Abstract
The presence of natural fractures in fractured media plays a vital role in the in-situ stress state, which is predominantly influenced by tectonic stresses and local perturbations. Fracture orientation, wellbore stability/orientation, and permeability anisotropy are strongly dependent on local stress variations. In this study, a discrete fracture network (DFN) was generated using the stochastic approach to investigate the correlation between geometrical properties of the fracture network (fracture density and length) and local stress variability. Afterward, the FLAC2D software was employed to propagate the stress redistribution in the simulation process. Considering the tensorial nature of stress, a stress field was calculated under orthogonal static boundary conditions. A tensor-based mathematical formulation was applied to compute total stress variability, shear strain distribution, and total displacement in different fracture network realizations. The results demonstrated that the mean local stress perturbation, effective variance (as an indicator of total stress variability), shear strain distribution, and total displacement increased with the rise of stress ratio and fracture density. The aforementioned parameters decreased as the power-law length exponent of the DFN generator increased. Overall, it is concluded that the stress/strain variability and total displacement in a dense fracture network and large length fractures have their maximum values.
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Abbreviations
- DFN:
-
Discrete fracture network
- FLAC2D :
-
Fast Lagrangian Analysis of Continua in 2 Dimensional
- LSM:
-
Lattice spring model
- FEM:
-
Finite element method
- FDM:
-
Finite difference method
- DEM:
-
Distinct element method
- DDA:
-
Discontinuous deformation analysis
- FEM-DEM:
-
Combined finite-distinct element method
- ADFNE:
-
Alghalandis Discrete Fracture Network Engineering
- UDEC:
-
Universal Distinct Element Code
- 3DEC:
-
3Dimention Distinct Element Code
- SR:
-
Stress ratio
- PDF:
-
Probability density function
- CCDF:
-
Complementary cumulative distribution function
- CDF:
-
Cumulative distribution function
- m :
-
Weibull distribution of Young's modulus index
- n(l):
-
Number of fractures
- a :
-
Power-law length exponent
- α :
-
Density term
- l :
-
Fracture length (m)
- l min :
-
The smallest fracture length (m)
- l max :
-
The largest fracture length (m)
- P ij :
-
Fracture frequency, i is the sample dimension and j is the measurement dimension
- P 30 :
-
Fracture density, the quantity of fractures per unit volume
- P 20 :
-
Fracture density, the quantity of fractures per unit area (m−2)
- P 10 :
-
Fracture density, the quantity of fractures per unit length
- P 32 :
-
Fracture intensity, the fracture persistence (total fracture length) per unit volume
- P 21 :
-
Fracture intensity, the fracture persistence (total fracture length) per unit area
- P WS :
-
Calculated fracture density using a window sampling approach (m−2)
- N :
-
Number of fractures
- A :
-
Area (m2)
- L :
-
Size of the model (m)
- E R :
-
Rock Elastic Modulus (GPa)
- υ R :
-
Rock Poisson's Ratio
- σ tR :
-
Rock Tensile Strength (MPa)
- C R :
-
Rock Cohesion (MPa)
- ϕ R :
-
Rock Friction Angle (°)
- E F :
-
Fracture Elastic Modulus (GPa)
- υ F :
-
Fracture Poisson's Ratio
- σ tF :
-
Fracture Tensile Strength (MPa)
- μ F :
-
Fracture Friction Coefficient
- ψ F :
-
Fracture Dilation angle (°)
- k n :
-
Fracture Normal Stiffness (GPa/m)
- C F :
-
Fracture Cohesion (MPa)
- d :
-
Mesh element size (m)
- 2c :
-
Vertical fracture length (m)
- \(\sigma_{xx}^{b}\) :
-
Boundary stress (MPa)
- σ xx :
-
Normal stress (MPa)
- S i :
-
Stress tensor field
- \(\bar{S}\) :
-
Mean stress field
- d(S i, \(\bar{S}\)):
-
Euclidean distance (Local stress perturbation) (MPa)
- md(S, \(\bar{S}\)):
-
Mean local stress perturbation (MPa)
- \(\left\| . \right\|\) :
-
Euclidean Norm
- V e :
-
Effective variance (MPa2)
- Ω:
-
Stress vector’s covariance matrix
- \(\left| . \right|\) :
-
Determinant of matrix Ω
- p :
-
Dimension of the stress tensor
- s i :
-
Stress vector
- s̄ :
-
Mean stress vector
- vech(.):
-
Half-vectorization function
- \(S_{xx}^{\infty } /S_{yy}^{\infty }\) :
-
Stress ratio
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Acknowledgements
The authors want to thank Mr. Roozbeh Imani Kalehsar for all the support and assistance that he gave in this project. This work would have not materialized without his help and support.
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Appendix
Appendix
Numerical methods for geomechanical modeling of fractured rocks are divided into two general approaches of continuum and discontinuum. The priority of using continuum and discontinuum models depends on the scale of the problem and the complexity of the fracture systems. The continuum approach works better on large-scale problems, while the discontinuum method is more commonly used in complex and multi-fracturing fracture network models.
1.1 Continuum models
Continuous models examine the rock as a continuous set using the most important methods of this model, namely the finite element method (FEM) or the finite difference method (FDM), which may also be used for fractured rocks with a small and large number of fractures. If the system contains several discontinuities associated with only a small amount of displacement/rotation, discrete fractures can be modeled by interface elements. However, it is difficult to study it in the dynamic state and displacement problems of natural fracture systems. Suppose the fracture density in a single fracture network is high. In that case, the modeling range may be divided into a limited number of grid blocks assigned with equivalent properties due to homogenization techniques. Equivalent properties such as bulk modulus and strength parameters are usually calculated using experimental formulas (taking into account the degradation effect due to pre-existing fractures) or analytical solutions (based on crack tensor theory) (Sitharam et al. 2001; Oda 1984). The simulation results may be sensitive to the detachment of fracture network blocks, especially when the block size is significantly smaller than the initially introduced volume (Sitharam et al. 2001).
1.2 FLAC2D Software
FLAC2D software performs the Lagrangian analysis of the continuous environment using the finite difference method, which can be used to simulate the behavior of earth structures, rock structures, or other objects that may reach plastic at the yield time. The environment and materials are created and simulated by the elements that together form a network. Each element, under the applied forces and boundary conditions of the problem and according to the considered behavioral model, may have linear or nonlinear stress–strain behavior, and the material can reach the yield point and become plastic, and the element network can become deformed. Since the analysis in this software is based on the explicit Lagrangian analysis method, plastic conditions and instability can be modeled with great accuracy. Also, this software has many capabilities such as considering the interaction of soil and structure, calculating large deformations, different behavioral models of soil, considering pore water pressure, and so on. This program has 11 behavioral models (null model, 3 elastic models, and 7 plastic models) for different materials. In FLAC2D, where the materials follow the Mohr-Columbus behavioral model, the factor of safety is automatically calculated for each model. The calculation is based on the resistance reduction method, which performs a set of modeling by changing the resistance properties to determine the conditions under which instability occurs (FLAC2D UsM Itasca Consulting Group. Inc).
1.3 Discontinuum model and distinct element method (DEM)
Discontinuum models include the discrete component method with explicit solutions and the discontinuous deformation analysis (DDA) method with implicit solutions. In discontinuum models, fractured rocks are considered as a set of blocks (discrete components) bounded by some intersecting discontinuities. The most important software for discrete component methods is UDEC and 3DEC software.
The method of calculating the discrete element method is divided into four main steps (Jing and Stephansson 2007):
-
The connection between blocks is identified and updated by detecting the space.
-
The contact force between the discrete components is calculated based on their relative positions.
-
The acceleration created by the force imbalance for each component is calculated using Newton's second law.
-
Speed and displacement are determined by the convergence of time with the new position.
The discrete approach method can determine the stress/strain characteristics of intact rocks, pre-existing fracture opening/shearing, and the interaction between multiple blocks and fractures.
In UDEC software, after constructing the model geometry and creating a joint in it, only the main body of the fractures is preserved after meshing. The most important advantage of FLAC2D software is the complete preservation of the dead ends of the fractures and not only preserving the main body. The Fig.
A sample of DFN model before (the left) and after (the right) the fracture system regularization
13 shows the change of the fracture network in a discrete fracture network sample in FLAC2D and UDEC software, which in FLAC2D software, as in the main case, there is no change in the fracture network pattern, but in UDEC software, the fracture network pattern is changed due to the regularization system.
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Khodaei, M., Biniaz Delijani , E., Dehghan, A.N. et al. Stress/strain variability in fractured media: a fracture geometric study. Geotech Geol Eng 39, 5339–5358 (2021). https://doi.org/10.1007/s10706-021-01838-4
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DOI: https://doi.org/10.1007/s10706-021-01838-4