Table of ContentsCHAPTER 1- Modeling in Chemical and Subsurface Energy Resources Engineering 1.1 Introduction1.2 Motivation1.3 Development of mathematical models1.4 Modeling simplification1.5 Organization of the book and guidance for readers1.6 SummaryCHAPTER 2-Fundamentals of Mathematical Modeling 2.1 Introduction2.2 Basics2.3 Fundamental laws2.4 Coordinate systems for heat, energy, and momentum balances2.5 Lumped versus distributed models§ Example 2.1: Lumped heat transfer analysis of a hot metallic ball§ Example 2.2: A continuous stirred tank (CST)§ Example 2.3: Tubular reactor§ Example 2.4: Setting boundary conditions for composite media§ Example 2.5: Scaling analysis2.6 References for further reading2.7 Additional problemsCHAPTER 3 -Review of Ordinary Differential Equations 3.1 Introduction3.2 Definitions3.3 Integrating factor method3.4 Bernoulli’s ordinary differential equation§ Example 3.1: Autocatalytic reaction§ Example 3.2: Bernoulli’s ODE3.5 Riccati’s ordinary differential equation§ Example 3.3: Riccati’s ordinary differential equation3.6 Second-order ordinary differential equations§ Example 3.4: Application of the method of undetermined coefficients§ Example 3.5: Application of the method of undetermined coefficients§ Example 3.6: Application of the method of variation of parameters3.7 Laplace transform§ Example 3.7: Laplace transform of a time domain function§ Example 3.8: Converting a Laplace domain function into partial fractions§ Example 3.9: Laplace transform of a time domain function§ Example 3.10: Inversion to the time domain§ Example 3.11: Application of the shift theorem§ Example 3.12: Laplace transform of discontinuous forcing function§ Example 3.13: Laplace transform of discontinuous forcing function§ Example 3.14: Solution of an IVP by Laplace transform§ Example 3.15: Solution of an IVP by Laplace transform§ Example 3.16: Solution of an IVP with a discontinuous forcing function§ Example 3.17: Solution of an IVP with Dirac delta function forcing function§ Example 3.18: Application of the convolution integral in solving IVPs§ Example 3.19: Application of convolution in solving integro-differential IVPs3.8 Cauchy–Euler equation§ Example 3.20: Solution to a non-homogeneous Cauchy–Euler equation3.9 Higher-order ordinary differential equations§ Example 3.21: Solution to a higher-order ODE§ Example 3.22: Solution to a higher-order ODE3.10 References for further reading3.11 Additional problemsCHAPTER 4 -Boundary Value Problems 4.1 Introduction4.2 Regular Sturm-Liouville problem4.3 Singular boundary value problems§ Example 4.1: Determining the type of BVPs4.4 Periodic boundary value problems4.5 Orthogonality property4.6 Boundary conditions4.7 Determination of eigenvalues and eigenfunctions for various boundary conditions4.8 A general note on the sign of eigenvalues4.9 Evaluation of norm for y” +ly=0 subject to various boundary conditions4.10 References for further reading4.11 Additional problemsCHAPTER 5- Separation of Variables 5.1 Classification of partial differential equations5.2 Outline5.3 Separation of variables for homogeneous PEDs with homogeneous boundary conditions§ Example 5.1: Heat conduction in a solid with Dirichlet boundary conditions§ Example 5.2: Heat conduction in a solid with Neumann boundary conditions§ Example 5.3: Heat conduction (or diffusion) problem with periodic boundary conditions§ Example 5.4: Diffusion and reaction in solids and porous materials-Robin BCs§ Example 5.5: Drying and hydration in a porous concrete slab§ Example 5.6: Heat conduction in a slab with a convective boundary condition5.4 Separation of variables for homogeneous PEDs with non-homogeneous boundary conditions§ Example 5.7: Heat conduction in a solid with non-homogeneous BCs§ Example 5.8: Pressure diffusion in a 1D reservoir with non-homogeneous BCs§ Example 5.9: Transient heat conduction with convective boundary conditions§ Example 5.10: Transient heat conduction with convective boundary conditions§ Example 5.11Diffusive mixing of CO2 in water for carbon storage applications§ Example 5.12: CO 2diffusion and adsorption in tight shale§ Example 5.13: Simplified diffusion model for CO2 leakage§ Example 5.14: Base state solution relevant to viscous fingering in porous media§ Example 5.15: Transient methane diffusion and degradation§ Example 5.16: Pressure diffusion in one-dimensional porous media5.5 Separation of variables for non-homogeneous PEDs§ Example 5.17: Conduction with uniform heat generation§ Example 5.18: Conduction with uniform heat generation and convective boundary§ Example 5.19: Space-dependent source models in heat conduction§ Example 5.20: Heat conduction with spatially varying internal heating in a closed domain5.6 Application of separation of variables to multi-dimensional diffusion-type problems§ Example 5.21: Diffusion/conduction-reaction in a 2D domain§ Example 5.22: Diffusion/conduction-reaction in a 3D domain5.7 References for further reading5.8 Additional problemsCHAPTER 6 ¾ Eigenfunction Expansion Method6.1 Introduction6.2 Outline6.3 Application of the method of eigenfunction expansion§ Example 6.1: Heat conduction with a non-homogeneous source§ Example 6.2: Pressure diffusion in porous media§ Example 6.3: Heat conduction with space-dependent internal heating§ Example 6.4: Transient heating from a linearly distributed, decaying internal heat source§ Example 6.5 Heat conduction with internal heat sources of various forms§ Example 6.6: Diffusion and chemical reaction with an instantaneous plane solute source in a closed system§ Example 6.7: Diffusive spread and degradation of a plane source pollutant6.4 References for further reading6.5 Additional problemsCHAPTER 7- Advection-Diffusion-Reaction Problems 7.1 Introduction7.2 A note on boundary conditions for advection-diffusion-reaction type problems7.3 Boundary value problems of advection-diffusion-reaction7.4 Application of separation of variables to advection-diffusion-reaction problems§ Example 7.1: Advection-diffusion transport in porous media with Dirichlet BCs§ Example 7.2: Advection-diffusion-reaction transport in porous media§ Example 7.3: Reduced-order modeling of gas absorption in a viscous liquid film§ Example 7.4: Reinfiltration in fractured porous media§ Example 7.5 Solute transport with dispersion and chemical reactions in porous media§ Example 7.6: Gas adsorption in packed columns7.5 Solution of advection-diffusion-reaction problems using transformation7.6 References for further reading7.7 Additional problemsCHAPTER 8-Steady-State Problems 8.1 Introduction8.2 Solution of the Laplace equation§ Example 8.1: Temperature distribution in a 2D rectangular domain§ Example 8.2: Steady-state conduction in a 2D domain with first- and second-type BCs§ Example 8.3: Steady-State heat conduction in a rectangular plate with adjoining BCs§ Example 8.4: Steady-State heat conduction in a rectangular plate without adjoining BCs§ Example 8.5 Pressure-driven flow in a Hele-Shaw cell8.3 Solution of the Poisson equation§ Example 8.6: Poisson equation in a rectangular domain§ Example 8.7: Steady-state heat conduction in a rectangular bar with internal heat generation8.4 Laplace equation in cylindrical coordinates§ Example 8.8: Steady-state heat conduction in cylindrical coordinates8.5 References for further reading8.6 Additional problemsCHAPTER 9- Diffusion Problems Involving Bessel Functions 9.1 Introduction9.2 Bessel functions9.3 Airy ordinary differential equation9.4 The general form of diffusion (conduction) in cylindrical polar coordinates§ Example 9.1: Transient radial heat conduction in a solid cylinder with constant temperature at the outer boundary§ Example 9.2: Transient radial heat conduction in a solid cylinder with external heating§ Example 9.3: Transient mass diffusion in a drying ceramic rod§ Example 9.4 Transient radial heat conduction in a long cylindrical shell with a Dirichlet BC§ Example 9.5 Transient heat conduction in a shell with convective inner and insulated outer boundaries§ Example 9.6: Transient pressure diffusion in a bounded reservoir with a constant flux well§ Example 9.7: Radial pressure diffusion in a bounded aquifer surrounding an oil reservoir§ Example 9.8: Heat and mass diffusion in a quarter-cylinder geometry§ Example 9.9: Mass transfer in a half-cylinder geometry with a convective outer boundary§ Example 9.10: Steady-state moisture diffusion in a semi-cylindrical material during drying§ Example 9.11: 3D Steady-state heat conduction in a solid cylinder§ Example 9.12: Counter-current imbibition in a cylindrical matrix block§ Example 9.13: Water saturation dynamics in a cylindrical water-wet rock sample§ Example 9.14: Transient moisture diffusion in a semi-cylindrical shell9.5 Useful formulas9.6 References for further reading9.7 Additional problemsCHAPTER 10- Diffusion and Conduction in a Sphere 10.1 Introduction10.2 The general form of diffusion (conduction) in spherical coordinates§ Example 10.1: Diffusion in a solid sphere with a Dirichlet-type outer boundary§ Example 10.2: Surfactant diffusion and adsorption kinetics§ Example 10.3: Nutrient release from coated fertilizer granules (spherical shell)10.3 References for further reading10.4 Additional problemsCHAPTER 11 - Wave Equation 11.1 Introduction11.2 Wave equation and boundary conditions11.3 Application of the separation of variables to the wave equation§ Example 11.1: Wave equation with Dirichlet boundaries§ Example 11.2: Wave equation with Neumann boundaries§ Example 11.3: Vibrations of a rectangular drumhead§ Example 11.4: Vibrations of a circular drumhead§ Example 11.5: Wave dynamics in a semi-cylindrical drum§ Example 11.6: Wave propagation in a circular drumhead§ Example 11.7: Wave propagation in a drumhead with a Neumann BC at the outer radius§ Example 11.8: Wave propagation in a drumhead with a Neumann BC at the outer radius and general initial conditions11.4 References for further reading11.5 Additional problemsCHAPTER 12- Laplace Transform 12.1 Introduction12.2 Preliminaries12.3 Application of the Laplace transform§ Example 12.1: Solute diffusion in a thin film§ Example 12.2: Diffusion of contaminants in soil with constant surface concentration§ Example 12.3: Linear flow in hydraulically fractured wells and aquifers§ Example 12.4: Mass transfer through a polymer membrane§ Example 12.5: Conduction in a slab with sinusoidal IC and fixed-temperature boundaries§ Example 12.6: 1D Wave equation in a semi-infinite domain§ Example 12.7: Heat conduction in a metal rod with periodic heating§ Example 12.8: Diffusional mass transfer in carbonated water flooding§ Example 12.9: Pressure diffusion during water injection into a linear semi-infinite aquifer§ Example 12.10: Diffusion-deposition of nanoparticles for controlling viscous fingering§ Example 12.11: Reinfiltration in fractured porous media§ Example 12.12: Molecular diffusion coefficients in bulk liquids and porous media§ Example 12.13: Molecular diffusion coefficients using the pressure decay test§ Example 12.14: Transient diffusional mass transfer from gas bubbles into stagnant liquid§ Example 12.15: Transient diffusional mass transfer from a gas phase into a liquid droplet§ Example 12.16: Self-diffusion and adsorption kinetics of surfactants§ Example 12.17: Simultaneous measurement of diffusion in bulk liquids and porous media§ Example 12.18: A reduced-order model of advection-dispersion-reaction for gas absorption into a viscous liquid film§ Example 12.19: Advection-diffusion-reaction with various applications§ Example 12.20: Pressure diffusion in a bounded radial aquifer for water-influx calculation in oil and gas reservoirs§ Example 12.21: Steady-state heat conduction in a fixed-bed catalytic reactor12.4 Inverse Laplace transform using the residue theorem§ Example 12.22: 1D transient heat conduction with an instantaneous heat pulse§ Example 12.23: Diffusion of a solute in a thin gel layer§ Example 12.24: Dissolution kinetics and interphase mass transfer of CO2from carbonated water into oil droplets§ Example 12.25: Matrix-fracture transfer in fractured reservoirs with constant flux§ Example 12.26: Matrix-fracture transfer in fractured reservoirs with linear pressure decline§ Example 12.27: Molecular diffusion of gases in liquids with interfacial resistance§ Example 12.28: 1D wave equation with Dirichlet BCs§ Example 12.29: 1D wave equation with time-dependent BC12.5 Numerical Laplace inversion methods12.6 References for further reading12.7 Additional problemsCHAPTER 13- Application of Duhamel’s Theorem 668-75713.1 Introduction13.2 Duhamel’s theorem13.3 Application to PDEs with time-dependent non-homogeneity§ Example 13.1: Pressure diffusion in a slab-shaped matrix block§ Example 13.2: Diffusion-reaction with Neumann boundary conditions at both ends§ Example 13.3: Heat extraction from fractured geothermal reservoirs§ Example 13.4: Transient moisture diffusive transport with surface evaporation§ Example 13.5: Counter-current imbibition in porous media§ Example 13.6: Transient heat conduction in fractured geothermal reservoirs§ Example 13.7: Conduction with decaying sinks and sources§ Example 13.8: Diffusion in biofilms for wastewater treatment§ Example 13.9: Reinfiltration in fractured porous media13.4 Application of Duhamel’s theorem with a jump boundary condition§ Example 13.10: Pressure diffusion in a linear aquifer during water disposal with a step change in injection rate§ Example 13.11: Pressure transient in a vertical well intersecting a hydraulic fracture§ Example 13.12: Two-rate pressure transient testing§ Example 13.13: Diffusion with gradual and sudden concentration changes at the boundary in a semi-infinite domain§ Example 13.14: Transient heat conduction in a finite-length solid with jump boundary conditions13.5 Notes13.6 References for further reading13.7 Additional problemsCHAPTER 14- Fourier Transform 758-82814.1 Introduction14.2 Fourier transform for infinite domains§ Example 14.1: Diffusion with exponentially decaying IC in an infinite domain§ Example 14.2: Diffusion in an infinite domain with an IC described by the Heaviside function§ Example 14.3: Diffusion in an infinite domain with an IC described by the Dirac delta function14.3 Fourier transform for semi-infinite domains§ Example 14.4: Diffusion/conduction in a semi-infinite domain with a Dirichlet BC§ Example 14.5: Diffusion in a semi-infinite domain with Dirac delta IC and Dirichlet BC§ Example 14.6: Diffusion in a semi-infinite domain with Heaviside IC and Dirichlet BC§ Example 14.7: Diffusion in a semi-infinite domain with Dirac delta IC and Neumann BC§ Example 14.8: Diffusion in a semi-infinite domain with Heaviside IC and Neumann BC§ Example 14.9: Pressure transient of a hydraulically fractured well with a constant rate§ Example 14.10: Pressure transient of fractured wells-Transformation of Neumann to Dirichlet§ Example 14.11: Gas exsolution from liquids -Transformation of Robin to Dirichlet§ Example 14.12: Diffusional mass transfer with interface resistance in gas absorption and carbon capture-Transformation of Robin to Dirichlet boundary condition14.4 Fourier transform for finite domains§ Example 14.13: 1D heat diffusion for thermal conductivity measurement§ Example 14.14: 1D transient heat conduction with an instantaneous heat pulse§ Example 14.15: Counter-current capillary imbibition in water-wet reservoirs§ Example 14.16: Measurement of the CO2 diffusion coefficient for carbon storage§ Example 14.17: Measurement of the CO2 diffusion coefficient using a closed chamber§ Example 14.18: Pressure diffusion in a slab-shaped matrix block in naturally fractured reservoirs subject to constant flux§ Example 14.19: Gas flow in bounded 2D porous media with source term§ Example 14.20: Steady laminar flow in a rectangular microfluidic channel14.5 References for further reading14.6 Additional problemsCHAPTER 15 -Green’s Function Method 829-90615.1 Introduction15.2 Solution of non-homogeneous time-dependent diffusion-type problems15.3 Application of Green’s function technique§ Example 15.1:1D diffusion equation in an infinite domain with source/sink§ Example 15.2: 1D diffusion equation in a semi-infinite domain§ Example 15.3: 1D pressure diffusion equation in a semi-infinite medium-Neumann BC§ Example 15.4: CO2 diffusion and adsorption in gas shale§ Example 15.5: Conduction with a spatially varying source§ Example 15.6: Solute transport in membrane separation processes§ Example 15.7: Moisture diffusion and evaporation in drying processes§ Example 15.8: Diffusion/conduction and reaction with Neumann BC at both ends§ Example 15.9: Transient heat conduction in a solid cylinder with external heating§ Example 15.10: Pressure diffusion in a bounded reservoir with constant flux production§ Example 15.11: Pressure diffusion in a bounded radial aquifer§ Example 15.12: Transient CO2 diffusion and reaction in a porous spherical sorbent§ Example 15.13: Controlled drug release and degradation in biodegradable microspheres§ Example 15.14: Controlled release in biodegradable microspheres15.4 Notes15.5 References for further reading15.6 Additional problemsCHAPTER 16-Integral Transform Method 907-95916.1 Introduction16.2 Integral transform16.3 Application of the integral transform§ Example 16.1: Pressure diffusion in single-phase fluid flow in a confined domain§ Example 16.2: 1D diffusion in a finite domain with Neumann boundary conditions§ Example 16.3: Conduction with decaying sources and sinks§ Example 16.4: Transient radial heat conduction in a solid cylinder with external heating§ Example 16.5: Pressure diffusion in a bounded radial aquifer for water influx calculations in oil and gas reservoirs§ Example 16.6: Moisture diffusion and evaporation in the drying of a porous solid cylinder§ Example 16.7: Transient CO2 diffusion and reaction in a porous spherical sorbent for carbon capture§ Example 16.8: Heat conduction in a spherical shell with time-dependent heat generation§ Example 16.9: Counter-current imbibition in a cylindrical matrix block with a time-dependent boundary condition§ Example 16.10: Cooling of a cylindrical object in a fluid16.4 References for further reading16.5 Additional problemsCHAPTER 17 - Similarity Method 960-100617.1 Introduction17.2 Determination of the similarity variable§ Example 17.1: One-dimensional diffusion equation with Dirichlet BC§ Example 17.2: One-dimensional diffusion equation with Neumann BC§ Example 17.3: One-dimensional diffusion equation with Robin BC§ Example 17.4: One-dimensional radial diffusion equation with line source BC§ Example 17.5: Mass transfer and diffusion of CO2 from a gas bubble into the surrounding water with a Dirichlet-type BC§ Example 17.6: Gas diffusion from a porous-walled spherical cavity with Neumann BC§ Example 17.7: Gas diffusion from a porous-walled spherical cavity with Robin BC§ Example 17.8: Spherical point source that emits a substance proportional to t½§ Example 17.9: Counter-current imbibition in porous media with flux proportional to t-½§ Example 17.10: Graetz-type problem§ Example 17.11: Oseen–Lamb vortex§ Example 17.12: Gas diffusion-induced liquid swelling17.3 Useful scaling transformation for non-linear PDEs§ Example 17.13: Scaling transformation for a non-linear PDE§ Example 17.14: Scaling transformation for gas absorption with a second-order reaction17.4 References for further reading17.5 Additional problemsTable of Laplace Transform 1007-1011Table of Useful Series Equivalents 1012-1018
