Finite Element Programming in Non-linear Geomechanics and Transient Flow

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Finite Element Programming in Nonlinear Geomechanics and Transient Flow
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Contents
Introduction
I. Basics of the finite element method
	1 Fundamental equations of poro-elasticity and fluid flow through porous media
		1.1 Force, displacement, stress, strain, and displacement–strain relations
			1.1.1 Three-dimensional stresses
				Strain
			1.1.2 For three-dimensional problems
			1.1.3 Displacement–strain relations
				1.1.3.1 Number of variables
		1.2 Equation of equilibrium and stress–strain relation
			1.2.1 Equation of equilibrium
			1.2.2 Stress–strain relations for isotropic linearly elastic materials
			1.2.3 Number of variables and equations
			1.2.4 Stress–strain relations for porous media
		1.3 Fluid flow through porous media
		1.4 Matrix expression
	2 Finite element methods
		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
	3 Finite element method with analytical integration using simple elements
		3.1 Discretization using 3D tetrahedral elements
		3.2 Analytical integrations
		3.3 Assembling the elements
		3.4 Nodal forces
		3.5 Body forces
	4 Finite element method with isoparametric elements
		4.1 Isoparametric elements
		4.2 Brick elements
			4.2.2 Twenty-node brick element
			4.2.3 Elements compatible with the 8-node and 20-node brick elements
			4.2.4 Ten-node parabolic tetrahedron element
		4.3 Infinite element
			4.3.1 Twelve-node infinite element
	5 Numerical integration
		5.1 Gaussian integration
		5.2 Integration formula for triangle and tetrahedron shape functions
	6 Solution of linear simultaneous equations
		6.1 Matrix transformation for the boundary condition given by local coordinates
		6.2 Solution of linear simultaneous equations
			6.2.1 Gaussian elimination
			6.2.2 Frontal method
				6.2.2.1 Variable bandwidth elimination method
			6.2.3 Conjugate gradient method
	7 Convergence and error analysis
		7.1 Theoretical estimation of error
		7.2 Numerical evaluation of error
	8 Application of the finite element method to nonlinear geological materials
		8.1 Standard triaxial rock test equipment and typical test results
			8.1.1 Stress and strain invariants
			8.1.2 Nonlinear elastic coefficients
		8.2 Nonlinearity at a low-stress state
		8.3 Shear-type nonlinear strain
			8.3.1 Yield stress
			8.3.2 Maximum stress yield theory: Rankine theory
			8.3.3 Maximum strain yield theory (Saint-Venant theory)
			8.3.4 Maximum shear stress yield theory (Tresca)
			8.3.5 Max octahedral shear stress yield theory (von Mises yield criterion)
			8.3.6 Mohr’s linear yield theory
			8.3.7 Mohr’s nonlinear yield theory
			8.3.8 Drucker–Prager yield theory
			8.3.9 Lade yield theory
		8.4 Yield envelope fitted to real polyaxial stress–strain empirical data
			8.4.1 Mohr–Coulomb
			8.4.2 Drucker–Prager
			8.4.3 Lade model
			8.4.4 Solenhofen limestone
			8.4.5 Dunham dolomite
			8.4.6 Fuji River sand
		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
	9 Coupling geomechanics and transient fluid flow
		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
			9.4.1 Full coupling
			9.4.2 Sequential coupling
				9.4.2.1 Fixed strain method
				9.4.2.2 Fixed total stress method
				9.4.2.3 Drained split method
				9.4.2.4 Undrained split method
		9.5 Stability of the sequential methods
			9.5.1 One-dimensional compaction problem coupled with fluid flow and geomechanics
			9.5.2 Stability analysis of the coupled problem
			9.5.3 Stability of drained split method
			9.5.4 Stability of undrained split method
			9.5.5 Stability of fixed strain method
			9.5.6 Stability of fixed total stress method
			9.5.7 Numerical example of one-dimensional compaction problem using the fixed total stress method
		9.6 Sequential coupling with commercially available reservoir models
			9.6.1 Sequential calculation of flow and geomechanics with uniaxial compaction assumption
				9.6.1.1 Approximation of reservoir section
					9.6.1.1.1 Approximation of overburden and underburden deformation
				9.6.1.2 Single-phase or multiple-phase problems assuming uniaxial compaction
				9.6.1.3 Calculation procedure
					9.6.1.3.1 Calculations of the displacement and stress and pore pressure changes in the overburden and underburden section
			9.6.2 Single or multiphase problems without assuming uniaxial compaction
				9.6.2.1 Fixed strain method
				9.6.2.2 Fixed total stress method
			9.6.3 One-step undrained method for short period production problems
			9.6.4 General multiphase problems
		Further reading
II. Applications of Flow3D and Geo3D to real field problems
	10 Pressure profile around perforations—field problems using Flow3D
		10.1 Pressure profile around a single perforation
			10.1.1 Analytical solution
		10.2 Numerical solution for pressure distribution around a single perforation
			10.2.1 Pressure distribution around a perforation without permeability damage
				10.2.1.1 Parameters for perforation flow model: permeability is uniform over entire domain
			10.2.2 Pressure distribution around a perforation with damaged permeability
				10.2.2.1 Parameters for perforation flow model: permeability is damaged
			10.2.3 Gas flow
		10.3 Pressure distribution around a perforation for gravel packed well
		10.4 Quantitative analysis of the effect of perforation interaction on flow efficiency
			10.4.1 Evaluation of flow performance
				10.4.1.1 Quantification of flow performance
				10.4.1.2 Effect of perforation pattern and shot density
				10.4.1.3 Permeability anisotropy impact on perforations
				10.4.1.4 Effect of perforation length
				10.4.1.5 Effect of perforation diameter
				10.4.1.6 Effect of mud damage and perforation damage
				10.4.1.7 Effect of sand production
		Nomenclature
		References
	11 Evaluation of mechanical stability of perforations using Geo3D
		11.1 Stability of perforations during oil and gas production
			11.1.1 Field applications
		11.2 Field observation of sand-production problems
			11.2.1 How fluid flow affects sand production from a three-dimensional perforation cavity
			11.2.2 How drawdown necessary to produce without sand changes during reservoir life
			11.2.3 Why sand problems often occur after water cut
				11.2.3.1 What can be learned from a step-flow test?
			11.2.4 Why sand flow is high when production is restarted
				11.2.4.1 Perforation design minimizing sand problems
		11.3 A quick method to forecast the possibility of sand problems: Perforation stability analysis using TWC or TPS test equi...
		11.4 Concluding remarks
		Nomenclature
		Further reading
	12 Numerical methods for the borehole breakout problems using Geo3D
		12.1 Rock failure and failure theories
			12.1.1 Total failure, local failure, and internal failure
				12.1.1.1 Evaluation of local failures
				12.1.1.2 Local failure criteria
					12.1.1.2.1 Mohr–Coulomb
					12.1.1.2.2 Drucker–Prager
					12.1.1.2.3 Mogi
					12.1.1.2.4 Modified Lade (expressed by Russell Ewy)
					12.1.1.2.5 Modified Lade (expressed by Eekelen)
					12.1.1.2.6 Lade model
			12.1.2 Critical plastic failure theory
		12.2 Failure envelopes from empirical results
			12.2.1 Castlegate sandstone
			12.2.2 Rozbark sandstone and sensitivity analysis
			12.2.3 Dunham dolomite
			12.2.4 Mizuho trachyte
			12.2.5 Shirahama sandstone
			12.2.6 Izumi sandstone
			12.2.7 Horonai sandstone
			12.2.8 Yubari shale
			12.2.9 Yamaguchi marble
				12.2.9.1 Summary of the fitness of failure envelope to the polyaxial experimental data
		12.3 Stress state around an inclined well drilled through inclined formation
			12.3.1 Analytical solution to calculate stress state around a borehole
			12.3.2 Calculation of principal stresses
			12.3.3 Zoback’s breakout angle estimation
			12.3.4 Breakout angle and depth for Mogi and Lade failure theories
				12.3.4.1 Mogi’s failure theory
				12.3.4.2 Breakout depth
					12.3.4.2.1 Lade and Drucker–Prager failure theory
			12.3.5 Definition of UCS
			12.3.6 Breakout measurements
		12.4 Comprehensive analysis of stress state around a borehole with temperature, swelling, and pore pressure change for laye...
			12.4.1 General statement
			12.4.2 Parameter study with typical field conditions
				12.4.2.1 Stress state around a borehole for the base borehole conditions
					12.4.2.1.1 Stress state: the base condition shown in Table 12.1
				12.4.2.2 Stress state for well inclination (γ)
				12.4.2.3 Stress state for pore pressure change due to osmosis effect
				12.4.2.4 Stress state due to swelling effect
				12.4.2.5 Stress state due to temperature effect
				12.4.2.6 Stress state due to overbalance effect
				12.4.2.7 Stress state due to drawdown effect
				12.4.2.8 Stress state due a hard bedding layer (γ=0 degrees)
				12.4.2.9 Stress state due to a soft bedding layer existing (γ=0 degrees)
				12.4.2.10 Stress state with a 60-degree hard bedding layer (γ=60 degrees)
				12.4.2.11 Stress state due to a soft bedding layer with 60-degree inclination (γ=60 degrees)
				12.4.2.12 Stress state for formation with transversely isotropic elastic material
				12.4.2.13 Stress state around a borehole in depleted reservoir condition
					12.4.2.13.1 Stress state around borehole under the depleted reservoir condition
					12.4.2.13.2 Stress state around borehole under the depleted reservoir condition with swelling effect
					12.4.2.13.3 Stress state around borehole under the depleted reservoir condition with sandstone nonlinearity effect
		12.5 Failure theories to predict breakout angle around a borehole
			12.5.1 Rock failure criteria
				12.5.1.1 Mohr–Coulomb
				12.5.1.2 Mogi
				12.5.1.3 Drucker–Prager
				12.5.1.4 Lade
			12.5.2 Failure function to predict breakout angle with various controllable parameters
			12.5.3 Breakout angle predicted by the four rock failure criteria of vertical well in the base reservoir conditions
				12.5.3.1 The base reservoir condition
				12.5.3.2 Well inclination effect
					12.5.3.2.1 Well azimuth=0 degrees, well angle=30 degrees
					12.5.3.2.2 Well azimuth=0 degrees, well angle=60 degrees
					12.5.3.2.3 Well azimuth=0 degrees, well angle=90 degrees
					12.5.3.2.4 Well azimuth=90 degrees, well angle=30 degrees
					12.5.3.2.5 Well azimuth=90 degrees, well angle=60 degrees
					12.5.3.2.6 Well azimuth=90 degrees, well angle=90 degrees
				12.5.3.3 Osmosis effect
				12.5.3.4 Swelling effect
				12.5.3.5 Temperature effect
				12.5.3.6 Overbalance effect
				12.5.3.7 Drawdown effect
				12.5.3.8 Hard bedding layer
					12.5.3.8.1 Hard bedding layer (γ=0 degrees)
					12.5.3.8.2 Hard bedding layer (γ=60 degrees)
				12.5.3.9 Soft bedding layer
					12.5.3.9.1 Soft bedding layer (γ=0 degrees)
					12.5.3.9.2 Soft bedding layer (γ=60 degrees)
				12.5.3.10 Formation with transversely isotropic elastic material
					12.5.3.10.1 Well angle=0 degrees
					12.5.3.10.2 Well angle=30 degrees
					12.5.3.10.3 Well angle=60 degrees
					12.5.3.10.4 Well angle=90 degrees
			12.5.4 Breakout angle predicted by the four rock failure criteria of horizontal well in depleted reservoir condition
				12.5.4.1 Depleted reservoir condition-base case
				12.5.4.2 Osmosis effect
				12.5.4.3 Swelling effect
				12.5.4.4 Temperature effect
				12.5.4.5 Overbalance effect
				12.5.4.6 Drawdown effect
				12.5.4.7 Sandstone nonlinearity effect
					12.5.4.7.1 Summary
		12.6 Effect of controllable parameters on safe mud window design
			12.6.1 Introduction of safe mud weight window
			12.6.2 Optimal safe mud weight window design with various controllable parameters
				12.6.2.1 Polar diagram of safe mud window in the base reservoir condition
				12.6.2.2 The base reservoir condition without any other effect: the normal faulting regime
				12.6.2.3 The base reservoir condition without any other effect: the strike-slip faulting regime
				12.6.2.4 The base reservoir condition without any other effect: the reverse faulting regime
				12.6.2.5 Critical mud weight polar diagram for shale swelling
					12.6.2.5.1 0.1% and 0.3% shale swelling under the normal faulting regime
					12.6.2.5.2 0.2% shale swelling under the strike-slip faulting regime
					12.6.2.5.3 0.2% shale swelling under the reverse faulting regime
				12.6.2.6 Safe mud weight window due to temperature effect
				12.6.2.7 Safe mud weight window due to hard bedding layer effect
					12.6.2.7.1 Hard bedding layer with inclination (γ=0 degrees)
				12.6.2.8 Safe mud weight window due to soft bedding layer effect
					12.6.2.8.1 Soft bedding layer with inclination (γ=0 degrees)
				12.6.2.9 Safe mud weight window due to formation with transversely isotropic elastic material effect
		12.7 Conclusion
		Nomenclature
		References
		Further reading
	13 Casing collapse for hydrostatic and geotechnical loads—Geo3D analysis
		13.1 Casing collapse for hydrostatic load
			13.1.1 Type A: the standard casing failure criterion applied to simple loading
				13.1.1.1 Equivalent yield strength σyieldee for nonzero axial stress and nonzero internal pressure
					13.1.1.1.1 Procedure
			13.1.2 Type B: failure criterion under geotechnical load
				13.1.2.1 Empirical method to predict the critical-strain and the limit-load to induce a buckling for compression without la...
					13.1.2.1.1 Experimental setup
					13.1.2.1.2 Empirical results
					13.1.2.1.3 Analyses of limit load of casings
					13.1.2.1.4 Summary for the uniaxial compression tests
				13.1.2.2 Critical strain initiating a local wrinkle for axial compression with lateral restraint
					13.1.2.2.1 Experimental setup
				13.1.2.3 Critical strain and stress initiating necking with axial extension
					13.1.2.3.1 Experimental setup
				13.1.2.4 Analysis of the experimental results
					13.1.2.4.1 Stress–strain curves
					13.1.2.4.2 Ultimate strength and deformation
					13.1.2.4.3 Minimum elongation of a casing
					13.1.2.4.4 Summary of the uniaxial extension tests
				13.1.2.5 Deformation of casings with a line external load
					13.1.2.5.1 Experimental results
					13.1.2.5.2 Simulation and analyses
					13.1.2.5.3 Field applications
					13.1.2.5.4 A field application example for liner stability of cased hole
					13.1.2.5.5 Reservoir description
		13.2 Concluding remarks
		Nomenclature
		Further reading
	14 Three-dimensional reservoir compaction problems with coupled Geo3D code
		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 ...
			14.4.1 Methods to evaluate the subsidence and compaction
			14.4.2 Subsidence and compaction prediction using the sequential coupling method using the fixed total stress method
			14.4.3 Subsidence and compaction prediction for undrained elastic moduli
			14.4.4 Subsidence and compaction prediction for drained elastic moduli
			14.4.5 Field applications
				14.4.5.1 Magnitude of stress and pore pressure changes and strain induced by compaction
		14.5 Parametric analysis of subsidence and compaction
			14.5.1 Earth deformation during reservoir compaction
			14.5.2 Subsidence and horizontal movement maps
		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...
		D.1 Calculation procedure
		D.2 Input file “Coefi.inp”
Index
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