VolumeBased Modelling of Fault Reactivation in Porous Media Using a ViscoElastic Proxy Model
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Abstract
The injection of fluids into the subsurface takes place in the context of a variety of engineering applications such as geothermal power generation, disposal of wastewater, \(\hbox {CO}_{2}\) storage and enhanced oil recovery. These technologies involve not only the underground emplacement of fluids in a geologic formation but also affect the stress state of these rocks. If the rock’s strength is surpassed, these stress changes can even lead to failure. In this context, we present a conceptual approach to model fault reactivation in porous media. As a starting point for developing and implementing this approach, the already existing combined hydro and geomechanical model within the opensource simulator DuMu\(^{{x}}\) was chosen. For the evaluation of shear slip on the fault plane, the classical Mohr–Coulomb failure criterion is used. Based on the energy balance from Kanamori (Earthquake thermodynamics and phase transformations in the earth’s interior, international geophysics, vol 76. Academic Press, London, pp 293–305, 2001), where a slip event on fault is described as a transformation of elastic energy into seismic waves, heat and an amount of energy required to cause fracture, we interpret failure as a dissipation of elastic energy. Furthermore, seismic data allow to infer a constant stress drop over a wide range of scales (Abercrombie and Leary in Geophys Res Lett 20(14):1511–1514, 1993). These findings are incorporated into our model by altering the material properties during the slip event. In detail, the linear elastic material law is replaced by a viscoelastic behaviour, which reproduces the characteristics mentioned above. This, in turn, leads to additional displacements, which are interpreted as the slip on the fault plane. Our results indicate that this pragmatic approach is capable of modelling fault reactivation without resolving the fault as a discrete surface but as a elements representing a fault zone instead.
Keywords
Fault reactivation Multiphase modelling Geomechanics1 Introduction
The number of injection wells in the USA is estimated to be above 500,000 (USEPA 2001; Lustgarten and Schmidt 2012). Without doubt, such fluid injections into the subsurface influence the stress state of respective rock formations. Highly sensitive seismometers can detect microearthquakes resulting from hydraulic fracturing operations, geothermal stimulations or wastewater injections. The created pathways are often a desired result to enhance the productivity of the reservoir exploitation, but they can also pose environmental risks.
For hydraulic fracturing operations, the concern has been raised that induced fractures could propagate through the overburden and connect the hydrocarbonbearing formations with shallow groundwater aquifers. Such a contamination of underground sources for drinking water can usually be ruled out due to the distance between the shale formations and the aquifers that contain drinkable water (Zoback et al. 2010). A study by Fisher and Warpinski (2012) revealed that injectioninduced seismicity is usually very confined. Only the presence of subvertical faults allows an upward migration of the seismicity. Besides the risk of contamination, the potential for reactivating faults and creating earthquakes can become crucial for the public perception of a project (Zoback et al. 2010; Leucht et al. 2010; Reith et al. 2013). This highlights the necessity to increase the knowledge base of coupled hydraulic and geomechanical modelling to better understand the relevant processes that lead to fault reactivation.
To ensure a broad applicability of a modelling approach designed for this purpose, the simulation code should cover multiphase flow in combination with the mechanical response of the rock formation. For this purpose, two simulators can be coupled, such as TOUGH2 and Code_Aster by Rohmer and Seyedi (2010) or TOUGH2 and FLAC3D by Rutqvist and Tsang (2002), Cappa and Rutqvist (2011). The latter combination has recently been expanded towards the modelling of fault reactivation by hydraulic fracturing (Rutqvist et al. 2013).
The aim of this study is to extend a given coupled multiphase flow and geomechanical approach towards the analysis of fault reactivation within a single software code, i.e. the opensource simulator DuMu\(^{{x}}\). We work along the hypothesis that this goal can be achieved by using a volumebased approach, which does not require to model the fault as a discrete surface but as a elements representing a fault zone instead. The modelling approach is then applied to a scenario setup presented in Rutqvist et al. (2013) to study the potential of fault reactivation by hydraulic fracturing operations.
2 Modelling Concept
2.1 Hydrogeomechanical Model

All participating solid materials are consolidated in a single solid phase s

Small deformations of the solid (application of small deformation theory valid)

Incompressible solid matrix and brine phase (Solid density \(\rho _{s}\) \(=\) const., brine density \(\rho _{w}\) \(=\) const.)

Isothermal conditions (solving the energy balance is not required)

Quasistationary conditions (inertia forces are neglected)

All the participating phases are immiscible within each other.
2.1.1 Governing Equations
2.1.2 Constitutive Equations and Supplementary Constraints
 Linear elastic behaviour of the rock, i.e. the stress \(\varvec{\sigma }\) increases linearly and reversibly with the strain \(\varvec{\varepsilon }\) (according to Hooke’s law). Two independent elastic constants relate the stress and strain in an isotropic case. Here, we use the Lamé constant \(\lambda \) and the shear modulus G:$$\begin{aligned} \varvec{\sigma } = \lambda \, \mathrm {tr}[\varvec{\varepsilon }] + 2 \,G \, \varvec{\varepsilon } \end{aligned}$$(17)
 The strain \(\varvec{\varepsilon }\) is derived from the displacement vector as follows$$\begin{aligned} \varvec{\varepsilon } = \frac{1}{2} (\mathrm {grad} \, \mathbf {u} + \mathrm {grad^{T}} \mathbf {u} ) \end{aligned}$$(18)
 The sum of the fluid saturations adds up to one$$\begin{aligned} \varSigma _{\alpha } \, S_{\alpha } = 1 \end{aligned}$$(19)
 The pressures are connected via the capillary pressure (calculated here according to the Brooks and Corey relation (Brooks and Corey 1964))$$\begin{aligned} p_{n} = p_{w} + p_{c}(S_{\alpha }) \end{aligned}$$(20)
2.2 Solution Procedure
For all simulations, the numerical toolbox DuMu\(^{{x}}\) (Flemisch et al. 2011; Becker et al. 2015) was used, which is based on the Distributed and Unified Numerics Environment DUNE (Bastian et al. 2008a, b). The system with two fluid phases and the solid as the porous medium with all equations described above was already implemented as the socalled el2pmodel. The discretisation is described in detail by Darcis (2013), and thus, we will give only a short summary of important characteristics of the model.
Two different discretisation schemes are used for the two balance equations. While Eq. 5 is discretised with a vertexcentred finite volume scheme, the socalled box method (Helmig 1997), the standard Galerkin finite element method is used for the momentum balance of the solid (Eq. 6).
As reported by Darcis (2013), with this approach spurious pressure oscillations can be avoided. Kim (2010) also observed stability issues and suggested a staggered grid approach. A similar—quasistaggered—effect is achieved here by the different weighting functions of the box and the standard Galerkin finite element method, allowing a nodalbased approach without explicitly using staggered grids.
2.2.1 Failure Evaluation
2.2.2 Fault Reactivation
Emerging technologies such as hydraulic fracturing, CO\(_2\) storage in the subsurface or geothermal stimulation have created a need to expand existing methods capable of modelling coupled multiphase flow and poromechanics towards the simulation of fracture initiation, growth and slip in general. Different approaches have been developed to achieve this goal: Phasefield models represent a fracture interface by describing the change from broken and the intact rock diffusely with an order parameter (Mikelic et al. 2015; Hofacker and Miehe 2013; Miehe et al. 2010). Other approaches depict faults as discrete surfaces by means of zerothickness elements using a penalty method (Ferronato et al. 2008) or a Lagrange multiplier formulation (Jha and Juanes 2014).
If the focus is shifted from the modelling of discrete fractures and fault surfaces towards the simulation of fault zones and their potential reactivation, the way these structures are geometrically represented changes: Accounting for the fact that faults zones are complex features consisting of a fault core and a damage zone, Rutqvist et al. (2013) choose to model the fault not as a surface but as a fault zone instead, using a socalled ubiquitous joint model. A comparison between different fault modelling approaches by Cappa and Rutqvist (2011) revealed that both a zerothickness interface and finitethickness solid elements produce similar results. Thus, we choose to use the latter approach using finite thickness elements for representing faults, as this fits well into our existing implementation and is also applicable to a variety of fault architectures ranging from single surfaces to complex fault zones (Cappa and Rutqvist 2011).
In a similar fashion, the description of the relevant physics during a slip event varies: It remains under discussion whether the coefficient of friction \(\mu _{\mathrm {frict}}\) is a function of the slip rate and a state variable (accounting average maturity of contact asperities) such as used by Jha and Juanes (2014) or the slip weakening is rate independent (Garagash and Germanovich 2012) and can be modelled by transferring no normal and shear stresses (Ferronato et al. 2008) or by a sudden reduction in the coefficient of friction (Cappa and Rutqvist 2011; Rutqvist et al. 2013).
We propose a different approach based on energetic considerations: During a shear slip event on a fault, previously builtup stress is released and transformed into seismic wave energy, thermal energy due to friction on the fault plane and energy required to cause fracture (Kanamori 2001). Thus, the shear stress on the fault plane is reduced. Following this, we can define the stress drop \(\Delta \sigma _{\mathrm {failure}}\) that denotes the difference in stress before and after a slip event. Generally, this \(\Delta \sigma _{\mathrm {failure}}\) is considered to range from 0.1 to 1 MPa. Such a confined range of values for the stress drop is not only assumed for large earthquakes (Aki 1972; Thatcher and Hanks 1973; Kanamori and Anderson 1975), but observational data indicate that this is also true for small events (Abercrombie and Leary 1993).
This leads to a new concept for modelling shear failure on an existing fault: If pressure margin \(p_{\mathrm {sm}}\) for shear failure is surpassed, slip on the fault reduces the stress by a value assumed to be constant. We assume that the energies, into which the elastic energy is transformed, are either dissipated or negligible for the stress redistribution, and thus, we consider them as not relevant for the phenomena described by our model. Accordingly, we propose to model shear failure simply as a sink of elastic energy. In the following, we will show that this can be modelled conceptually comparable to some of the approaches described above, as in either case a shear stress reduction is achieved by modifying the material properties.
2.2.3 The Maxwell Material as an Energetic/Phenomenological Equivalent
A purely elastic spring and a purely viscous damper in series (socalled Maxwell material) are introduced as a phenomenological equivalent of such an elastic energy sink description.
From Eqs. 29 and 30 follows a time dependence of the reversible and the irreversible displacement: While the reversible strain decreases, the irreversible deformation increases with time. Energywise, elastic energy initially stored in the spring is shifted to the dashpot and transformed into heat. It becomes apparent that the Maxwell material behaves energetically similar to the effect of failure described in Sect. 2.2.2: in both cases, elastic energy is dissipated. Thus, we propose to use the Maxwell material as an energetic/phenomenological equivalent to shear failure.
2.3 Simplifications and Assumptions Used for the Shear Failure Process

In reality, shear failure would start where the critical pressure is surpassed first. But as we resolve the time discretely during the simulation despite its continuous character, it can happen that for the previous timestep, the critical margin for shear failure \(p_{\mathrm {sm}}\) was below zero for all elements, but is surpassed for several elements after the actual timestep. In order to avoid this, the time discretisation is refined the closer \(p_{\mathrm {sm}}\) gets to zero. This enables us to resolve the onset of fault reactivation and the cell where failure starts quite precisely.

For our example, the fault is represented by finitethickness elements on quadrilateral grids. This means that slip and stress drop happen only in the direction of the coordinate axes and on planes spanned by the coordinate axes, respectively.

Once failure has been detected, the direction of maximal shear stress is determined. This gives the plane of shear failure, on which the shear stress is reduced.

The whole failure process is modelled within one timestep. This means parameters acting during the failure process such as the speed of the propagating slip are not resolved. Instead, only the effect on the stress field is modelled at the start and the end of a timestep.

Furthermore, we assume that the failure process is considerably faster than the flow processes. During the simulation, this is achieved by reducing the timestep length significantly if failure happens.
2.4 Modelling the Shear Failure Process
As already mentioned, this work focuses as a first step on the effect of shear failure; hence, the dissipation of elastic energy due to the viscoelastic behaviour during a slip event is limited to shear stress.
2.4.1 Calculation of Viscosity, Slip and Shear Dilation
Vividly speaking, this means that the porous medium becomes “softer”, which will lead to a increased displacement after the timestep of failure has finished. This additional deformation results from the now active dashpot and represents the irreversible portion of the total displacement. Equivalent to the slip on a fault plane, this irreversible displacement persists as plastic deformation after the failure process is finished. We interpret the displacement in direction of the maximum effective normal stress as the slip and calculate an additional irreversible displacement normal to the fault plane from the dilation angle \(\gamma \) to account for deformation due to shear dilation. In a threedimensional setup, the displacement in the remaining direction is assumed to be elastic and reversible.
2.4.2 Necessity of a Permanent Stress Drop
3 Simulation Setup
The setup of this scenario is based on the study of Rutqvist et al. (2013) on fault reactivation during hydraulic fracturing operation, for which the coupled thermohydromechanical simulator TOUGHFLAC (Rutqvist et al. 2002; Rutqvist 2011) was used. Assuming a fault zone which is intensively jointed oriented parallel to the fault plane, the fault was described with the ubiquitous joint model. The potential for shear failure was evaluated using the Mohr–Coulomb criterion and strain softening strength properties were assumed consistent with a slipweakening fault model.
The twodimensional domain is 2 km \(\times \) 2 km in size, located at a depth of 500–2500 m and discretised as illustrated by Fig. 2. It consists of the Marcellus play with a 30mthick gasbearing shale layer in between other lowpermeability formations. These formations could consist of grey shale or limestone and are assumed to have the same mechanical and hydraulic properties as the gasbearing shale. A fault of 1 km length cuts through these layers with a dip angle of 80\(^{\circ }\). The fault is assumed to be initially impermeable, and a possible reactivation of this fault due to a fluid injection is the subject of this example.
For the in situ stress field, the vertical stress is assumed to be the maximum principal stress corresponding with the significant depth of the domain, and the minimum principal stress is oriented horizontally and parallel to the injection well. In agreement with the assumptions of Rutqvist et al. (2013), the vertical stress gradient is calculated from the overburden density of 2700 kg/m\(^3\) based on the measured density of all formations overlying the Marcellus shale by Starr et al. (2011). A horizontaloververtical stress ratio of \(R = \sigma _{\mathrm {h}}/\sigma _{\mathrm {v}} = 0.6\) is used consistent with observations by Cipolla et al. (2010).
For the rock formations in the domain, a Young’s modulus of 30 GPa and a Poisson ratio of 0.2 are chosen. These values are derived from laboratory measurement of the Barnett Shale from Tutuncu et al. (2010), but Rutqvist et al. (2013) argue that due to lack of experimental data, these values can be used for the Marcellus shales as well. The fault’s Young’s modulus is set to a significantly lower value of 5 GPa, and zero cohesion was assigned. For the fault’s coefficient of friction, we assumed the same value of 0.6 as Rutqvist et al. (2013) used for the joints. For our fault description, we do not distinguish between joints and matrix within the faults zone. Furthermore, instead of setting a residual value for the coefficient of friction after the slip, this approach uses the constant stress drop to model the processes during fault reactivation.
Material properties of Scenario 1
Parameters  Shale  Fault 

Porosity, \(\phi \) (–)  0.01  0.01 
Rock density, \(\rho _\mathrm {s}\) (kg/m\(^3\))  2700  2700 
Permeability, k (m\(^2\))  \(1 \times 10^{19}\)  \(1 \times 10^{16}\) 
Young’s modulus, E (GPa)  30  5 
Poisson’s ratio, \(\nu \) (–)  0.20  0.25 
Biot’s coefficient \(\alpha \) (–)  1  1 
Cohesion (MPa)  –  0 
Coefficient of friction, \(\mu _{\mathrm {frict}}\) (–)  –  0.6 
Dilation angle, \(\gamma \) (\(^{\circ }\))  –  10 
Stress drop, \(\Delta \sigma \) (MPa)  –  0.1 
4 Results and Discussion
A second set of plots shows the slip on the fault plane in detail (Fig. 6). While all other plots cover the example of a stress drop \(\sigma _{\mathrm {failure}}\) = 0.1 MPa, here the evolution of the slip is also shown for \(\sigma _{\mathrm {failure}}\) = 0.2 MPa. It becomes apparent that for the lower stress drop, two slip events shortly after another occurred, while \(\sigma _{\mathrm {failure}}\) = 0.2 MPa leads to only one event with about twice the displacement of the first failure on the fault. Fig. 6 also illustrates how the failing cell deforms: It is sheared in ydirection (consistent with the preset stress field), and some opening in xdirection resulting from the implemented shear dilation is observable.
Figure 7 provides details about the evolution of pressure, shear stress, porosity and permeability during the slip events. The shear deformation of the element decreases porosity and permeability slightly, but the resulting dilation then leads to an increase in porosity and permeability. While one might expect that a higher porosity results in a pressure drop, but the opposite is true due to water flowing into the cell facilitated by the increased permeability.
The described behaviour is plausible and comparable to the results of the corresponding scenario from Rutqvist et al. (2013). But naturally, the modelling of such a scenario contains a significant number of influential variables, which, if not specified exactly similar, hinder reproducibility. In our case, this is true for the injection parameters: While the mass rate per volume (in kg/(s m\(^3\))) was taken from Rutqvist et al. (2013), the exact location and number of the injection cells leave room for adjustments. We assume that this is the reason why the pressure at which fault reactivation occurs is similar (around 25 MPa in our case compared to around 26 MPa for the case with a fault’s permeability of \(1 \times 10^{16}\) m\(^2\) in Rutqvist et al. (2013)), but the onset is different (2 vs. 3 h after the injection started). There is also discrepancy in the magnitude of the slip event: Both the rupture length (23.50 m) and the slip amount (\(2.744\) mm) are much higher in case of Rutqvist et al. (2013) compared to only one cell failing with less than a tenth of a millimetre of slip. This might be attributed to the different injection setup, but the evaluation of the failure presumably plays also a role. During the development of the model, we noticed a significant dependence of the magnitude of fault reactivation on the temporal discretisation close to the exceedance of failure criterion. As mentioned in Sect. 2.3, we used a timestep refinement to appropriately resolve the onset of shear failure on the fault. Furthermore, Rutqvist et al. (2013) performed a parameter study on the impact of the residual friction angle. One outcome of this study was a range of stress drops from 2.5 up to 6 MPa depending on the residual friction angle. As visible in Fig. 5b, b), the shear stress in our case is well below these values, and thus, only a smaller stress drop is possible. Nonetheless, the correlation between stress drop and the resulting slip appears to be qualitatively comparable: The assigned stress drop of 0.1 and 0.2 MPa produces slips of 0.0062 and 0.0126 mm, while stress drops of 2.5 up to 6.0 MPa of Rutqvist et al. (2013) correspond with slips of 1.037 and 2.744 mm, respectively.
Our simulation also produces a rupture length different from the one presented in Rutqvist et al. (2013): While up to 23.50 m are reported by Rutqvist et al. (2013), just one element fails in our case and the rupture does not propagate into one of the neighbouring elements. Instead, the same element fails a second time. This behaviour is a result of the fact that in the current implementation of the model the element regains its full strength after failure has finished. We believe that once failure comes to a halt, only a partial regain in strength is more plausible due to some permanent damage caused by the rupturing. This could be achieved by making the elastic constants dependent on a damage variable. In this sense, Lyakhovsky and BenZion (2008), Lyakhovsky et al. (2011) and Shalev and Lyakhovsky (2013) presented a model combining poroelastic deformation and damage evolution with groundwater flow. The concept couples the rock properties with the damage variable, resulting in a strongly nonlinear stress–strain relationship. This could serve our purpose of a more plausible description of the failed element’s strength and would also allow some degradation of the elastic properties depending on the damage variable prior to the slip on the fault plane. Such an implementation would make an already failed element less likely to fail again and transfer stress to the neighbouring elements. A simple test where the shear modulus regained only 90 per cent of its original value leads to failure of a neighbouring element and, thus, indicates that this is a promising approach for further development of this model concept.
5 Conclusions and Outlook
This paper introduces a numerical modelling concept to detect shear failure and resulting fault reactivation, where the slip event is represented by a viscoelastic proxy model. The aim of this study was to extend a given fully coupled multiphase flow and geomechanical approach towards the analysis of fault reactivation within the within the opensource simulator DuMu\(^{{x}}\). By evaluating failure based on the Mohr–Coulomb criterion and by using viscoelasticity as an equivalent to the shear stress reduction on the fault plane, this could be achieved. The approach was checked against a wellestablished scenario of fault reactivation due to hydraulic fracturing by Rutqvist et al. (2013), where a reduced coefficient of friction was used to model shear failure on the fault plane. Although the results do not match perfectly, the comparison demonstrates that our modelling approach is able to reproduce qualitatively the relevant characteristics such as slip and shear dilation as well as the feedback on the hydraulic properties. This supports the argument that both approaches model the dissipation of elastic energy during the slip event in conceptually comparable fashion, but are dependent on various influential parameters. Among these, the time discretisation close to the exceedance of the failure criterion and the corresponding onset of shear failure as well as the drop in the coefficient of friction or the shear stress need to be investigated in detail, preferably in the context of a benchmark study.
Future work should also apply the approach to multiphase examples, of which the implemented model is capable. For the sake of comparability, we choose the scenario of Rutqvist et al. (2013) over a multiphase injection scenario.
In addition, the main focus of the approach was on the modelling of shear failure on a fault, but a comprehensive approach must include tensile failure, too. Thus, such an expansion of the model towards a representation of tensile failure will be the subject of further research.
Notes
Acknowledgments
The International Research Training Group NUPUS is gratefully acknowledged for enabling this work within its framework. The authors further acknowledge the German Research Foundation DFG, the Netherlands Organization for Scientific Research NWO and the Norwegian Research Council NRC for funding NUPUS.
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