Abstract
The modeling methodology described and demonstrated in this book is capable of efficient and accurate simulation of fluid flow, deformation, seismicity, and transmissivity evolution in large two-dimensional discrete fracture networks. Appropriate stress conditions and constraints on displacements are applied on elements depending on whether they are open, sliding, or stationary. Results are convergent to grid refinement, and discretization settings required for acceptable accuracy were identified. A variety of techniques that enable efficiency and realistic model behavior—such as adaptive domain adjustment, crack trip region adjustment, and the strain penalty method—have been developed and tested. The model can be used for direct solution of fracture contact problems in a way that has minimal memory requirement, excellent efficiency, and desirable scaling with problem size.
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Abbreviations
- A :
-
Fracture surface area, m2
- activelist :
-
List of elements maintained for use in adaptive domain adjustment
- a :
-
Element half-length, m
- a const :
-
Element half-length for initial discretization, m
- a frac :
-
Fracture half-length, m
- a min :
-
Minimum element half-length, m
- a rs :
-
Rate-state friction, coefficient for velocity term, unitless
- BE,σ, BD,σ, BE,τ, B D,τ :
-
Matrices of interaction coefficients, MPa/mm
- b :
-
Rate-state friction, coefficient for state term, unitless
- checklist :
-
List maintained of elements that are fully included in the problem domain with adaptive domain adjustment
- cstress :
-
An simulation option to include the stress caused by normal displacement of a closed element
- D :
-
Cumulative shear displacement discontinuity, mm
- D cum :
-
Cumulative rapid slip during a seismic event, mm
- D E,eff , D e,eff :
-
Effective cumulative displacement discontinuity, mm
- D e,eff,max, D E,eff,max :
-
Maximum effective cumulative sliding displacement used for calculating aperture, mm
- D s , D n :
-
Variables used in the description of the stress penalty method, referring to shear and normal displacement discontinuity (equivalent to D and E), mm
- d :
-
Distance between the centers of two elements, m
- d c :
-
Rate-state friction, characteristic weakening distance, m
- dt :
-
Duration of a time step, s
- E :
-
Void aperture, mm
- E 0 :
-
Reference void aperture, mm
- E hfres :
-
Residual closed aperture, mm
- E open :
-
Open aperture, physical separation between walls, mm
- e :
-
Hydraulic aperture, mm
- e 0 :
-
Reference hydraulic aperture, mm
- e hmat :
-
Relative error for h-matrix approximation, unitless
- e j :
-
Relative difference between simulations, defined in a variety of ways
- e proc :
-
Process zone hydraulic aperture, mm
- f 0 :
-
Rate and state friction term, unitless
- G :
-
Shear modulus, GPa
- G adj :
-
Olson (2004) adjustment factor to interaction coefficients, unitless
- h :
-
Out of plane fracture width, or height, m
- I :
-
Unit matrix, unitless
- itertol :
-
Convergence tolerance for iterative coupling, MPa
- J :
-
Iteration matrix, various forms
- J mech,thresh :
-
Threshold parameter for including mechanical interaction terms in the iteration matrix, unitless
- K I :
-
Stress intensity factor, MPa-m1/2
- K I,crithf :
-
Critical stress intensity factor for propagation of a new fracture, MPa-m1/2
- K I,crit :
-
Critical stress intensity factor for propagation of opening on a preexisting fracture, MPa-m1/2
- K frac :
-
Fracture stiffness, MPa−1
- K hf :
-
Stiffness of closed, newly formed fractures, MPa−1
- k :
-
Permeability, m2
- l c ,l o ,l s ,l f :
-
Parameters used for discretization refinement, m, unitless, unitless, and unitless
- M 0 :
-
Seismic moment, N-m
- M w :
-
Moment magnitude, unitless
- mechtol :
-
Convergence tolerance for the shear stress residual equations, MPa
- nochecklist :
-
List maintained of elements that are not fully included in the problem domain with adaptive domain adjustment
- nocstress :
-
An simulation option not to include the stress caused by normal displacement of a closed element
- opentol :
-
Tolerance for including elements with adaptive domain adjustment, MPa
- P :
-
Pressure, MPa
- P init :
-
Initial formation fluid pressure MPa
- P inj :
-
Injection pressure, MPa
- P prodmin :
-
Minimum production pressure, MPa
- P injmax :
-
Maximum injection pressure, MPa
- Q :
-
Calculated injection rate, kg/s
- q :
-
Mass flow rate, kg/s
- q flux :
-
Mass flux, kg/(s-m2)
- q injmax :
-
Maximum injection rate, kg/s
- q prodmax :
-
Maximum production rate, kg/s
- R :
-
Residual equation, various units
- S :
-
Specified total injection rate, kg/s
- S 0 :
-
Cohesion, MPa
- S 0,open :
-
Cohesion term for open elements, MPa
- s a :
-
Mass source term per area, kg/(s-m2)
- s :
-
Mass source term, kg/s
- slidetol :
-
Tolerance for including elements with adaptive domain adjustment, MPa
- T :
-
Transmissivity, m3
- T hf,fac :
-
Factor for calculating residual transmissivity of newly formed fractures, m2
- T g :
-
Geometric transmissibility between elements, m3
- T s :
-
Stress tensor, MPa
- t :
-
Time, s
- v :
-
Sliding velocity, m/s
- v 0 :
-
Rate-state friction, reference velocity, m/s
- v s :
-
Shear wave velocity, m/s
- X :
-
Vector of unknowns, various forms
- x :
-
Dummy variable to specify an arbitrary direction, m
- ΔD, ΔE :
-
Change in displacement discontinuity during a time step, mm
- Δσ n,strainadj , Δσ s,strainadj :
-
Stress (normal or shear) applied in the strain penalty method, MPa
- δ :
-
Parameter used in adaptive time stepping, sum of absolute value of change in shear stress and effective normal stress, MPa
- δ strainadj :
-
The largest value of Δσk,strainadj during a time step, MPa
- ε :
-
Strain tensor, unitless
- ε n, ε s :
-
Displacement discontinuity normal or shear strain, used for high strain penalty method, unitless
- ε n,lim , ε s,lim :
-
Limit to displacement discontinuity strain (normal or shear), unitless
- ε tol :
-
User specified relative tolerance for h-matrix assembly, unitless
- η :
-
Radiation damping coefficient, MPa/(m/s)
- η targ :
-
Target change in stress change parameter for adaptive time stepping, MPa
- η targ,strainadj :
-
Target change in εk for time stepping, unitless
- θ :
-
Rate-state friction, state variable, s
- μ d :
-
Dynamic coefficient of friction, unitless
- μ f :
-
Coefficient of friction, unitless
- μ l :
-
Fluid viscosity, Pa-s
- ρ :
-
Density, kg/m3
- π :
-
Mathematical constant Pi, unitless
- σ ’ n :
-
Effective normal stress, MPa
- σ n :
-
Normal stress, MPa
- σ n,Eref , σ n,eref :
-
Reference fracture stiffness, mm
- σ yy :
-
Remote compressive stress in the y direction, MPa
- σ xy :
-
Remote shear stress, MPa
- σ xx :
-
Remote compressive stress in the x direction, MPa
- τ :
-
Shear stress, MPa
- υ p :
-
Poisson’s ratio, unitless
- φ E,dil , φ e,dil :
-
Aperture dilation angle, °
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McClure, M., Horne, R. (2013). Conclusions. In: Discrete Fracture Network Modeling of Hydraulic Stimulation. SpringerBriefs in Earth Sciences. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00383-2_5
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DOI: https://doi.org/10.1007/978-3-319-00383-2_5
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