The study of 1D spring-block model of earthquake dynamics with consideration of water effects in preexisting fault deals with new forms of frictional force. An analytical study of the equation of motion enables us to establish that motion of geological fault is accelerated by water pressure. In the same setting the critical value of frictional velocity for which appears the discontinuous (first-order) transition from a stick-slip behavior to a creep motion strongly depends on water pressure. The investigation also displays the magnitude and probability of events as a function of water pressure; these two quantities decrease and increase, respectively, with the variation of water pressure.

Despite significant advances made in the study of geological fault structures and plate tectonics, our understanding of the physical mechanisms responsible for the initiation, propagation, and termination of earthquake rupture remains unfinished. Burridge and Knopoff [

The aim of this paper is to study Giovanni’s modified single block model, by considering the new shape of frictional forces (that contains water pressure), and investigate the effects of water pressure on the earthquake dynamics. The content of this paper is organized as follows.

Section

Section

Section

Section

The last section is devoted to discussion and conclusion.

Figure

(a) Fault with permeable side and impervious side. (b) Very high pressure of water. (c) Flux of water in the fault.

Since our investigation focused on the role of water on the fault’s dynamics, we hereby propose a new form of frictional force that takes into consideration water characteristic in the process. The model below has a great advantage of being analytically tractable, so that one hopes that a thorough understanding of such a simple model might in turn shed further light on the basic principles governing real earthquakes. We will for convenience choose the frictional force such as

Spring-block model for earthquakes.

When the force due to the spring exceeds the threshold force denoted by

For this model to be relevant for real earthquakes, dimensionless water pressure must be less than one (i.e.,

We start the analysis by considering first the linearized version of the motion equation. In view of (

For convenience we have introduced the parameter

Equation (

Now, our attention is focused on the creep motion of the block; for this we consider the case (i) and then the solution of (

Effects of water pressure on the magnitude of maximum velocity (

During the description of earthquake, it is important to define the position of the block at the end of the slip. But, when the block ceases to move, it occupies an unspecified position. However, determination of this position requires the knowledge of the time

Next we investigate the situation when

The main objective of this section is to investigate the behavior of the block in stick-slip motion. Considering the case (ii):

Since the velocity

Several friction models have been recently considered in the literature [

Initially, when

Figures

(a) Effects of water pressure on the closed time of linear motion part for (

From here it is easy to understand that water can considerably reduce the time of transition (from a stick-slip behavior to a creep motion). Although the block spends a very long time in this linear regime, considerable motion will occur only for times close to the instant

A curve given the evolution of

The block displacement

The curve of Figure

Following the study of water effects on the first-order transition, it is worthy to investigate the influence of water on energy release during an earthquake.

The objective of this section is to investigate the effects of water pressure on the earthquake magnitude

Hereafter, we consider the magnitude expression formulated by Kanamori et al. [

Wherein

The effect of water on the magnitude is investigated by plotting the magnitude

Magnitude

From Figure

Now let us examine the influence of pressure on the probability of earthquakes occurrence; as defined by Gutenberg and Richter in 1944 [

It appears from Figure

Probability of events in terms of water pressure with

It is important to remark that the value of frictional velocity which leads to the transition at

In this paper, we have studied water effects on the dynamics of a 1D spring-block model for earthquakes by performing the Vasconcelos frictional force [

The authors do not have any conflict of interests regarding the publication of this paper.

Professor F. B. Pelap is grateful to Professor Giovani L. Vasconcelos for valuable discussions during his last visit in the University of Pernembuco, Brazil.