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Mathematical and Numerical Modeling in Porous Media

Applications in Geosciences

Mathematical and Numerical Modeling in Porous Media by Martin A. Diaz Viera
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About the book series
Editorial board of the book series
About the editors

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

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

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

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. Ramírez-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

5 Dynamic porosity and permeability modification due to microbial growth using a coupled flow and transport model in porous media
(M.A. Díaz-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

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. Ramírez-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

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

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

Section 3: Statistical and stochastic characterization

9 A 3D geostatistical model of Upper Jurassic Kimmeridgian facies distribution in Cantarell oil field, Mexico
(R. Casar-González, M.A. Díaz-Viera, G. Murillo-Muñetón, L. Velasquillo-Martínez, J. García-Hernández & 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

10 Trivariate nonparametric dependence modeling of petrophysical properties
(A. Erdely, M.A. Díaz-Viera &V. Hernández-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

11 Joint porosity-permeability stochastic simulation by non-parametric copulas
(V. Hernández-Maldonado, M.A. Díaz-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

12 Stochastic simulation of a vuggy carbonate porous media
(R. Casar-González &V. Suro-Pérez)
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

13 Stochastic modeling of spatial grain distribution in rock samples from terrigenous formations using the plurigaussian simulation method
(J. Méndez-Venegas & M.A. Díaz-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

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

Section 4:Waves

15 On the physical meaning of slow shear waves within the viscosity-extended Biot framework
(T.M. Müller & 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 α and β matrices
15.A.2 Inertial regime

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

Subject index
Book series page


CRC Press; July 2012
369 pages; ISBN 9780203113882
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Title: Mathematical and Numerical Modeling in Porous Media
Author: Martin A. Diaz Viera; Pratap Sahay; Manuel Coronado; Arturo Ortiz Tapia
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