The influence of fluid injection on tectonic fault sliding and seismic event generations was studied by a multi-degree-of-freedom rate-and-state friction model with a two-parametric friction law. A system of blocks (up to 25 blocks) elastically connected to each other and connected by elastic springs to a constant-velocity moving driver was considered. Variation of the pore pressure due to fluid injection led to variation of effective stress between the first block and the substrate. Initially the block system was in a steady-sliding state; then, its state was changed by the pore pressure increase. The influence of the model parameters (number of blocks, spring stiffness, velocity weakening parameter) on the seismicity variations was considered. Various slip patterns were obtained and analysed.

Despite the fact that the rate-and-state model of friction was proposed in the second half of the previous century, the interest in it has increased in recent years. The rate-and-state model (Gu et al., 1984; Dieterich, 1992; Abe and Kato, 2013) was adopted as a quite appropriate basis for describing seismic processes in the Earth's crust and for modelling relevant geophysical systems. Currently, it is believed that this model describes the seismic process most adequately.

Brace and Byerlee (1966) proposed considering unstable frictional sliding along tectonic faults as a model of earthquakes. The model included a suggestion that a cohesion existing in some parts of tectonic fault prevents free slipping along it and leads to an accumulation of a shear stress to a critical level, after which the slip and the earthquake occur.

Peculiarities of the friction force dependence on the duration of the stationary state of the contact and on the velocity of the motion along the fault were examined by Dieterich (Dieterich, 1992). Gu et al. (1984) experimentally investigated various modes of the frictional movements and determined empirical constants whose values are used in many modern variants of the rate-and-state equation.

The rate-and-state equation was considered by Hobbs (Hobbs, 1990) by means of non-linear dynamics methods. Change in the friction was studied as a function of displacement and velocity at a variation of the stiffness coefficient in the rate-and-state equation. A similar approach was implemented by Erickson et al. (2008); they examined an appearance of chaotic solutions in the one-parameter velocity-dependent friction equation.

Radial flow in a homogeneous reservoir.

Abe and Kato (Abe and Kato, 2013, 2014) examined two- and three-degree-of-freedom spring-block models with a one-parameter rate-and-state friction law and obtained different slip patterns for such system. By varying stiffness parameters, they obtain periodic recurrence of the seismic and aseismic events and several types of seismicity chaotic behaviour.

Pore pressure change at the boundary between the first block and substrate.

Turuntaev et al. (2012) showed using the Grassberger–Procaccia method (Grassberger and Procaccia, 1983) that the man-made impact underground leads to an increase in the “regularity” of the seismic regime. To explain the increase in the seismic regime regularity, a model of the fault motion defined by the two-parameter velocity dependent friction law was considered.

Cumulative number of events vs. number of blocks.

In the presented paper, we consider the two-parameter type of the friction law in a multi-degree-of-freedom spring-block model and change the value of critical shear stress in the rate-and-state equation, suggesting that this is the value varied by human impact (by fluid injection and corresponding pore pressure change). Here we use the classical pore-elastic model of radial filtration of injected fluid to calculate the typical pore pressure change.

Cumulative number of events vs. number of blocks.

Cumulative number of events vs. number of blocks.

Cumulative number of events vs. number of blocks.

The tectonic fault model proposed by Burridge and Knopov (Burridge and
Knopov, 1966) looks like a system of blocks elastically connected to each
other (Fig. 1a, b). Each block moves under the net action of elastic forces
from adjacent blocks and driver and friction force from the stationary
substrate. Here, the multi-degree-of-freedom system is investigated. Every
block of mass

Maximum seismic moment of event vs. number of blocks.

Maximum seismic moment of event vs. number of blocks.

We assume that the friction shear stress at the block boundary obeys the following two-parameter friction law:

Maximum seismic moment vs. number of blocks.

Maximum seismic moment vs. number of blocks.

Cumulative seismic moment vs. number of blocks.

Cumulative seismic moment vs. number of blocks.

As was shown by Gu et al. (1984), if

Cumulative seismic moment vs. number of blocks.

Cumulative seismic moment vs. number of blocks.

Values of coefficients

Event cumulative number dependence on the stiffness of the interblock link.

Event cumulative number dependence on the stiffness of the interblock link.

If

The event maximum seismic moment dependence on the stiffness of the interblock link.

The event maximum seismic moment dependence on the stiffness of the interblock link.

The parameters in all the simulations were the following:

Maximum velocity dependence on the stiffness of the interblock link.

Maximum velocity dependence on the stiffness of the interblock link.

To estimate the pore pressure change, we considered radial flow of fluid in
an infinite homogeneous reservoir of constant thickness from the injection
well with a negligibly small radius (Fig. 2). The initial reservoir pressure
was assumed to be the same everywhere and equal to

The block system cumulative seismic moment dependence on the stiffness of the interblock link.

The block system cumulative seismic moment dependence on the stiffness of the interblock link.

Block velocity variations in time for the system consisting of 20 blocks in Case 1.

Block velocity variations in time for the system consisting of 20 blocks in Case 1.

Block velocity variations in time for the system consisting of 20 blocks in Case 2.

Seismic event occurrences in time for the system consisting of 20 blocks in Case 2.

So we got the standard diffusivity equation, where

The solution of this equation with the above initial and boundary conditions reads as

The values of parameters used in the calculations were close to the
parameters of the Basel project (Häring et al., 2008):

Seismic event occurrences in time for the system consisting of 20 blocks in Case 2.

To study the influence of the number of blocks in the multi-degree-of-freedom
spring-block system on characteristics of the simulated seismicity (the total
number of events, the maximum and cumulative seismic moments) for the
“natural” and “induced” cases, the calculations for 2, 3, 4, 5, 10, 15,
20 and 25 blocks were made for the same motion durations – 1 million
seconds. The time restriction was related to the computational complexity of
25 block simulations. During this time, the pressure changed significantly in
Set 2 (near 11 MPa). The number of events, the maximum seismic moment of one
event and the cumulative seismic moment of all events and all blocks are
shown in Figs. 4–15. The calculations were made for different ratios of
stiffness of the springs between the blocks

Time variation of seismic activity.

Time variation of seismic activity.

It can be seen that if the pore pressure did not change (Set 1, Figs. 5, 7, 9, and 11), the number of events grows almost linearly with the increase in the number of blocks for all values of stiffness of springs between the blocks in both cases (1 and 2); the maximum seismic moment of the events decreases with the increase in the number of blocks.

Time variation of seismic activity.

However, for small values of

Now, let us consider the change in the behaviour of the system consisting of
20 blocks with the change in the stiffness of the link between the blocks

Case 2 is characterized by slower changes in the velocity with time than
Case 1 (Fig. 26). That is why there is no clear dependence of the number of
events and the block maximum velocity on the interblock link stiffness. Such
behaviour becomes more evident with a decrease in parameter

Our model demonstrates that the influence of the interblock link stiffness on the behaviour of the studied systems is very strong. By changing the stiffness, we may get periodic or chaotic motion of the system and occurrence of the first strong seismic event almost immediately after the injection start or after a relatively long time (compare Figs. 27 and 28); furthermore, the main seismic activity may occur at the moment of the injection start, when the pressure gradient is the highest, or in the post-injection phase. In Figs. 27–30 the seismic activity variations in the form of the number of events per 10 days (left vertical axis) and the ratio of the cumulative seismic moment of events to the average cumulative seismic moment per 10 days (right vertical axis) are shown for both “natural” (Set 1) and “induced” (Set 2) seismicity. The “natural” seismic activity variations have almost the same amplitudes during all considered time intervals, while the “induced” seismic activity variations depend on interblock link stiffness: in the case of small stiffness the amplitude of the seismic activity during injection is almost the same as in the post-injection period (Fig. 29). When the stiffness becomes higher, the seismicity during injection becomes twice greater than the post-injection activity (Fig. 30); the further increase in the interblock link stiffness leads to a significant increase in the post-injection activity (Figs. 31 and 32).

The recurrence maps of the seismic event sequences are shown in Figs. 33 and
34. It could be seen that for

Time variation of seismic activity.

Iteration map of recurrence intervals of seismic events,

Iteration map of recurrence intervals of seismic events;

The problem of the influence of the fluid injection on the tectonic fault sliding and generation of the seismic events was studied by numerical calculations of the peculiarities of motions of a system of blocks (consisting of up to 25 blocks) elastically connected to each other and connected by elastic springs to a constant-velocity moving driver (the multi-degree-of-freedom spring-block model). The rate-and-state friction model with the two-parametric friction law was adopted for description of the friction between the blocks and the substrate. Initially the block system was in steady-sliding state; then, its state was disturbed by the pore pressure increase. The influences of the model parameters (the number of the blocks, the spring stiffness, the velocity weakening parameter) on the process of the model seismicity variations were considered.

It was shown that the considered spring-block system could exhibit different types of motion with different patterns. The motion could be periodic or chaotic; the magnitude of the seismic events depends on fragmentation of the fault system (the number of blocks in the considered model) and may have different values. The analysis shows that the stiffness of the link between the blocks affects significantly the behaviour of the model and the resulting seismicity, so the main seismic activity could appear directly after the start of the fluid injection or in the post-injection phase. Such influence of the injection on seismicity could be observed in the real cases. However, the parameters in the rate-and-state model are known only from laboratory experiments, and it is hard to believe that one should use the same values to describe the real-scale phenomena. Yet our study showed that it is possible to select more suitable parameters that will allow one to match results of calculations and data of real observations. It can be concluded that the considered model has the potential to be used for the estimations of the possible fluid-induced seismicity activity variations.

Results of calculations are available by e-mail request.

The authors declare that they have no conflict of interest.

The financial support of the Russian Foundation for Basic Research (project no. 16-05-00869) is acknowledged. Edited by: A. Dyskin Reviewed by: two anonymous referees