Mathematical and Numerical Modeling in Porous Media

About the book series Editorial board of the book series Preface Acknowledgements About the editors Contributors Section 1: Fundamental concepts 1 Relative permeability (T.J.T. Spanos) 1.1 Introduction 1.2 Darcy's equation 1.3 Heterogeneity 1.4 Lubrication theory 1.5 Multiphase flow in porous media 1.6 Dispersion 1.7 Few comments about the associated thermodynamics 1.8 Conclusions 1.A Appendix 1.A.1 Solid properties 1.A.2 Fluid properties 1.A.3 Reciprocity References 2 From upscaling techniques to hybrid models (I. Battiato & D.M. Tartakovsky) 2.1 Introduction 2.2 From first principles to effective equations 2.2.1 Classification of upscaling methods 2.2.2 Flow: From Stokes to Darcy/Brinkman equations 2.2.3 Transport: From advection-diffusion to advection-dispersion equation 2.3 Applicability range of macroscopic models for reactive systems 2.3.1 Diffusion-reaction equations: mixing-induced precipitation processes 2.3.2 Preliminaries 2.3.3 Upscaling via volume averaging 2.3.4 Advection-diffusion-reaction equation 2.4 Hybrid models for transport in porous media 2.4.1 Intrusive hybrid algorithm 2.4.2 Taylor dispersion in a fracture with reactive walls 2.4.3 Hybrid algorithm 2.4.4 Numerical results 2.4.5 Non-intrusive hybrid algorithm 2.5 Conclusions References 3 A tensorial formulation in four dimensions of thermoporoelastic phenomena (M.C. Suarez Arriaga) 3.1 Introduction 3.2 Theoretical and experimental background 3.3 Model of isothermal poroelasticity 3.4 Thermoporoelasticity model 3.5 Dynamic poroelastic equations 3.6 The finite element method in the solution of the thermoporoelastic equations 3.7 Solution of the model for particular cases 3.8 Discussion of results 3.9 Conclusions References Section 2: Flow and transport 4 New method for estimation of physical parameters in oil reservoirs by using tracer test flow models in Laplace space (J. Ramirez-Sabag, O.C. Valdiviezo-Mijangos & M. Coronado) 4.1 Introduction 4.2 Numerical laplace transformation of sample data 4.3 The laplace domain optimization procedure 4.4 The real domain optimization procedure 4.5 The optimization method 4.6 The validation procedure 4.6.1 Employed mathematical models 4.6.2 Generation of synthetic data 4.6.3 Result with synthetic data 4.7 Reservoir data cases 4.7.1 A homogeneous reservoir (Loma Alta Sur) 4.7.2 A fractured reservoir (Wairakei field) 4.8 Summary and concluding remarks References 5 Dynamic porosity and permeability modification due to microbial growth using a coupled flow and transport model in porous media (M.A. Diaz-Viera &A. Moctezuma-Berthier) 5.1 Introduction 5.2 The flow and transport model 5.2.1 Conceptual model 5.2.2 Mathematical model 5.2.3 Numerical model 5.2.4 Computational model 5.3 Numerical simulations 5.3.1 Reference study case description: a waterflooding test in a core 5.3.2 Modeling of secondary recovery by water injection 5.3.3 Modeling of enhanced recovery by water injection with microorganisms and nutrients 5.3.4 Porosity and permeability modification due to microbial activity 5.4 Final remarks References 6 Inter-well tracer test models for underground formations having conductive faults: development of a numerical model and comparison against analytical models (M. Coronado, J. Ramirez-Sabag & O. Valdiviezo-Mijangos) 6.1 Introduction 6.2 Description of the analytical models 6.2.1 The closed fault model 6.2.2 The open fault model 6.3 The numerical model 6.4 Numerical results 6.5 Comparison of the analytical models against numerical simulations 6.5.1 Injection-dominated flow case 6.5.2 Fault-dominated flow case 6.5.3 Closed fault case 6.6 Summary and final conclusions References 7 Volume average transport equations for in-situ combustion (A.G. Vital-Ocampo & O. Cazarez-Candia) 7.1 Introduction 7.2 Study system 7.2.1 Local mass, momentum and energy equations 7.2.2 Jump conditions 7.3 Average volume 7.4 Average equations 7.5 Physical model 7.6 Equations for in-situ combustion 7.7 Numerical solution 7.8 Solution 7.9 Results 7.10 Conclusions 7.A Appendix 7.A.1 Oil vaporization References 8 Biphasic isothermal tricomponent model to simulate advection-diffusion in 2D porous media (A. Moctezuma-Berthier) 8.1 Introduction 8.2 Model description 8.2.1 General considerations 8.2.2 Mathematical model 8.2.3 Numerical model 8.2.4 Solution of the system 8.2.5 Management of the partials derivatives 8.2.6 Solution scheme 8.2.7 Treating the boundary conditions 8.2.8 Initial conditions for the fluid flow and the tracer systems 8.3 Validation of biphasic flow system 8.4 Conclusions References Section 3: Statistical and stochastic characterization 9 A 3D geostatistical model of Upper Jurassic Kimmeridgian facies distribution in Cantarell oil field, Mexico (R. Casar-Gonzalez, M.A. Diaz-Viera, G. Murillo-Muneton, L. Velasquillo-Martinez, J. Garcia-Hernandez & E. Aguirre-Cerda) 9.1 Introduction 9.2 Methodological aspects of geological and petrophysical modeling 9.2.1 The geological model 9.2.2 The petrophysical model 9.3 Conceptual geological model 9.3.1 Geological setting 9.3.2 Sedimentary model and stratigraphic framework 9.3.3 The conceptual geological model definition 9.3.4 Analysis of the structural sections 9.3.5 Description of the stratigraphic correlation sections 9.3.6 Lithofacies definition 9.4 Geostatistical modeling 9.4.1 Zone partition 9.4.2 Stratigraphic grid definition 9.4.3 CA facies classification 9.4.4 Facies upscaling process 9.4.5 Statistical analysis 9.4.6 Geostatistical simulations 9.5 Conclusions References 10 Trivariate nonparametric dependence modeling of petrophysical properties (A. Erdely, M.A. Diaz-Viera &V. Hernandez-Maldonado) 10.1 Introduction 10.1.1 The problem of modeling the complex dependence pattern between porosity and permeability in carbonate formations 10.1.2 Trivariate copula and random variables dependence 10.2 Trivariate data modeling 10.3 Nonparametric regression 10.4 Conclusions References 11 Joint porosity-permeability stochastic simulation by non-parametric copulas (V. Hernandez-Maldonado, M.A. Diaz-Viera &A. Erdely-Ruiz) 11.1 Introduction 11.2 Non-conditional stochastic simulation methodology by using Bernstein copulas 11.3 Application of the methodology to perform a non-conditional simulation with simulated annealing using bivariate Bernstein copulas 11.3.1 Modeling the petrophysical properties dependence pattern, using non-parametric copulas or Bernstein copulas 11.3.2 Generating the seed or initial configuration for simulated annealing method, using the non-parametric simulation algorithm 11.3.3 Defining the objective function 11.3.4 Measuring the energy of the seed, according to the objective function 11.3.5 Calculating the initial temperature, and the most suitable annealing schedule of simulated annealing method to carry out the simulation 11.3.6 Performing the simulation 11.3.7 Application of the methodology for stochastic simulation by bivariate Bernstein copulas to simulate a permeability (K) profile. A case of study 11.4 Comparison of results using three different methods 11.4.1 A single non-conditional simulation, and a median of 10 non-conditional simulations of permeability 11.4.2 A single 10% conditional simulation, and a median of 10, 10% conditional simulations of permeability 11.4.3 A single 50% conditional simulation, and a median of 10, 50% conditional simulations of permeability 11.4.4 A single 90% conditional simulation, and a median of 10, 90% conditional simulations of permeability 11.5 Conclusions References 12 Stochastic simulation of a vuggy carbonate porous media (R. Casar-Gonzalez &V. Suro-Perez) 12.1 Introduction 12.2 X-ray computed tomography (CT) 12.3 Exploratory data analysis of X-Ray computed tomography 12.4 Transformation of the information from porosity values to indicator variable 12.5 Spatial correlation modeling of the porous media 12.6 Stochastic simulation of a vuggy carbonate porous media 12.7 Simulation annealing multipoint of a vuggy carbonate porous media 12.8 Simulation of continuous values of porosity in a vuggy carbonate porous medium 12.9 Assigning permeability values based on porosity values 12.10 Application example: effective permeability scaling procedure in vuggy carbonate porous media 12.11 Scaling effective permeability with average power technique 12.12 Scaling effective permeability with percolation model 12.13 Conclusions and remarks References 13 Stochastic modeling of spatial grain distribution in rock samples from terrigenous formations using the plurigaussian simulation method (J. Mendez-Venegas & M.A. Diaz-Viera) 13.1 Introduction 13.2 Methodology 13.2.1 Data image processing 13.2.2 Geostatistical analysis 13.3 Description of the data 13.4 Geostatistical analysis 13.4.1 Exploratory data analysis 13.4.2 Variographic analysis 13.5 Results 13.6 Conclusions References 14 Metadistances in prime numbers applied to integral equations and some examples of their possible use in porous media problems (A. Ortiz-Tapia) 14.1 Introduction 14.1.1 Some reasons for choosing integral equation formulations 14.1.2 Discretization of an integral equation with regular grids 14.1.3 Solving an integral equation with MC or LDS 14.2 Algorithms description 14.2.1 Low discrepancy sequences 14.2.2 Halton LDSs 14.2.3 What is a "metadistance" 14.2.4 Refinement of mds 14.3 Numerical experiments 14.3.1 Fredholm equations of the second kind in one integrable dimension 14.3.2 Results in one dimension 14.3.3 Choosing a problem in two dimensions 14.3.4 Transformation of the original problem 14.3.5 General numerical algorithm 14.3.6 MC results, empirical rescaling 14.3.7 Halton results, empirical rescaling 14.3.8 MDs results, empirical rescaling 14.3.9 MC results, systematic rescaling 14.3.10 Halton results, systematic rescaling 14.3.11 MDs results, systematic rescaling 14.3.12 Accuracy goals 14.3.13 Rate of convergence 14.4 Conclusions References Section 4:Waves 15 On the physical meaning of slow shear waves within the viscosity-extended Biot framework (T.M. Muller & P.N. Sahay) 15.1 Introduction 15.2 Review of the viscosity-extended biot framework 15.2.1 Constitutive relations, complex phase velocities, and characteristic frequencies 15.2.2 Properties of the slow shear wave 15.3 Conversion scattering in randomly inhomogeneous media 15.3.1 Effective wave number approach 15.3.2 Attenuation and dispersion due to conversion scattering in the slow shear wave 15.4 Physical interpretation of the slow shear wave conversion scattering process 15.4.1 Slow shear conversion mechanism as a proxy for attenuation due to vorticity diffusion within the viscous boundary layer 15.4.2 The slow shear wave conversion mechanism versus the dynamic permeability concept 15.5 Conclusions 15.A Appendix 15.A.1 alpha and ss matrices 15.A.2 Inertial regime References 16 Coupled porosity and saturation waves in porous media (N. Udey) 16.1 Introduction 16.2 The governing equations 16.2.1 Variables and definitions 16.2.2 The equations of continuity 16.2.3 The equations of motion 16.2.4 The porosity and saturation equations 16.3 Dilatational waves 16.3.1 The Helmholtz decomposition 16.3.2 The dilatational wave equations 16.3.3 The dilatational wave operator matrix equation 16.3.4 Wave operator trial solutions 16.4 Porosity waves 16.4.1 The porosity wave equation 16.4.2 The dispersion relation 16.4.3 Comparison with pressure diffusion 16.5 Saturation waves 16.5.1 The wave equations 16.5.2 The dispersion relation 16.6 Coupled porosity and saturation waves 16.6.1 The dispersion relation 16.6.2 Factorization of the dispersion relation 16.7 A numerical illustration 16.7.1 The porosity wave 16.7.2 The saturation wave 16.8 Conclusion References Subject index Book series page