Finite Element Programming in Non-linear Geomechanics and Transient Flow

By Nobuo Morita

Finite Element Programming in Non-linear Geomechanics and Transient Flow delivers a textbook reference for both students and practitioners alike, with provided codes to understand and modify. Starting with the fundamentals, the reference covers the basics of finite element methods, including coupling geomechanics and transient fluid flow. The next phase moves from theory into practical application from programs Flow3D and Geo3D, utilizing source codes to solve real field challenges. Stability of perforations during oil and gas production, sand production problems, rock failure, casing collapse, and reservoir compaction problems are just some examples.

Next, the reference elevates to hands-on experience, sharing source codes with additional problems engineers can work on independently. This gives students and engineers a starting point to modify their own code in a fraction of the time.

• Helps users understand finite element programs such as Flow3D and Geo3D to solve geomechanics problems, including casing stability, reservoir compaction challenges, and sand production
• Bridges the gap between theory, applications and source codes to help readers develop or modify their own computer programs with provided source codes
• Includes cases studies and practice examples that illustrate real-world applications

• Language: English
• Publisher: Elsevier Science
• Release date: Aug 21, 2021
• ISBN: 9780323911139

Nobuo Morita

Nobuo Morita is a professor at the Harold Vance Department of Petroleum Engineering, Texas A&M University, College Station, Texas. He teaches courses on boundary element methods for application to petroleum engineering problems, non-linear mechanics and finite element methods for geomechanics. He is a supervising professor of the Texas A&M Geomechanics Joint Industry Project. He holds sand control, borehole stability and hydraulic fracturing workshops twice per year around the world and has provided consulting services in major oil companies around the world. He was previously employed by ConocoPhillips for 14 years.

Book preview

Finite Element Programming in Nonlinear Geomechanics and Transient Flow

Nobuo Morita

Table of Contents

Cover image

Title page

Copyright

Introduction

I: Basics of the finite element method

Chapter 1. Fundamental equations of poro-elasticity and fluid flow through porous media

Abstract

1.1 Force, displacement, stress, strain, and displacement–strain relations

1.2 Equation of equilibrium and stress–strain relation

1.3 Fluid flow through porous media

1.4 Matrix expression

Chapter 2. Finite element methods

Abstract

2.1 Discretization using the virtual work principle

2.2 Discretization using the minimization of total potential energy

2.3 Discretization using the residual method

2.4 Discretization of the set of flow equations through porous media using the residual method

Chapter 3. Finite element method with analytical integration using simple elements

Abstract

3.1 Discretization using 3D tetrahedral elements

3.2 Analytical integrations

3.3 Assembling the elements

3.4 Nodal forces

3.5 Body forces

Chapter 4. Finite element method with isoparametric elements

Abstract

4.1 Isoparametric elements

4.2 Brick elements

4.3 Infinite element

Chapter 5. Numerical integration

Abstract

5.1 Gaussian integration

5.2 Integration formula for triangle and tetrahedron shape functions

Chapter 6. Solution of linear simultaneous equations

Abstract

6.1 Matrix transformation for the boundary condition given by local coordinates

6.2 Solution of linear simultaneous equations

Chapter 7. Convergence and error analysis

Abstract

7.1 Theoretical estimation of error

7.2 Numerical evaluation of error

Chapter 8. Application of the finite element method to nonlinear geological materials

Abstract

8.1 Standard triaxial rock test equipment and typical test results

8.2 Nonlinearity at a low-stress state

8.3 Shear-type nonlinear strain

8.4 Yield envelope fitted to real polyaxial stress–strain empirical data

8.5 Incremental form of nonlinear stress strain for application of the finite element method

8.6 Application of the Newton–Raphson method to nonlinear problems

8.7 Calculation method of λ, Dep

8.8 Implementation

8.9 Construction of constitutive relations from triaxial data

Chapter 9. Coupling geomechanics and transient fluid flow

Abstract

9.1 Fundamental equations for isotropic poro-elasticity problems

9.2 Discretization using the virtual work principle

9.3 Discretization of transient flow equations through porous media

9.4 Coupling geomechanics and transient fluid flow

9.5 Stability of the sequential methods

9.6 Sequential coupling with commercially available reservoir models

Further reading

II: Applications of Flow3D and Geo3D to real field problems

Chapter 10. Pressure profile around perforations—field problems using Flow3D

Abstract

10.1 Pressure profile around a single perforation

10.2 Numerical solution for pressure distribution around a single perforation

10.3 Pressure distribution around a perforation for gravel packed well

10.4 Quantitative analysis of the effect of perforation interaction on flow efficiency

Nomenclature

References

Chapter 11. Evaluation of mechanical stability of perforations using Geo3D

Abstract

11.1 Stability of perforations during oil and gas production

11.2 Field observation of sand-production problems

11.3 A quick method to forecast the possibility of sand problems: Perforation stability analysis using TWC or TPS test equipment

11.4 Concluding remarks

Nomenclature

Further reading

Chapter 12. Numerical methods for the borehole breakout problems using Geo3D

Abstract

12.1 Rock failure and failure theories

12.2 Failure envelopes from empirical results

12.3 Stress state around an inclined well drilled through inclined formation

12.4 Comprehensive analysis of stress state around a borehole with temperature, swelling, and pore pressure change for layered and orthotropic formations

12.5 Failure theories to predict breakout angle around a borehole

12.6 Effect of controllable parameters on safe mud window design

12.7 Conclusion

Nomenclature

References

Further reading

Chapter 13. Casing collapse for hydrostatic and geotechnical loads—Geo3D analysis

Abstract

13.1 Casing collapse for hydrostatic load

13.2 Concluding remarks

Nomenclature

Further reading

Chapter 14. Three-dimensional reservoir compaction problems with coupled Geo3D code

Abstract

14.1 Introduction of reservoir compaction problems

14.2 Strain nuclei method

14.3 Analytical solution at the center of a reservoir for a radial reservoir

14.4 Subsidence, pore pressure, and stress change in the overburden formation using the finite element method coupled with transient flow and geomechanics models

14.5 Parametric analysis of subsidence and compaction

References

Further reading

Appendix A. Apparent elastic modulus with pore fluid

III: Programming of the finite element methods

Appendix B. Computer program structure for transient fluid flow problems through porous media

B.1 Program Flow3D

B.2 Input example FLOW3D

Appendix C. Program Geo3D

C.1 Input example for Geo3D

C.2 Input example for Geo3D

Appendix D. Program COEFI for determining the parameters of the nonlinear constitutive relations from a set of triaxial test results

D.1 Calculation procedure

D.2 Input file Coefi.inp

Index

Copyright

Gulf Professional Publishing is an imprint of Elsevier

50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States

The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, United Kingdom

Copyright © 2021 Elsevier Inc. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions.

This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

Library of Congress Cataloging-in-Publication Data

A catalog record for this book is available from the Library of Congress

ISBN: 978-0-323-91112-2

For Information on all Gulf Professional Publishing publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Charlotte Cockle

Senior Acquisitions Editor: Katie Hammon

Editorial Project Manager: Aleksandra Packowska

Production Project Manager: Anitha Sivaraj

Cover Designer: Victoria Pearson

Typeset by MPS Limited, Chennai, India

Introduction

Sedimentary rocks are formed by the accumulation of solids and fluids. When oil or gas is extracted from the sedimentary rock or when water is injected, stress change and fluid flow occur. For some cases, fluid flows without significantly deforming the rock. For other cases, stress change occurs without disturbing fluid flow. Or, for some other cases, both fluid flow and geomechanics are tightly coupled. This book is written to enhance the ability of programming the finite element methods in geomechanics for geophysicists and oil and gas development engineers.

1. The finite element method for 3D transient flow problems is developed and the Flow3D program code is attached.

2. The finite element method for 3D nonlinear geomechanics problems is developed and the Geo3D program code is attached.

3. Several methods to couple the fluid flow and geomechanics are developed. Subroutines are attached to couple Flow3D and Geo3D.

4. Necessary procedures are described to couple Geo3D and commercial finite difference flow code such as ECLIPSE.

Since the procedures of applying the finite element methods to field projects are important, the latter half of the book describes the field projects that were performed for oil industries. The field applications will enlighten the readers on how fluid is flowing around a well, how sand is produced from a production well, how instability occurs in the drilling hole, how casing is deformed, and how subsidence and compaction occur during oil and gas extraction.

It takes several years to develop a 3D finite clement program suitable for research and field applications. The transient 3D flow and Geo3D codes in this textbook have been used for various field problems. Currently, many finite element programs are available. However, good 3D transient flow codes and the mechanical finite clement codes suitable for nonlinear geomechanics are scarce. To enhance research and field analysis, the reader may create new codes by adding several new subroutines to Flow3D and Geo3D codes attached in this book.

I

Basics of the finite element method

Outline

Chapter 1 Fundamental equations of poro-elasticity and fluid flow through porous media

Chapter 2 Finite element methods

Chapter 3 Finite element method with analytical integration using simple elements

Chapter 4 Finite element method with isoparametric elements

Chapter 5 Numerical integration

Chapter 6 Solution of linear simultaneous equations

Chapter 7 Convergence and error analysis

Chapter 8 Application of the finite element method to nonlinear geological materials

Chapter 9 Coupling geomechanics and transient fluid flow

Chapter 1

Fundamental equations of poro-elasticity and fluid flow through porous media

Abstract

This chapter describes the fundamental equations of poro-elasticity and fluid flow through porous media.

Keywords

Force; displacement; stress; strain; poro-elasticity; effective stress; neutral stress; darcy's law; continuity equation; phase behaviour equation

1.1 Force, displacement, stress, strain, and displacement–strain relations

The normal stress and the shear stress are the normal and shear forces (Fig. 1.1) per unit area defined by

(1.1)

Figure 1.1 Force vector.

1.1.1 Three-dimensional stresses

Conventionally, the normal stresses are expressed by and the shear stresses are expressed by . The planes perpendicular to x, y, and z coordinates are called x, y, and z planes, respectively. The first subscript is the plane on which the stresses act and the second subscript is the stress direction. For three-dimensional problems (Fig. 1.2), the stresses are represented by the following nine stress components:

Figure 1.2 3D stresses.

Strain

The deformation varies from point to point, and it is split into compressional and shearing deformations as shown in Fig. 1.3. The normal strain component is defined as the change in length per unit length and the shear strain is defined as the change in angle between two original orthogonal directions. Conventionally, the normal strain is expressed by and the shear strain or angular deformation is expressed by .

(1.2)

Figure 1.3 Visual illustration of strains.

1.1.2 For three-dimensional problems

The normal strains and the shear strains are expressed by and with subscripts for the face applied and the direction of the deformation (Fig. 1.4).

(1.3)

Figure 1.4 Displacement and deformation of a square structure.

1.1.3 Displacement–strain relations

Strains are expressed by

(1.4)

1.1.3.1 Number of variables

Stress: 9 components

σxx, σyy, σzz, τxy, τyz, τzx, τyx, τzy, τxz

Strain: 9 components

εxx, εyy, εzz, γxy, γyz, γzx, γyx, γzy, γxz

Displacement: 3 components

u, v, z

The total number of variables is 21.

The stress and strain components are reduced due to the symmetry for nonsingular force condition. Then, the following equations are applied:

(Proof for static problems)

The moment around O (Fig. 1.5) is

(1.5)

Figure 1.5 Moment balance around the origin O for a static structure.

Hence, τxy=τyx holds. Similarly, other shear stress components are also symmetric.

If the shear stresses and strains are symmetric, then the number of the unknowns is reduced to 15 as follows:

Stress: 6 components

σx, σy, σz, τxy, τyz, τzx

Strain: 6 components

εx, εy, εz, γxy, γyz, γzx

Displacements: 3 components

u, v, w

Totally, the variables are reduced to 15.

1.2 Equation of equilibrium and stress–strain relation

1.2.1 Equation of equilibrium

The forces in the x direction for a cubic body are given by

(1.6)

The sum of the forces is zero if they are in equilibrium. That is,

(1.7)

Similarly, for y and z directions, the forces are in equilibrium. Hence,

(1.8)

(1.9)

1.2.2 Stress–strain relations for isotropic linearly elastic materials

Hook’s law for linearly elastic material assumes that if an axial stress is applied along the x coordinate, the strain in the x coordinate becomes proportionally large (Fig. 1.6). The proportional constant is called as Young’s modulus.

Figure 1.6 Concept of Poisson’s ratio.

The strains in y and z directions increase if the strain in the x direction is reduced. The ratio of the strains in y and z directions and the strain in the x direction is called as Poisson’s ratio denoted by . Or,

(1.10)

The above strains are generated when the stress is applied only in x direction. Similar equations are applied to forces in y and z directions. Hence,

(1.11)

For y, z directions,

(1.12)

(1.13)

If a shear stress is applied as shown in Fig. 1.7, the shear strain increases proportionally; hence,

(1.14)

where the proportional constant G is called as the shear modulus.

Figure 1.7 Concept of shear modulus.

1.2.3 Number of variables and equations

So far, the variable definitions are described and the relations between variables are derived. The followings are the summary of the number of the equations and the number of the variables.

15 variables used for fundamental elasticity

Stress: 6 components

σx, σy, σz, τxy, τyz, τzx

Strain: 6 components

εx, εy, εz, γxy, γyz, γzx

Displacement: 3 components

u, v, w

And, 15 equations are derived:

15 equations for linear elasticity problems

Displacement–strain relations

(1.15)

Equations of equilibrium are given by

(1.16)

Stress–strain relations are given by

(1.17)

Note that for the tensor calculations, the shear strains are halved since the following definition is more convenient for calculations:

Hence, the variables can be determined by solving the set of the above equations if the boundary conditions are specified.

1.2.4 Stress–strain relations for porous media

Some equations used for nonporous media are applied to porous media. Both the equations of equilibrium and strain–displacement relations have the same form as the nonporous media. However, the stress–strain relations for porous media must include the deformation induced by pore pressure. For porous media, uniform and hydrostatic pressure is applied to each grain if the pore pressure is increased without increasing the effective stress. Suppose the pore pressure p is applied on each grain. The pore pressure is applied on all the solid surfaces uniformly so that the stress induced within the solid is also uniform if the solid material is uniform. The solid reduces the size proportionally; hence the porosity remains the same as shown in Fig. 1.8.

Figure 1.8 Pore pressure applied on the grain surfaces.

Suppose the hydrostatic stress is applied everywhere within the solid, that is, uniform stress σX═σy═σZ═ーｐ is applied everywhere within the solid, then the strain induced by the hydrostatic stress is given by

(1.18)

where Em and νm are matrix Young’s modulus and Poisson’s ratio, respectively. The hydrostatic stress applied throughout the grains is called neutral stress since the grains shrink proportionally with pressure p; hence, the shape of each grain does not change. Since the shape of each grain does not change, the porosity remains unchanged.

Now, let apply external stress without changing the pore pressure. Then, the additional stresses are given by

(1.19)

The above stresses are called effective stresses. Note that the compressive stress is negative. If the compressive stress is positive, the sign in front of p becomes minus. With the effective stress, the grain shape changes where the external stresses are transmitted through the grain–grain contact points. The bulk stress–strain relation is given by the following equations:

(1.20)

where Young’s modulus and Poisson’s ratio are the bulk Young’s modulus and the bulk Poisson’s ratio. The strain induced by the neutral stress and the effective stress is superimposed.

Stress–strain relation for porous media:

(1.21)

The shear stress is not affected by the neutral stress so that the effective stresses are written by the following tensor form:

(1.22)

A more detailed discussion follows since the effect of the neutral stress is actually very complex and the normal rocks contain various minerals. The rock grains deform proportionally only if all the grains consist of the same minerals and all the pores are connected so that the pore pressure is the same throughout the pores. Then, all the grains are proportionally shrunk due to the neutral stress; hence, the porosity remains the same. If the porosity remains the same, the rock properties are not significantly affected by the neutral stress. The permeability slightly changes even with the same porosity since the pore capillaries become slightly smaller. However, such changes are normally small enough so that the changes are negligible.

The proportional change in grain shape is actually considered to be ideal since the actual rock consists of heterogeneous grains and disconnected voids. Actually, each grain consists of different minerals so that the deformation of each grain is not uniform even with the same neutral stress. Some voids are also isolated so that the pore pressure in the isolated voids may be different from the pore pressure at the surrounding pores. Rocks have laminations during the sedimentation process, inducing vertical changes in grain properties. All these cause distortion of formation when the neutral stress is applied. In fact, it is not difficult to construct an artificial rock that does not deform uniformly with the neutral stress. However, passt tests show that 99% of pores are connected for the standard sedimentary rock; hence, the uniform grain deformation due to the neutral stress is a good approximation.

The displacement–strain relations and the equation of equilibrium are the same regardless of the porous or nonporous media if the total stress concept is used. Hence, if the pore pressure is specified, the number of variables is 15 and the number of equations is also 15 for porous media. To find the solution, the effective stresses or the total stresses may be selected for the stress method. However, selecting the total stress as the primary variables may reduce the confusion. The effective stress can be later calculated once the total stress is found.

1.3 Fluid flow through porous media

According to Darcy’s law, the volumetric rate of flow per unit cross-sectional area is proportional to the pressure gradient and inversely proportional to the fluid viscosity. The equation is given by

(1.23)

where K is the proportional coefficient called as permeability and is the non-Darcy coefficient (Fig. 1.9).

Figure 1.9 Flow in and out from a small hexahedron.

The continuity equation is derived based on the material balance which states that the volumetric rate of flow into the small volume element minus the flow out from the element is the accumulated liquid volume in the element. Or,

(1.24)

or

or

(1.25)

Simplifying the above equation gives fluid flow equations.

Fluid flow equations

(1.26)

Assuming the liquid compressibility is small, we have

(1.27)

The boundary conditions are given by

(1.28)

In the above equations, the variables are , , and p, and the number of equations is 5; hence, if the boundary conditions are given, the system of the equations may be solved.

1.4 Matrix expression

A matrix expression is used for developing equations for the finite element method.

The displacement–strain relation is given by

(1.29)

Equation of equilibrium is given by:

where

(1.30)

The stress–strain relation is given by

(1.31)

For fluid flow through porous media, the matrix expression is given by

Darcy’s flow

(1.32)

where , , and are scalar.

Continuity equation

(1.33)

Or

(1.34)

Phase behavior: assuming the liquid compressibility is small, we have

(1.35)

Chapter 2

Finite element methods

Abstract

The finite element methods are developed in this chapter using the virtual work principle, minimization of total potential energy, and the residual method.

Keywords

Finite element method; virtual work principle; virtual displacement; virtual force; strain energy; equilibrium; potential energy; residual method

2.1 Discretization using the virtual work principle

Virtual work principle: The virtual work principle uses two forms of general principle, those of virtual displacement and virtual forces. The virtual displacement form states that for a body in equilibrium with body forces and applied boundary forces, the sum of the energy of the applied loads and the strain energy stored during the virtual displacement is equal to zero. That is, the following equation holds.

(2.1)

where work done by the boundary stresses and internal work done by stresses and body forces.

(Proof)

The strain energy induced by virtual displacement is given by

(2.2)

The following equation is used to modify Eq. (2.2)

(2.3)

Then,

(2.4)

Using the Gauss divergence theory , Eq. (2.4) becomes

(2.5)

The left term of the last equation is the work done by the surface stress and is written by

(2.6)

Hence,

(2.7)

If a body is in equilibrium with applied boundary forces, then,

(2.8)

We now derive a set of discretized equations based on the virtual displacement principle. Suppose now a body is in equilibrium state with body forces and applied boundary forces. Or, the following equation holds.

Suppose u is the displacement in the body. The approximate form of u is obtained using the displacements at element nodes with a proper interpolation function.

(2.9)

The displacement strain relation becomes

(2.10)

where

(2.11)

(2.12)

(2.13)

Now, consider virtual displacement at the nodes. Then, the displacement and strain are given by (Fig. 2.1)

(2.14)

(2.15)

Figure 2.1 Virtual displacements at nodes.

The stress strain relation is given by

(2.16)

(2.17)

The work done from external stress Ti is given by

(2.18)

The strain energy stored in the body is given by

(2.19)

According to the virtual work principle, if the body is in equilibrium, the following equation holds.

Hence,

(2.20)

(2.21)

Since Eq. (2.21) holds for arbitrary virtual displacement, the following equation is obtained.

(2.22)

Hence,

The finite element method

(2.23)

where

The above equation is the discretized form of the fundamental equations of elasticity. It applies to one element. If it is assembled for all the elements, a set of linear equations are obtained with respect to the nodal displacements.

2.2 Discretization using the minimization of