# A continuum model for coupled stress and fluid flow in discrete fracture networks

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## Abstract

We present a model coupling stress and fluid flow in a discontinuous fractured mass represented as a continuum by coupling the continuum simulator TF_FLAC^{3D} with cell-by-cell discontinuum laws for deformation and flow. Both equivalent medium crack stiffness and permeability tensor approaches are employed to characterize pre-existing discrete fractures. The advantage of this approach is that it allows the creation of fracture networks within the reservoir without any dependence on fracture geometry or gridding. The model is validated against thermal depletion around a single stressed fracture embedded within an infinite porous medium that cuts multiple grid blocks. Comparison of the evolution of aperture against the results from other simulators confirms the veracity of the incorporated constitutive model, accommodating stress-dependent aperture under different stress states, including normal closure, shear dilation, and for fracture walls out of contact under tensile loading. An induced thermal unloading effect is apparent under cold injection that yields a larger aperture and permeability than during conditions of isothermal injection. The model is applied to a discrete fracture network to follow the evolution of fracture permeability due to the influence of stress state (mean and deviatoric) and fracture orientation. Normal closure of the fracture system is the dominant mechanism where the mean stress is augmented at constant stress obliquity ratio of 0.65—resulting in a reduction in permeability. Conversely, for varied stress obliquity (0.65–2) shear deformation is the principal mechanism resulting in an increase in permeability. Fractures aligned sub-parallel to the major principal stress are near-critically stressed and have the greatest propensity to slip, dilate and increase permeability. Those normal to direction of the principal stress are compacted and reduce the permeability. These mechanisms increase the anisotropy of permeability in the rock mass. Furthermore, as the network becomes progressively more sparse, the loss of connectivity results in a reduction in permeability with zones of elevated pressure locked close to the injector—with the potential for elevated pressures and elevated levels of induced seismicity.

## Keywords

Coupled simulation Discrete fracture network Geothermal reservoir Fracture permeability Stress-dependent aperture## 1 Introduction

Geothermal energy is a potentially viable form of renewable energy although the development of enhanced geothermal system (EGS) as a ubiquitous source has considerable difficulties in developing an effective reservoir with sustained thermal output. One key solution in generating an effective reservoir is to stimulate using low-pressure hydraulic-shearing or elevated-pressure hydraulic-fracturing. In either case, the stimulation relies on changes in effective stresses driven by the effects of fluid pressures, thermal quenching and chemical effects, each with their characteristic times-scales. Thus incorporating the effects of coupled multi-physical processes (Thermal–Hydraulic–Mechanical) exerts an important control on the outcome (Taron and Elsworth 2010). Therefore an improved understanding of coupled fluid flow and geomechanical process offers the potential to engineer permeability enhancement and its longevity—and thus develop such reservoirs at-will, regardless of the geological setting (Gan and Elsworth 2014a, b). This is a crucial goal in the development of EGS resources.

The subsurface fluids flow through geological discontinuities such as faults, joints and fractures and this significantly increases the complexity of the constitutive relationship linking fluid flow and solid deformation. To better represent the discontinuities in the simulation of fractured masses, a joint element model was developed based on the finite-element method for the coupled stress and fluid flow analysis by Goodman (1968), allowing the jointed rock treated as a system of blocks and links. Following this modeling approach in discretizing the fractured mass, a coupled finite element simulator by incorporating Biot’s three-dimensional theory of consolidation was developed in 1982 (Biot 1941; Noorishad et al. 1982). It presented another approach to delineate the nonlinear deformation of fractures. A coupled boundary element—finite element procedures were developed to extend the analysis of both the fluid flow and deformation in discrete fractured masses into three dimensions in 1986 (Elsworth 1986). The revised direct boundary element model is capable to discretize individual fractures and assess their interactions with the adjacent fractures. A finite element model was developed to simulate the mixed mode of induced fluid-driven fracture propagation in rock mass, and the interaction with the natural fractures in 1990 (Heuze et al 1990). Since then, there are many available simulators which are capable to simulate coupled hydraulic–mechanical–thermal influence within the deformable fractured medium (Rutqvist and Stephansson 2003; Kohl et al. 1995).

The current available models are primarily divided into equivalent continuum approach and discontinuum approach. Current discontinuum models include boundary element methods (Ghassemi and Zhang 2006; McClure and Horne 2013), distinct element methods (Fu et al. 2013; Min and Jing 2003; Pine and Cundall 1985). The discontinuum approach for fractured masses assumes that the rock is assembled from individual blocks delimited by fractures. The fractures can be represented by either explicit discrete element along fractures, or by two elements—rock blocks with interfaces between them (Zhang and Sanderson 1994). Given the fact that rocks and fractures are explicitly characterized, the discontinuum approach is able to investigate the small-scale behavior of fractured rock mass, which could be more realistic in replicating the in situ behavior. However, the discontinuum model application in large reservoir scales and long term prediction demand greater computational efficiency and time.

Conversely, the major assumption for the equivalent continuum approach is that the macroscopic behavior of fractured rock masses and their constitutive relations can be characterized by the laws of continuum mechanics. The equivalent continuum approach has the advantage of representing the fractured masses at large scale, and propose simulation results of long term response. The behaviors of fractures are implicitly included in the equivalent constitutive model and modulus parameters. The most central effort in developing equivalent continuum approach is the crack tensor theory (Oda 1986). Oda developed a set of governing equations for solving the coupled stress and fluid flow with geological discontinuities, which contains the fracture size, fracture orientation, fracture volume, and fracture aperture. The rock mass is treated as an anisotropic elastic porous medium with the corresponding elastic compliance and permeability tensors.

Recent available continuum simulator TFREACT was developed by Taron (Taron et al. 2009), which couples analysis of mass and energy transport in porous fractured media (TOUGH) and combines this with mechanical deformation [FLAC3D (Itasca 2000)] with extra constitutive models including permeability evolution, dual-porosity poroelastic response. The purpose in this work is to extend the analysis of fractured flow and fracture deformation into the randomly distributed fractured rocks, and provide a tractable solution to optimize the potential flow and production in spatially large fractured networks. To better represent the permeability evolution due to the influence of stress, the constitutive model about the stress-dependent permeability evolution should be extended to include the scenario, which the two fracture walls are out of contact under tensile loading. The crack tensor theory is employed to determine the mechanical properties of fractured rock masses in the tensor forms, differentiating the responses of the fluid flow and stress from both the intact rocks and fractured rock masses. The constitutive model of stress-dependent permeability evolution accommodates three different stress states, including normal closure, shear dilation, and for fracture walls out of contact under tensile loading. The accuracy of the developed simulator is assessed by addressing the interaction of fracture opening and sliding deformation in response to fluid injection inside a long single fracture. Via assessing the simulation results against other discontinuum simulators, it is proved that the developed equivalent continuum model is a feasible approach to simulate the coupled thermal-hydro-mechanical behavior in discrete fractured rock masses. A series of parameter tests were conducted on the different boundary stress conditions, fracture orientation angle, fracture density. Results about the fluid transport and fracture aperture evolution implied the behavior of permeability anisotropy and flow channeling.

## 2 Constitutive model development

To implement an equivalent continuum model accommodating the fractured mass, four constitutive relations require to be incorporated. These are the formulations for a crack tensor, a permeability tensor, a model for porosity representing the fracture volume, and a model for stress-dependent fracture aperture.

### 2.1 Crack tensor

*E*, as

*D*, and components of crack tensors \(F_{ij}\), \(F_{ijkl}\) respectively.

*fracnum*is the number of fractures truncated in an element block, \(\delta_{ik}\) is the Kronecker’s delta. The related basic components of crack tensor for each crack intersecting an element are defined \(F_{ij}\) as below (Rutqvist et al. 2013),

*b*is the aperture of the crack, \(V_{e}\) is the element volume, and \(n\) is the unit normal to each fracture. Therefore the formula for the total elastic compliance tensor \(T_{ijkl}\) of the fractured rock can be expressed as,

*K*and shear modulus

*G*for the fractured rock mass are formulated as below,

### 2.2 Permeability tensor and aperture evolution

When the fluid pressure in the fracture exceeds the normal stress across the fracture, the two walls of the fracture are separated and the effective normal stress is zero (Crouch and Starfield 1991) since \(P_{f} = \sigma_{n}\).

*D*is the fracture half length, and \(\eta\) is a geometrical factor which depends on the crack geometry and assumptions related to slip on the patch (Dieterich 1992). In this study, \(\eta\) is defined to represent a circular crack and is given as \(\frac{7\pi }{24}\). Therefore the equation for the fracture opening displacement \(b_{open}\) is formulated as below,

### 2.3 Porosity model

### 2.4 Numerical simulation workflow

*T*) and pore pressure (\(P_{f}\)) in TOUGH. Subsequently, initial data describing the fracture network, including fracture orientation, trace length, aperture, and modulus are input into a FORTRAN executable. The compliance tensor transfers the composite fracture modulus with the equilibrium pore pressure distribution into FLAC3D to perform the stress–strain simulation. The revised undrained pore pressure field is then redistributed, based on principles of dual porosity poromechanics, and applied to both fracture and matrix. The stress-dependent fracture aperture is calculated and updated based upon the failure state of the fracture. Since the fracture permeability and composite modulus of the fractured medium are both mediated by the magnitude of fracture aperture, the fracture aperture is iteratively adjusted in the simulation loop to define the revised fracture permeability and to alter the compliance tensor of the composite fractured mass.

## 3 Model verification

In this section, the discrete fracture network (DFN) model is verified by comparing the results against other DFN models. The model in this study (Kelkar et al. 2015) is an injection well intersecting a single fracture that dilates and slips in response to fluid injection.

### 3.1 Model setup

Data used in the simulation (McTigue 1990)

Parameter (unit) | Magnitude |
---|---|

Shear modulus [G (GPa)] | 15 |

Poisson’s ratio (\(\nu\)) | 0.25 |

Undrained Poisson’s ratio | 0.33 |

Matrix permeability [\(k_{m}\) (m | \(4.0 \times 10^{ - 19}\) |

Matrix porosity (\(\phi_{m}\)) | 0.01 |

Biot’s coefficient (\(\alpha\)) | 0.44 |

Water viscosity [\(\mu\) (Pa s)] | \(3.547 \times 10^{ - 4}\) |

Fluid compressibility (\({\text{MPa}}^{ - 1}\)) | \(4.2 \times 10^{ - 4}\) |

Thermal expansion coefficient of solid [\(\alpha_{s}\) (\({\text{K}}^{ - 1}\))] | \(2.4 \times 10^{ - 5}\) |

Thermal diffusivity of intact porous rock [\(c^{T}\) (\({\text{m}}^{2} /{\text{s}}\))] | \(1.1 \times 10^{ - 6}\) |

Fluid density [\(\rho_{w}\) (\({\text{kg}}/{\text{m}}^{3}\))] | 1000 |

Heat capacity of fluid [\(c_{w}\) (\({\text{J}}/{\text{kg}}\,{\text{K}}\))] | 4200 |

Initial reservoir temperature ( | 420 |

Injection water temperature ( | 400 |

Initial joint normal stiffness [\(k_{n}\) (GPa/m)] | 0.5 |

Initial joint shear stiffness [\(k_{s}\) (GPa/m)] | 50 |

Fracture aperture initial [\(b_{ini}\) (mm)] | 1 |

In-situ stress (MPa)—y direction | 20 |

In-situ stress (MPa)—x direction | 13 |

Initial reservoir pore pressure (MPa) | 10 |

Injection rate (\({\text{m}}^{3} /{\text{s}}\))/m thickness of reservoir | \(6.0 \times 10^{ - 8}\) |

Friction angle, dilation angle | \(30{{^\circ }}\), \(2.5{{^\circ }}\) |

Fracture cohesion [ | 0 |

### 3.2 Results discussion

In this section, the results are compared between the contrasting conditions of isothermal injection and non-isothermal injection. To isolate the impact of the induced thermal stress effect in the aperture evolution, the liquid viscosity is set as constant for both the isothermal and non-isothermal injection conditions.

^{3D}is able to match results for the evolution of fracture aperture.

^{3D}and the other discrete fracture models. In this response, the normal aperture grows slowly under initial pressurization and most rapidly as the walls lose contact and the full geometric stiffness of the fracture is mobilized as the fracture walls lose contact. The walls of the fracture are out of contact after ~60 days of injection. Due to permeability enhancement in the fracture, the injection pressure gradient in the fracture decreases after the fracture opens.

^{3}~ 10 %.

### 3.3 Multi-fracture validation

^{3D}and the original TFREACT (Fig. 13), which aims at validating the correctness of the developed equivalent continuum model in simulating the interaction of multi-fractures. In Fig. 12, there are two sets of orthogonal fractures (red and green lines) ubiquitously distributed in a 200 m × 200 m × 10 m reservoir. The fractures are separated at a constant spacing 4 m. The injector is centrally located in the reservoir with a constant injection rate of 0.01 m

^{3}/s. The reservoir properties are described in Table 1.

^{−1}in this model. Figure 13 shows the evolution of fracture aperture over time. There is good agreement in the magnitude of fracture aperture between the continuum TFREACT and the equivalent continuum TF_FLAC

^{3D}. The injection pressure results are either validated in Fig. 14. When the fracture pressure increases, the decreased effective normal stress prompts the enhancement of fracture aperture.

## 4 DFN application

In realistic reservoir, fractures networks usually are non-uniformly distributed. Complex fracture pattern yields heterogeneous fluid flow and heat thermal drawdown (Bear 1993; Tsang and Neretnieks 1998), and the development of anisotropic fracture permeability, which could induce flow channeling in the major fractures (Chen et al. 1999; Min et al. 2004). In this section, a reservoir model with two discrete sets of fractures is constructed to address the evolution of fracture permeability, due to the influence of stress state and fracture orientation.

### 4.1 Fracture network generation

The reservoir properties in the DFN model are the same and are defined in Table 1. In the base case, the maximum principal stress is 20 MPa (N–S direction) and the minimum principal stress is 13 MPa (E–W direction). The initial reservoir pressure is uniform at 10 MPa. The water is injected at a constant rate of \(5.0 \times 10^{ - 6} \,{\text{m}}^{3} /{\text{s}}\).

### 4.2 Effect of applied stress

Sensitivity tests of applied stress boundaries in the fracture aperture evolution

Test category | \(\sigma_{x}\) (MPa) | \(\sigma_{y}\) (MPa) | \(\sigma_{x} /\sigma_{y}\) |
---|---|---|---|

Stress ratio | 13 | 20 | 0.65 |

19.5 | 30 | 0.65 | |

26 | 40 | 0.65 | |

39 | 60 | 0.65 | |

Stress difference | 20 | 20 | 1 |

30 | 20 | 1.5 | |

40 | 20 | 2 |

### 4.3 Effect of fracture orientation

## 5 Conclusion

This work presents the development of an equivalent continuum model to represent randomly distributed fractured masses, and investigating the evolution of stress-dependent permeability of fractured rock masses. The model incorporates both mechanical crack tensor and permeability tensor approaches to characterize the pre-existing fractures, accommodating the orientation and trace length of fractures. Compared to other discrete fracture models, the advantages of this model are primarily represented in simulating the large scale reservoir including the long term coupled thermal–hydraulic–mechanical response, and also without any dependence in the fracture geometry and gridding. The accuracy of simulator has been evaluated against other discrete fracture models by examining the evolution of fracture aperture and injection pressure during injection. The simulator is shown capable of predicting the evolution of the fracture normal aperture, including the state of fracture normal closure, shear dilation, and out of contact displacements under tensile loading.

The influence of stress state (mean and deviatoric) and fracture orientation on the evolution of fracture permeability are assessed by applying the model to a DFN. Normal closure of the fracture system is the dominant mechanism in reducing fracture permeability, where the mean stress is augmented at a constant stress obliquity ratio of 0.65. Conversely, for varied stress obliquity (0.65–2) shear deformation is the principal mechanism resulting in an increase in permeability. Fractures aligned sub-parallel to the major principal stress are near critically-stressed and have the greatest propensity to slip, dilate and increase permeability. Those fractures normal to direction of the principal stress are subjected to the increasing normal stress, and reduce the permeability. These mechanisms increase the anisotropy of permeability in the rock mass. Furthermore, as the network becomes progressively more sparse, the loss of connectivity results in a reduction in permeability with zones of elevated pressure locked close to the injector—with the potential for elevated levels of induced seismicity.

## Notes

### Acknowledgments

This work is a partial result of support from the US Department of Energy under project DOE-DE-343 EE0002761. This support is gratefully acknowledged. We also acknowledge the data from the University of Oklahoma and University of Texas at Austin.

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