2D/3D Boundary Element Programming in Petroleum Engineering and Geomechanics

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Introduction
Fundamental elasticity equations
	Fundamental elasticity equations
	Boundary conditions
Fluid flow through porous media
	Fundamental equations of fluid flow through porous media
	Boundary condition
Tensor notation or index notation
Fundamental solutions
	Fundamental solutions for boundary element methods
	Fundamental solution for the fluid flow through porous media
	Fundamental solutions for boundary element methods for various problems
Boundary element methods
	Direct boundary element method (direct-BEM)
		Stress evaluation
	Indirect boundary element method (indirect-BEM)
	Displacement discontinuity method (DDM)
		Derivation of displacement discontinuity method
			Alternative method to reduce the singularity.
		Other expression of the generalized displacement discontinuity method for 3-D fracture problems
		2D fracture problems
			Analytical method to obtain Eq. (5.143)
		3D problems originally proposed by Crouch
	Dual boundary element method (dual BEM)
		Alternative method to reduce the degree of singularity
	Integral equation for poro-elasticity problems
		Fundamental equations
			Derivation of Eq. (5.208) using divergence theorem
Discretization of integral equation
	Discretization using the direct BEM with a constant strain
	Stress and strain evaluations within domain
	Boundary element method including volume integration with tetrahedral and eight-node solid elements
	Higher-order elements
		Interpolation function
		Discontinuous element
		Discretization using shape functions with high order polynomials
		Assembling GH matrix without crack elements
	Corner nodes
	High order discontinuous shape function
Numerical integration
	Numerical integration suitable to boundary element methods
		Five types of singular boundary integrations
		Definition of the Cauchy principal-value integral
			2D problems
			3D problems
		Definition of the Hadamard finite part integral
			2D problems
			3D problems
		Standard procedure for singular integrations
	Gauss integration (for integration over the element without singular point)
		Integration formula for triangle and tetrahedron shape functions
	Quasisingular and singular integrations
		Domain refinement
		Coordinate transformation-1 (polynomial transformation)
		Coordinate transformation-2: A self-adaptive coordinate transformation for quasisingular or singular integrations
		Coordinate transformation-3 Lachat-Watson transformation for weak singularity
		Coordinate transformation-4 PART method: Projection and Angular & Radial Transformation for quasisingular integrat ...
			Tangential integration with the Gaussian quadrature
			Radial integration with the Gaussian quadrature after radial transformation
		Application example
			Analytical solution
		Radial transformation similar to the PART method described by S.G. Mikhlin
		Cauchy principal-value integral for strong singularity O(1/r2)
		Application example
		Hadamard finite part integral: singular subtraction method for the finite part integral of the hypersingularity O( ...
	Comparison of accuracy of numerical integrations around a singular point
	Methods to avoid singular integrations using the rigid body movement and no flow conditions
		Finite region rigid-body-motion constraint
		Semiinfinite region rigid-body-motion constraint
		Flow problems
Solution of linear system of
equations
	Transformation from local coordinate to global coordinate
	System of linear equations
		Gaussian elimination
		Variable bandwidth elimination method
		Conjugate gradient method
Discretization of the system
of equations for fluid flow
through porous media
	Equations for fluid flow through porous media
	Fundamental solution for steady-state flow
	Boundary element method for unsteady-state flow through porous media
2D structure code (Direct BEM)
with stiffness matrix without
numerical integration
	Simple 2D elasticity computer program (usinganalytical integration for constructing H and G matrix)
	Coefficient matrix
		Nondiagonal matrix
		Diagonal coefficients
	Example code
		Input data for a simple structural problem
2D structure code (direct BEM)
with stiffness matrix with
numerical integration
	Discontinuous quadratic element for 2D elasticity problems
	Integration of stiffness matrix
	Code example
		Input data for a simple structural problem
2D displacement discontinuity
boundary element method
(DDM) suitable for crack problems
	Analytical formulation of 2D DDM
		Displacement and the stresses within the domain
	Flowchart
	Example of input data for line crack problems
		Input example
		Circle without crack
		Input data
	Results
2D transient flow program using
time-dependent fundamental
solution
	2D transient flow program using 3-node discontinuous quadratic element
	Coefficients of the integral equations
	Example program
		Example input data
3D boundary element code
(direct BEM) for solid elasticity
problems
	3D elasticity program with quadratic continuous element
		Construct GH Matrix
	Program flow-chart
		Descriptions of the subroutines
		Model around a borehole
Analytical 3D displacement
discontinuity method:
Stress disturbance induced
by movement of fault planes and
fracture opening
	Displacement discontinuity method
	Stress shadow problems for a specified fracture surface pressure
3D static fracture model
using the dual boundary
element method
	Basic modes of crack surface displacements
		Stress and displacement around a crack tip
		Stress intensity factors
	Dual boundary element method (DBEM)
	Discretization of DBEM
		Hyper singular integration
		Special crack tip element
		Evaluation of the fracture intensity factor
	Integrations
	Computer program
		Flow chart
	Accuracy and applications
		3D fracture model around an inclined borehole using the dual boundary element method
			Model calibrations
			Penny shaped crack
			Slant elliptical-shaped crack
			Borehole displacement
	Shear-type borehole wall shifts induced during lost circulations
		Abstract
		Introduction
			Fracture model
			Borehole wall shift calculations
			Analysis and discussion for practical applications
	Conclusions
	Stress transformation for a slant fracture from an inclined borehole
	Main factors determining the lost circulation volume
3D Fracture propagation code
coupled with 2D finite element
flow and 3D DDM fracture code
	Flow equations for 2D flow of frac-fluid
	The variational method to discretize the flow equation
	Galerkin approximation
	Matrix form of fluid flow
	Curved fracture problems for uniform formation without modulus contrast
	Planer fracture for layered formation with modulus contrast
		Shape functions for crack elements
		Coupling and solution procedures
		Advancement of a fracture tip
	Flow chart and input example
		Test input
		Initial mesh
	Results
References
Online Appendix A
Online Appendix B
	Fundamental solution for isotropic bimaterials(L. Rongved, 1955)
	Calculation of Tkα11
	References
Online Appendix C: Example codes
	Chapter 10. Example code
	Chapter 11. Code example
	Chapter 12. Crouch method
	Chapter 13. Example program
	Chapter 14. Elast3D
	Chapter 15. Analytical 3D displacement discontinuous method for stress shadow problem induced by fracturing
	Chapter 16. 3D Curved static fracture with a Borehole
	Chapter 17. 3D fracture propagation program
	Chapter 18. Mesh generation preprocessing code