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Mathematical and Numerical Modeling in Porous Media

Applications in Geosciences

Edited by Martin A. Diaz Viera, Pratap Sahay, Manuel Coronado, Arturo Ortiz Tapia

CRC Press – 2012 – 370 pages

Series: Multiphysics Modeling

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    978-0-415-66537-7
    July 24th 2012

Description

Porous media are broadly found in nature and their study is of high relevance in our present lives. In geosciences porous media research is fundamental in applications to aquifers, mineral mines, contaminant transport, soil remediation, waste storage, oil recovery and geothermal energy deposits. Despite their importance, there is as yet no complete understanding of the physical processes involved in fluid flow and transport. This fact can be attributed to the complexity of the phenomena which include multicomponent fluids, multiphasic flow and rock-fluid interactions. Since its formulation in 1856, Darcy’s law has been generalized to describe multi-phase compressible fluid flow through anisotropic and heterogeneous porous and fractured rocks. Due to the scarcity of information, a high degree of uncertainty on the porous medium properties is commonly present. Contributions to the knowledge of modeling flow and transport, as well as to the characterization of porous media at field scale are of great relevance. This book addresses several of these issues, treated with a variety of methodologies grouped into four parts:

I Fundamental concepts

II Flow and transport

III Statistical and stochastic characterization

IV Waves

The problems analyzed in this book cover diverse length scales that range from small rock samples to field-size porous formations. They belong to the most active areas of research in porous media with applications in geosciences developed by diverse authors.

This book was written for a broad audience with a prior and basic knowledge of porous media. The book is addressed to a wide readership, and it will be useful not only as an authoritative textbook for undergraduate and graduate students but also as a reference source for professionals including geoscientists, hydrogeologists, geophysicists, engineers, applied mathematicians and others working on porous media.

Contents

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. 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

References

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

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. 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

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-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

References

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

References

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

References

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

References

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

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. 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

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

Name: Mathematical and Numerical Modeling in Porous Media: Applications in Geosciences (Hardback)CRC Press 
Description: Edited by Martin A. Diaz Viera, Pratap Sahay, Manuel Coronado, Arturo Ortiz Tapia. Porous media are broadly found in nature and their study is of high relevance in our present lives. In geosciences porous media research is fundamental in applications to aquifers, mineral mines, contaminant transport, soil remediation, waste storage,...
Categories: Computational Numerical Analysis, Hydrogeology & Groundwater, Geophysics