FLOW EQUATIONS, MOVING FRONTS AND INTERFACES, 17 STATIC AND DYNAMIC FILTRATION, SIMPLE FLOWS WITHOUT MUDCAKE, FLOWS WITH MOVING BOUNDARIES, COUPLED DYNAMICAL PROBLEMS: MUDCAKE AND FORMATION INTERACTION, DYNAMIC FILTRATION AND BOREHOLE FLOW RHEOLOGY, CONCENTRIC POWER LAW FLOWS WITHOUT PIPE ROTATION, CONCENTRIC POWER LAW FLOWS WITH PIPE ROTATION, FORMATION INVASION AT EQUILIBRIUM MUDCAKE THICKNESS, DYNAMIC FILTRATION IN ECCENTRIC BOREHOLES, 18 FORMATION TESTER APPLICATIONS, 19 ANALYTICAL METHODS FOR TIME LAPSE WELL LOGGING ANALYSIS, EXPERIMENTAL MODEL VALIDATION, CHARACTERIZING MUDCAKE PROPERTIES, POROSITY, PERMEABILITY, OIL VISCOSITY, AND PORE PRESSURE DETERMINATION, EXAMPLES OF TIME LAPSE ANALYSIS, 20 COMPLEX INVASION PROBLEMS: NUMERICAL MODELING, FINITE DIFFERENCE MODELING, EXAMPLE 20-1 LINEAL LIQUID DISPLACEMENT WITHOUT MUDCAKE, EXAMPLE 20-2 CYLINDRICAL RADIAL LIQUID DISPLACEMENT WITHOUT CAKE, EXAMPLE 20-3 SPHERICAL RADIAL LIQUID DISPLACEMENT WITHOUT CAKE, EXAMPLE 20-4 LINEAL LIQUID DISPLACEMENT WITHOUT MUDCAKE, INCLUDING COMPRESSIBLE FLOW TRANSIENTS, EXAMPLE 20-5 VON NEUMANN STABILITY OF IMPLICIT TIME SCHEMES, EXAMPLE 20-6 GAS DISPLACEMENT BY LIQUID IN LINEAL CORE WITHOUT MUDCAKE, INCLUDING COMPRESSIBLE FLOW TRANSIENTS, EXAMPLE 20-7 SIMULTANEOUS MUDCAKE BUILDUP AND DISPLACEMENT FRONT MOTION FOR INCOMPRESSIBLE LIQUID FLOWS, 21 FORWARD AND INVERSE MULTIPHASE FLOW MODELING, IMMISCIBLE BUCKLEY-LEVERETT LINEAL FLOWS WITHOUT CAPILLARY PRESSURE, MOLECULAR DIFFUSION IN FLUID FLOWS , IMMISCIBLE RADIAL FLOWS WITH CAPILLARY PRESSURE AND PRESCRIBED MUDCAKE GROWTH, IMMISCIBLE FLOWS WITH CAPILLARY PRESSURE AND DYNAMICALLY COUPLED MUDCAKE GROWTH, CUMULATIVE REFERENCES,