[5] The term criticality is used to describe the pore-pressure value that must be reached in order to trigger an earthquake. We assume a rectangular probability density function *f*(*C*) for the criticality *C*(*r*) of the medium at any point with position vector *r*, with the minimum and maximum criticality values *C*_{min} and *C*_{max}, respectively; then *f*(*C*) = 1/*dC*, with *dC* = *C*_{max} − *C*_{min}.

[6] We consider a point source that causes pore-pressure perturbations *p*(*r**,t*) which propagate according to the diffusion equation ∂*p*/∂*t* = *D*∇^{2}*p*, with *t* the time, and *D* the scalar hydraulic diffusivity; this equation is valid for an irrotational fluid displacement field [*Wang*, 2000], which we consider as an approximation for pore-pressure perturbations due to borehole fluid injections. The permeability *k* is related to diffusivity by *D* = *k*/(μ*S*), where *S* is the uniaxial specific storage coefficient, and μ is the fluid viscosity. Generally, for the Earth's crust *D* is between 10^{−4} and 10 m^{2}/s [*Talwani and Acree*, 1984; *Kuempel*, 1991; *Wang*, 2000; *Scholz*, 2002]. As long as pore pressure rises at a given point, an earthquake can be triggered. Thus the minimum monotonous majorant *g*(*r*,*t*) of pore pressure is the decisive parameter for triggering seismicity (Figure 1). The absolute value of the position vector equals *r* = , with *x*, *y*, *z* the Cartesian coordinates, and the point source placed at the origin of the coordinate system. The medium is subdivided into *N* cells. Then the probability that an event occurs in a cell will be given by the following value of the probability distribution function of the criticality:

For the whole volume *V* of the area of interest the cumulative number of triggered events *N*_{ev}(*t*) is defined as:

Using δ*V* to represent the volume in each cell, we can rewrite equation (2):

and the seismicity rate *R*(*t*) for the whole volume *V* (i.e., a sphere with radius *a*) is (in spherical coordinates):

Firstly, we consider time *t* only during fluid injection with duration *t*_{0}. For a step function point pore-pressure source of strength *q*, pore pressure *p*_{b} (*r,t*) rises monotonically during *t*_{0}; the subscript *b* denotes before *t*_{0}. Thus for *t* ≤ *t*_{0} it is *g*_{b}(*r,t*) = *p*_{b} (*r,t*), which in 3D is:

with *erfc* the complementary error function *erfc*(*x*) = 1 − *erf*(*x*) [e.g., see *Carslaw and Jaeger*, 1959]. By substituting the solution of the diffusion equation, equation (5), in equation (4), differentiating and integrating we receive for the seismicity rate *R*_{b}(*t*) up to *t*_{0} in 3D:

with *a* the radius of the sphere comprising the volume of the seismically active region, and *F* = *q/*(*dC* · δ*V* · ).

[7] Secondly, we consider the time t after the end of injection at *t*_{0}, where the pore pressure *p*_{a}(*r,t*) continues to increase as a function of time and distance from the injection point (Figure 1). At points where pressure reaches its maximum, no further seismic events can be triggered. Thus a zone of seismic quiescence is developing with *r* < *r*_{bf}(*t*). Here *r*_{bf}(*t*) is the back front of seismicity that *Parotidis et al.* [2004] described with the following equation:

Then using equations (4) and (7) we describe the rate *R*_{a}(*t*) after *t*_{0} in 3D according to

resulting in

with *u*_{1} = −*a*/ · exp (−*a*^{2}/(4*Dt*)); *u*_{2} = · *erf*(*a*/); *u*_{3}, *u*_{4} equal −*u*_{1}, −*u*_{2}, correspondingly, but *a* is substituted by *r*_{bf}. The *v* terms equal the *u* terms, but *t* is substituted by (*t* − *t*_{0}).